Symmergent Gravity, Seesawic New Physics, and their Experimental Signatures

TL;DR

This paper proposes a new emergent gravity framework called symmergent gravity, which addresses UV sensitivity and incorporates general relativity, predicts specific experimental signatures, and suggests a link to trans-Planckian supersymmetry.

Contribution

It introduces symmergent gravity, a novel approach where affine curvature emerges to restore gauge symmetries and incorporate gravity, linking UV sensitivity resolution with testable collider and cosmological signatures.

Findings

Gravity is Einstein with no higher-curvature terms.

Heavy NP scalars can produce cosmic and gamma rays.

UV cutoff may suppress the cosmological constant.

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 ({\it 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 $\Lambda^2_{SM}/\Lambda^2_{NP}$ if the SM is not to jump from $\Lambda_{SM}\approx 500\, {\rm GeV}$ to the NP scale $\Lambda_{NP}$. The zero coupling, certifying an undetectable NP, agrees with all the collider and dark matter bounds at present. The {\it seesawic} bound $\Lambda^2_{SM}/\Lambda^2_{NP}$, directly verifiable at colliders, implies that: {\it (i)} dark matter must have a mass $\lesssim \Lambda_{SM}$, {\it (ii)} Higgs-curvature coupling must be $\approx 1.3\%$, {\it (iii)} the SM RGEs must remain nearly as in the SM, and {\it (iv)} right-handed neutrinos must have a mass $\lesssim 1000\, {\rm TeV}$. These signatures serve as a concise testbed for symmergence.

Figures

Tables

Table 1. Table 1: Quantum corrections and problems they give cause to. The coefficients c ϝ , … , c g subscript 𝑐 italic-ϝ … subscript 𝑐 𝑔 c_{\digamma},\dots,c_{g} are loop factors that depend on Λ W / Λ U subscript Λ 𝑊 subscript Λ 𝑈 {\Lambda_{{W}}}/{\Lambda_{{U}}} . (CCP, with c ~ O = c O + ( 2 ​ c O ​ Λ W 2 + ∑ ϝ c ϝ ​ m ϝ 2 ) / Δ 2 subscript ~ 𝑐 𝑂 subscript 𝑐 𝑂 2 subscript 𝑐 𝑂 superscript subscript Λ 𝑊 2 subscript italic-ϝ subscript 𝑐 italic-ϝ subscript superscript 𝑚 2 italic-ϝ superscript Δ 2 {\tilde{c}}_{O}=c_{O}+(2c_{O}\Lambda_{W}^{2}+\sum_{\digamma}c_{\digamma}m^{2}_{\digamma})/\Delta^{2} , makes sense only in curved geometry hence the grey color.)

	$\delta S_{log}$	$\delta S_H$	$\delta S_V$	$\delta S_O$	Problem
$\delta V$	$\ne 0$	0	0	${\tilde{c}}_O{\Delta }^4$	CCP
$\delta m_H^2$	$\ne 0$	$c_H{\Delta }^2$	0	0	BHP
$\delta m_W^2$	$\ne 0$	0	$c_W{\Delta }^2$	0	BHP
$\delta m_Z^2$	$\ne 0$	0	$c_Z{\Delta }^2$	0	BHP
$\delta m_{\gamma }^2$	0	0	$c_{\gamma }{\Delta }^2$	0	CCB
$\delta m_g^2$	0	0	$c_g{\Delta }^2$	0	CCB

Equations

$$\text{BSM}=\text{gravity}(\text{GR})+\text{new QFT beyond the SM}\text{(NP)}$$

$$\delta m_H^2=c_H{\Lambda }^2$$

$$c_H=\frac{3}{16{\pi }^2{\left|⟨H⟩\right|}^2}\left(4m_t^2-2M_W^2-M_Z^2-m_H^2\right)\approx 5.14\times 10^{-2}$$

$${\Lambda }_W\approx 550\mathrm{GeV}$$

$$\text{constructing the SM effective action below}{\Lambda }_W\text{assuming that}$$

$$\text{there is nothing but the SM all the way up to}{\Lambda }_U\gg {\Lambda }_{LHC},$$

$$\text{gravity and neutralization of the destabilizing}{\Lambda }_U\text{sensitivities.}$$

$$S_{eff}\left(\eta \right)$$

$$+\delta S_O\left(\eta ,{\Delta }^2\right)+\delta S_H\left(\eta ,{\Delta }^2\right)+\delta S_V\left(\eta ,{\Delta }^2\right)$$

$${\Delta }^2={\Lambda }_U^2-{\Lambda }_W^2$$

$$\delta S_O$$

$$\delta S_H$$

$$\delta S_V$$

$$\frac{{\Lambda }_W}{{\Lambda }_U}$$

$${\eta }_{\mu \nu }\hookrightarrow g_{\mu \nu }$$

$${}^g{\Gamma }_{\mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \rho }\left({\partial }_{\mu }{g}_{\nu \rho }+{\partial }_{\nu }{g}_{\rho \mu }-{\partial }_{\rho }{g}_{\mu \nu }\right)$$

$$\displaystyle S_{eff}\Big{(}g\Big{)}-\int d^{4}x\sqrt{-g}\left\{{\tilde{M}}^{2}R+{\tilde{\alpha}}R^{2}+{\tilde{\beta}}R^{\mu\nu}R_{\mu\nu}+\cdots\right\}$$

$$\delta S_V\left(\eta ,{\Delta }^2\right)=\delta S_V\left(\eta ,{\Delta }^2\right)-I(\eta ,V)+I(\eta ,V)$$

$$I(\eta ,V)=\int d^{4}x\sqrt{-\eta }\frac{c_V}{2}\text{mbox}Tr\left\{{\eta }_{\mu \alpha }{\eta }_{\nu \beta }{V}^{\mu \nu }{V}^{\alpha \beta }\right\}$$

$$\delta S_V\left(\eta ,{\Delta }^2\right)$$

$$+\int d^{4}x\sqrt{-\eta }{c}_V\text{mbox}Tr\left\{V^{\mu }\left(-D_{\mu \nu }^2+{\Delta }^2{\eta }_{\mu \nu }\right)V^{\nu }\right\}$$

$$+\int d^{4}x\sqrt{-\eta }{c}_V\text{mbox}Tr\left\{{\partial }_{\mu }\left({\eta }_{\alpha \beta }{V}^{\alpha }{V}^{\beta \mu }\right)\right\}$$

$$\delta S_V\left(g,{\Delta }^2\right)$$

$$+\int d^{4}x\sqrt{-g}{c}_V\text{mbox}Tr\left\{V^{\mu }\left(-{\mathcal{D}}_{\mu \nu }^2+{\Delta }^2{g}_{\mu \nu }\right)V^{\nu }\right\}$$

$$+\int d^{4}x\sqrt{-g}{c}_V\text{mbox}Tr\left\{{\nabla }_{\mu }\left(g_{\alpha \beta }{V}^{\alpha }{V}^{\beta \mu }\right)\right\}$$

$${\Delta }^2{g}_{\mu \nu }\hookrightarrow {\mathbb{R}}_{\mu \nu }\left(\Gamma \right)$$

$$\delta S_V\left(g,\mathbb{R}\right)$$

$$+\int d^{4}x\sqrt{-g}{c}_V\text{mbox}Tr\left\{V^{\mu }\left(-{\mathcal{D}}_{\mu \nu }^2+{\mathbb{R}}_{\mu \nu }\left(\Gamma \right)\right)V^{\nu }\right\}$$

$$+\int d^{4}x\sqrt{-g}{c}_V\text{mbox}Tr\left\{{\nabla }_{\mu }\left(g_{\alpha \beta }{V}^{\alpha }{V}^{\beta \mu }\right)\right\}$$

$$\delta S_V\left(g,\mathbb{R},R\right)=\int d^{4}x\sqrt{-g}{c}_V\text{mbox}Tr\left\{V^{\mu }\left({\mathbb{R}}_{\mu \nu }(\Gamma )-R_{\mu \nu }(g)\right)V^{\nu }\right\}$$

$$\frac{{\Lambda }_W}{{\Lambda }_U}\text{must be held unchanged}$$

$${\Gamma }_{\mu \nu }^{\lambda }={}^g{\Gamma }_{\mu \nu }^{\lambda }+\mathcal{O}{\left(M_{Pl}^{-2}\right)}_{H,V}$$

$${\mathbb{R}}_{\mu \nu }(\Gamma )=R_{\mu \nu }(g)+\mathcal{O}{\left(M_{Pl}^{-2}\right)}_{H,V}$$

$$\delta S_V\left(g,\mathbb{R},R\right)=\int d^{4}x\sqrt{-g}\left\{0+\mathcal{O}{\left(M_{Pl}^{-2}\right)}_{H,V}\right\}$$

$$\int d^{4}x\sqrt{-\text{Det}\left(\frac{\mathbb{R}\left(\Gamma \right)}{{\Delta }^2}\right)}$$

Full text

Metric-affine gravity, general relativity, hierarchy problem, seesawly coupled particles.

1. I Introduction

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

3. II.1 UV Boundary

4. II.2 SM Effective Action

5. II.3 UV Sensitivity Problems

6. II.4 Impossibility of Renormalizing Away CCP, CCB and BHP

7. III Incorporation of Curvature into the Effective SM

8. III.1 How not to Incorporate: Curvature by Hand

9. III.2 How to Incorporate: Curvature by Gauge Symmetry Restoration

10. III.2.1 $\delta S_{V}\left(\eta,\Delta^{2}\right)$ in a New Light

11. III.2.2 $\delta S_{V}\left(g,\Delta^{2}\right)$ in a New “Curvature”

12. III.3 How Curvature Emerges: Elasticated Flat Spacetime

13. III.4 Symmergence Principle

14. IV Incorporation of GR and the Requisite NP

15. IV.1 Metric-Affine Gravity

16. IV.2 Necessity of NP

17. IV.3 Incorporation of GR

18. V Observational and Experimental Tests

19. V.1 CCB is suppressed

20. V.2 The UV boundary might have a say in CCP

21. V.3 Higher-curvature terms are absent

22. V.4 Higgs-curvature coupling is predicted

23. V.5 Couplings run nearly as in the SM

24. V.6 SM-NP coupling is not a necessity

25. V.7 Electroweak stability restricts SM-NP coupling

26. V.8 The NP can be probed in cosmic rays

27. V.9 The NP can be probed in colliders

28. V.9.1 NP Scalars and Vectors

29. V.9.2 Right-Handed Neutrinos

30. V.10 Dark matter weighs below weak scale

31. VI Conclusion

I 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 sm-sofar ; exotica . 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 veltman , exclusion of gravity grav , and absence of a dark matter candidate cdm , which are impossible to address without new physics beyond the SM (BSM). Schematically,

[TABLE]

in which the general relativistic (GR) structure of gravity is revealed by various experiments will and observations lam-cdm . The NP is under way at colliders exotica and dark matter searches cdm-current .

Quantum correction to the Higgs boson mass, quadratically sensitive to UV boundary veltman , 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 exotica 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 (Sec. II and Sec. III) and, at a later stage, it ends up with NP through the consistency of the induced gravitational constant (Sec. IV).

The GR must be incorporated into the SM will ; lam-cdm . This has been attempted with classical GR cgr (despite cgr2 ), quantized GR qgr1 (despite qgr2 ; bicep ), and emergent GR sakharov (see also emerge-verlinde ; emerge-raamsdonk ; emerge-lit ; terazawa ). The present work incorporates curvature into the SM effective action in flat spacetime (Sec. II) by building on the nascent ideas proposed in demir1 ; demir2 and subsequent developments voiced in talk-ppc ; talk-NP ; talk-hepac ; talk-qdis . It incorporates gravity in a way restoring the gauge symmetries broken by the UV cutoff hard-break (Sec. III B), in a way elasticating the rigid flat spacetime (Sec. III C), and in a way involving a nontrivial NP sector to induce the gravitational constant (Sec. IV). This gauge symmetry-restoring emergent gravity, symmergent gravity talk-NP ; talk-hepac ; talk-qdis in brief, sets up a framework (Sec. IV) in which

curvature arises as a manifestation of the elastication of the flat spacetime, 2. 2.

GR emerges along with the restoration of gauge invariance, 3. 3.

gravitational constant necessitates an NP sector, 4. 4.

SM + NP possesses exact fermi-bose balance and the induced gravitational constant is suggestive of a trans-Planckian SUSY breaking susy , 5. 5.

the UV boundary can be fixed, in principle, by suppression of the cosmological constant (with no immediate solution for the cosmological constant problem ccp though)

so that there arise a number of descriptive signatures with which symmergence can be probed via decisive experiments (Sec. V):

Higher-curvature terms are predicted to be absent. This excludes, for instance, $f(R)$ gravity fR and agrees well with the current cosmological data lam-cdm . 2. 2.

NP scalars with trans-GZK GZK VEVs are predicted to give cause to cosmic rays cosmic-ray (digluon) as well as gamma rays gamma-ray (diphoton) via certain Planck-suppressed higher-dimension operators. 3. 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 beyond ) 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 $m_{H}^{2}/m_{NP}^{2}$ then the Higgs boson mass $m_{H}$ remains within the allowed limits. This seesawic (seesaw-wise) structure leads to various testable features:

1. (a)

It is predicted that heavier the NP larger the luminosity needed to discover it. This distinctive feature can be probed at present coupling-LHC and future colliders coupling-FCC ; coupling-ILC . 2. (b)

It is also predicted that the right-handed neutrinos neutrino-portal must weigh below a 1000 TeV (see also vissani ). This bound can be tested at future colliders biz if not at the near-future SHiP experiment SHiP . 3. (c)

It turns out that the SM couplings (gauge and non-gauge) 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 cutoff-dimreg , can be tested at present coupling-LHC and future colliders coupling-FCC ; coupling-ILC . 4. (d)

Symmergence accommodates both ebony (having only gravitational interactions with the SM) demir2 ; ebony 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 cdm-current ; dm-LHC . 5. (e)

It is predicted that, in the SM, non-minimal Higgs-curvature coupling equals $1.3\%$ at one-loop and remains so unless the NP lies near the electroweak scale. This coupling, too small to drive the Higgs inflation Higgs-inflation , can serve as a testbed at collider experiments non-min-cinli if not in the astrophysical or cosmological environments.

The work is concluded in Sec. VI.

II 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.

II.1 UV Boundary

The Higgs mass-squared $m_{H}^{2}$, measured to be $\left(m_{H}^{2}\right)_{LHC}\approx(125\,{\rm GeV})^{2}$ at the LHC higgs-mass , is overwhelmed by the quantum correction veltman

[TABLE]

in which $\Lambda$ is the UV boundary of the SM, and

[TABLE]

is the loop factor. The correction (2), a one-loop SM effect, grows quadratically with $\Lambda$ and exceeds $\left(m_{H}^{2}\right)_{LHC}$ already at $\Lambda=\Lambda_{W}$, where

[TABLE]

lies just above the electroweak scale $|\langle H\rangle|\approx 246.22\ {\rm GeV}$. This low-lying $\Lambda_{W}$, which can be changed slightly by incorporating subleading $\log m_{H}/\Lambda$ corrections to (2), is a characteristic feature of the SM spectrum and the experimental result $\left(m_{H}^{2}\right)_{LHC}$. It implies that the SM must stop working at $\Lambda_{W}$. 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 demir2 ; talk-NP , that is, by first

[TABLE]

and then

[TABLE]

The first step sets up a UV boundary $\Lambda_{U}$ and reveals SM’s UV sensitivity. The second, on the other hand, uncovers NP via induction of gravity and fixes $\Lambda_{U}$ in terms of the NP scale.

II.2 SM Effective Action

In accordance with (5), integration of the fast modes (fields with energies $\gtrsim\Lambda_{{W}}$) out of the SM spectrum gives an effective action for slow modes $\digamma$ (fields with energies $\lesssim\Lambda_{{W}}$) eff-ac

[TABLE]

in which $\eta_{\mu\nu}$ is the flat metric, $H$ is the slow Higgs field, $V_{\mu}$ are the slow gauge fields (photon, gluon, $W$ and $Z$), and finally

[TABLE]

is the UV-EW gap. The tree-level SM action $S_{tree}$ and the logarithmic corrections $\delta S_{log}$ both lie below $\Lambda_{{W}}$. But, the other three

[TABLE]

pull the SM off the electroweak scale depending on how large $\Delta^{2}$ is. They tend to destabilize the SM and it is this destabilization that necessitates a neutralization mechanism. Their Wilson coefficients $c_{O,\cdots,V}$ involve only the ratio veltman

[TABLE]

as the measure of the EW-UV hierarchy.

II.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) ccp , caused by $\delta S_{O}$, exists only when gravity is present. The big hierarchy problem (BHP) veltman (gauge hierarchy problem) refers to quadratic UV sensitivities of the Higgs (from $\delta S_{H}$) and $W/Z$ (from $\delta S_{V}$) masses. The electric charge or color breaking (CCB) hard-break , on the other hand, arises from the photon and the gluon mass terms in $\delta S_{V}$ (purely quadratic in $\Delta$). The SM is impossible to make sense in the UV before these problems are satisfactorily resolved.

II.4 Impossibility of Renormalizing Away CCP, CCB and BHP

The SM is a renormalizable QFT. If so, why isn’t it possible to include $\delta S_{log}$, $\delta S_{O}$, $\delta S_{H}$, $\delta S_{V}$ into a renormalization of $S_{tree}$ to get rid of the problems in Table 1? Because $\Delta$ is physical. Indeed, $\Lambda_{U}$ can pertain to gravity or NP sectors veltman . Moreover, there is simply no place to hide $\delta m_{\gamma,g}^{2}=c_{\gamma,g}\Delta^{2}\neq 0$ since $(m_{\gamma,g}^{2})_{tree}=0$. These problems are therefore physical and their solutions entail physical changes on the SM (like inclusion of gravity).

III 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 (II.2) and how that incorporation affects the UV sensitivity problems in Table 1.

III.1 How not to Incorporate: Curvature by Hand

Gravity is incorporated into classical field theories in flat spacetime by first mapping the flat metric $\eta_{\mu\nu}$ into a putative curved metric $g_{\mu\nu}$ as

[TABLE]

in view of general covariance eqp , and then adding curvature of the Levi-Civita connection

[TABLE]

to make $g_{\mu\nu}$ dynamical. The curvature sector, added by hand, ignoring ghosts, can involve all curvature invariants h-curv . It can involve, for instance, the Ricci tensor $R_{\mu\nu}\left({}^{g}\Gamma\right)$ in traced ($R=g^{\mu\nu}R_{\mu\nu}\left({}^{g}\Gamma\right)$), squared ($R_{\mu\nu}\left({}^{g}\Gamma\right)R^{\mu\nu}\left({}^{g}\Gamma\right)$) or in any other invariant form. This curvature-by-hand method, a standard procedure for classical field theories, leads to

[TABLE]

when applied to the SM effective action in (II.2). The problem with this action is that ${\tilde{M}}$, ${\tilde{\alpha}}$, ${\tilde{\beta}}$, $\cdots$ are all inherently incalculable demir2 ; talk-NP ; talk-hepac . This is because matter loops have already been used up in forming the flat spacetime effective action $S_{eff}$, 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 eqp is not adequate for incorporating curvature into effective SM. In essence, what is needed is a separate covariance relation between the scales in $S_{eff}$ (say, $\Lambda_{{W}}^{2}$ or $\Delta^{2}$) and curvature talk-ppc so that gravity can be incorporated in a way involving no arbitrary, incalculable constants.

III.2 How to Incorporate: Curvature by Gauge Symmetry Restoration

The gauge part $\delta S_{V}\left(\eta,\Delta^{2}\right)$, 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 stueckelberg nor the spontaneous symmetry breaking ssb . It can, nonetheless, be tackled by furthering the nascent ideas proposed in demir2 (which attempts at restoring gauge symmetry within the GR with fixed $\Lambda_{{U}}^{2}+\Lambda_{{W}}^{2}$) and subsequent advancements voiced in talk-ppc ; talk-NP ; talk-hepac ; talk-qdis .

III.2.1 $\delta S_{V}\left(\eta,\Delta^{2}\right)$ in a New Light

It proves useful to start with the obvious identity demir2 ; talk-ppc

[TABLE]

in which the gauge-invariant kinetic construct

[TABLE]

is subtracted from and added back to $\delta S_{V}$. This construct, involving the loop factor $c_{V}$ and the field strength tensor $V_{\mu\nu}$, leads to

[TABLE]

if, at the right hand side of (16), $\delta S_{V}$ is replaced with (11), $``\!-I(\eta,V)"$ is left untouched, and yet $``\!+I(\eta,V)"$ is integrated by-parts to involve $D_{\mu\nu}^{2}={{D}}^{2}\eta_{\mu\nu}-D_{\mu}D_{\nu}-V_{\mu\nu}$ where $D_{\mu}$ is gauge-covariant derivative. This recast $\delta S_{V}$ gets to curved spacetime via (13) to become

[TABLE]

where ${\mathcal{D}}_{\mu}$ is the gauge-covariant derivative with respect to the covariant derivative $\nabla_{\mu}$ of the Levi-Civita connection ${}^{g}\Gamma^{\lambda}_{\mu\nu}$, and ${\mathcal{D}}_{\mu\nu}^{2}={\mathcal{D}}^{2}g_{\mu\nu}-{\mathcal{D}}_{\mu}{\mathcal{D}}_{\nu}-V_{\mu\nu}$.

III.2.2 $\delta S_{V}\left(g,\Delta^{2}\right)$ in a New “Curvature”

Is there a simple way of killing ${{\delta S_{V}\left(g,\Delta^{2}\right)}}$? Yes, there is. Indeed, ${{\delta S_{V}\left(g,\Delta^{2}\right)}}$ vanishes identically if $\Delta^{2}g_{\mu\nu}$ is replaced with $R_{\mu\nu}\left({}^{g}\Gamma\right)$. This encouraging feature, not to be confused with derivation of gravity from self-interacting spin-2 fields in flat spacetime deser-gravity , is pitiably problematic because $\Delta^{2}g_{\mu\nu}\,{{\hookrightarrow}}\,R_{\mu\nu}\left({}^{g}\Gamma\right)$ contradicts with $\eta_{\mu\nu}\,{{\hookrightarrow}}\,g_{\mu\nu}$. If it were not for this contradiction, emergence of curvature from $\Delta^{2}g_{\mu\nu}$ would solve the CCB demir2 ; talk-ppc ; talk-NP ; talk-hepac .

The contradiction can be avoided by introducing, for instance, a more general map talk-qdis

[TABLE]

in which ${\mathbb{R}}_{\mu\nu}(\Gamma)$ is the Ricci curvature of a symmetric affine connection $\Gamma^{\lambda}_{\mu\nu}$ (bearing no relationship to the Levi-Civita connection ${}^{g}\Gamma^{\lambda}_{\mu\nu}$). This new map removes contradiction because while ${\mathbb{R}}_{\mu\nu}\left(\Gamma\right)\leadsto\Delta^{2}g_{\mu\nu}$ fixes the affine connection, $g_{\mu\nu}\leadsto\eta_{\mu\nu}$ does the metric talk-qdis . It throws ${{\delta S_{V}\left(g,\Delta^{2}\right)}}$ in (19) into metric-affine geometry Palatini ; Palatini2 to give it the “mass-free” form

[TABLE]

whose by-parts integration results in

[TABLE]

under the condition that $c_{V}$ must be held unchanged or, equivalently, the EW-UV hierarchy

[TABLE]

while the affine curvature arises as in (20). This preservation is crucial for ensuring that the SM is indeed stabilized at $\Lambda_{W}\ll\Lambda_{U}$ veltman .

The CCB attains a solution only if ${{\delta S_{V}\left(g,{\mathbb{R}},R\right)}}$ in (22) is suppressed. And suppression is seen to necessitate $\Gamma^{\lambda}_{\mu\nu}$ to be close to ${}^{g}\Gamma^{\lambda}_{\mu\nu}$. Fortunately, $\Gamma^{\lambda}_{\mu\nu}$ does really lie close to ${}^{g}\Gamma^{\lambda}_{\mu\nu}$. Indeed, as already voiced in talk-qdis and as will be derived in equations (44) and (45) in Sec. IV.3 below, $\Gamma^{\lambda}_{\mu\nu}$ assumes the form

[TABLE]

as a solution to its equation of motion. It is clear that $\Gamma^{\lambda}_{\mu\nu}$ differs from ${}^{g}\Gamma^{\lambda}_{\mu\nu}$ only by Planck-suppressed ${\mathcal{O}}\!\left(M_{Pl}^{-2}\right)$ 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 Palatini ; Palatini3 , and the analysis in Sec. IV.3 will follow them.

Under the solution (24) for $\Gamma^{\lambda}_{\mu\nu}$, the Ricci curvature takes the form

[TABLE]

and its replacement into the action (22) leads to

[TABLE]

as a proof of the fact that the CCB is suppressed up to Planck-suppressed ${\mathcal{O}}\!\left(M_{Pl}^{-2}\right)$ 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.

III.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 correction $\delta S_{O}$ in (9), for instance, can no way curve it, even with infinite $\Delta$. 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 sakharov 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 $M_{Pl}^{2}\sim\Delta^{2}$ but leaves behind an ${\mathcal{O}}(\Delta^{4})$ vacuum energy and an ${\mathcal{O}}(\Delta^{2})$ 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

[TABLE]

takes the form

[TABLE]

after integrating out the affine curvature ${\mathbb{R}}_{\mu\nu}(\Gamma)$. The reason, as was first pointed out in demir1 , is that the equation of motion for the affine connection

[TABLE]

which follows from (27), is solved by

[TABLE]

with a dynamically-induced metric $g_{\mu\nu}$. This solution, the well-known Eddington solution eddington1 ; eddington2 , is precisely the map (20) imposed by gauge symmetry restoration (partly because curvature scale in (27) is set to $\Delta^{2}$).

The meaning of the 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 $\delta S_{O}(g)$, obtained from $\delta S_{O}(\eta)$ in (9) via (13), can be mapped into the affine action $\delta S_{O}(\Gamma)$ by integrating in ${\mathbb{R}}_{\mu\nu}\left(\Gamma\right)$ via the Eddington solution (30). This “integrating in” stage, illustrated in Fig. 1, represents the elastication of the flat spcetime demir1 . The problem here is that the affine geometry of $\delta S_{O}(\Gamma)$ is devoid of any metrical structure to accommodate $\delta S_{log}$ and $\delta S_{V}$. The remedy is to integrate out $\Delta^{2}$ to get the metric-affine action $\delta S_{O}(g,\Gamma)$ in Fig. 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 Sec. III.1 followed by (20) in Sec. III.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.

The emergence mechanism here is similar, at least in philosophy, to that of Sakharov sakharov 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 emerge-verlinde and entanglement gravity of Raamsdonk emerge-raamsdonk .

III.4 Symmergence Principle

The results of Sec. III.2.2 and Sec. III.3 above can be organized as a principle to apply to UV sensitivies of any given QFT. Indeed, it turns out that gravity can be incorporated into a flat spacetime QFT with UV-IR gap $\Delta$ and metric $\eta_{\mu\nu}$ by first letting eqp

[TABLE]

and then talk-qdis

[TABLE]

so that if

[TABLE]

then CCB gets suppressed and GR emerges as a gauge symmetry restoring emergent gravity or, briefly, symmergent gravity. This three-stage mainframe will define what shall be called symmergence in what follows.

IV 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.

IV.1 Metric-Affine Gravity

Affine gravity, as in $\delta S_{O}(\Gamma)$ in Fig. 1, involves only the affine connection eddington1 ; eddington2 . Metrical gravity, the GR itself, is based solely on the metric tensor. The metric-affine gravity (MAG), on the other hand, comprises, as in $\delta S_{O}(g,\Gamma)$ in Fig. 1, both metric and affine connection as two independent dynamical variables Palatini ; Palatini2 ; Palatini3 . The MAG reduces to the so-called Palatini formalism Palatini if connection doe 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 Palatini3 ; romano . Then, reduction to the GR can put constraints even on the matter sector. The sections below, starting with Sec. IV.3, will be devoted to the GR limit of the MAG arising from the SM effective action.

IV.2 Necessity of NP

Under (13) and (20) or under the emergence mechanism in Fig. 1, the Higgs and vacuum sectors of the flat spacetime effective action in (II.2) transform as

[TABLE]

after defining

[TABLE]

for convenience. The right-hand side of (34) forms the curvature sector of the SM effective action in metric-affine geometry Palatini ; Palatini2 ; Palatini3 . The apparent gravitational scale, $\frac{c_{O}}{2}\Lambda_{{W}}^{2}+\frac{1}{4}\sum_{\digamma}c_{\digamma}m_{\digamma}^{2}$, is wrong in both sign ($c_{O}<0$ in the SM) and size ($\sum_{\digamma}c_{\digamma}m_{\digamma}^{2}<\Lambda_{{W}}^{2}$ in the SM). It is for this reason that an NP sector (made up of entirely new fields $\digamma^{\prime}$) must be introduced so that the new gravitational scale

[TABLE]

can come out right thanks to either the NP spectrum ($c_{O^{\prime}}$) or the NP scale ($m_{\digamma^{\prime}}$). In fact, its one-loop value

[TABLE]

makes it clear that the NP must have either a crowded (${\mbox{Str}}\left[1\right]\sim 10^{35}$) talk-NP ; talk-hepac ; talk-qdis or a heavy (${\mbox{Str}}\left[m^{2}\right]\gtrsim M_{Pl}^{2}$) bosonic sector.

IV.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

[TABLE]

involves

[TABLE]

as an extension of $q_{\mu\nu}$ in (35) to the gauge and NP sectors. It gathers affine curvature terms from (22) (after extending it with the NP gauge bosons $V^{\prime}_{\mu}$ with loop factors $c_{V^{\prime}}$) and (34) (after augmenting it with the NP vacuum energy with the loop factor $c_{O^{\prime}}$ and all of the NP fields with loop factors $c_{\digamma^{\prime}}$).

The equation of motion for $\Gamma^{\lambda}_{\mu\nu}$, stipulated by stationarity of the action (38) against variations in $\Gamma^{\lambda}_{\mu\nu}$, turns out to be a metricity condition on $Q_{\mu\nu}$

[TABLE]

and possesses the generic solution Palatini ; Palatini3

[TABLE]

which expresses $\Gamma^{\lambda}_{\mu\nu}$ in terms of its own curvature ${\mathbb{R}}_{\mu\nu}(\Gamma)$, as evidenced by $Q_{\mu\nu}$ itself. This dependence on curvature is critical because if it is the case then (41) acts as a differential equation for $\Gamma^{\lambda}_{\mu\nu}$, and its solution carries extra geometrical degrees of freedom not found in the Levi-Civita connection ${}^{g}\Gamma^{\lambda}_{\mu\nu}$ Palatini3 . This means that ${\mathbb{R}}_{\mu\nu}(\Gamma)$ must disappear from $Q_{\mu\nu}$ for GR to be able to symmerge, and it does so if

[TABLE]

or, 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 beyond . All it does is to trim $Q_{\mu\nu}$ in (IV.3) down to

[TABLE]

in which

[TABLE]

is the apparent gravitational scale. Now, GR can symmerge because $\Gamma^{\lambda}_{\mu\nu}$ is an auxiliary field (algebraic solution) having no geometrical content beyond ${}^{g}\Gamma^{\lambda}_{\mu\nu}$ Palatini ; Palatini3 . Indeed, after (43), the affine connection in (41) takes the form

[TABLE]

to contain only the Levi-Civita connection and the scalars $\phi$ and vectors ${\mathcal{V}}_{\mu}$ in ${\mathring{Q}}_{\mu\nu}$. Given the enormity of $M$, this expression can always be expanded order by order in $1/M^{2}$ to get

[TABLE]

using the notation in (24) for the remainder of the expansion. The higher-order ${\mathcal{O}}\!\left(M^{-4}\right)_{\phi,{\mathcal{V}}}$ terms are straightforwardly computed from the exact expression (44) order by order in $1/M^{2}$. In accordance with (45), the affine curvature expands as

[TABLE]

where $\left(\nabla^{2}\right)_{\mu\nu}^{\alpha\beta}=\nabla^{\alpha}\,\nabla_{\mu}\delta^{\beta}_{\nu}+\nabla^{\alpha}\,\nabla_{\nu}\delta^{\beta}_{\mu}-\Box\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-\nabla_{\nu}\,\nabla_{\mu}g^{\alpha\beta}$ grav-prop . This expansion leads to the reductions

[TABLE]

[TABLE]

[TABLE]

so that the SM + NP affine action in (38) reduces to

[TABLE]

to yield the Einstein-Hilbert action for non-minimally coupled scalars $\phi=H,H^{\prime}$ non-min-curve . Here, ${\mathcal{O}}\!\left(M^{-2}\right)_{\phi,{\mathcal{V}}}$ terms can be constructed from the exact solution (44) order by order in $1/M^{2}$. Unlike the generic effective field-theoretic approach weinberg-eff , the remainder ${\mathcal{O}}\!\left(M^{-2}\right)_{\phi,{\mathcal{V}}}$ assumes a specific structure coordinated by (44) through ${\mathring{Q}}_{\mu\nu}$ in (43).

The GR gets properly incorporated if Planck scale is generated correctly. In view of (50), it is given by

[TABLE]

in which vacuum contribution becomes significant when $\sum_{H^{\prime}}c_{H^{\prime}}\langle H^{\prime\dagger}H^{\prime}\rangle\sim M^{2}$. The first part, identical to (IV.3), implies that the bosonic sector of the NP must be either crowded ($n_{b^{\prime}}\sim 10^{35}$ and $m_{\digamma^{\prime}}\gtrsim\Lambda_{W}$) demir2 ; talk-NP ; talk-hepac ; talk-qdis or heavy enough ($n_{b^{\prime}}\sim$ much less yet $m_{\digamma^{\prime}}\lesssim\Lambda_{U}$) for it to be able to induce the trans-Planckian scale $8\pi M_{Pl}$.

The NP sector, needed for induction of the gravitational scale as in (36) or (51), can have nontrivial structure. Indeed, given the fermi-bose balance of SM+NP and given also that ${\mbox{Str}}\left[m^{2}\right]$ in (51) is reminiscent of the mass sum rule in broken SUSY susy ; beyond , it becomes apparent that the SM and NP, with the seesawic couplings in (59), may well be the remnants of a trans-Planckian SUSY broken around $8\pi M_{Pl}$. 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 susy to generate the nonzero sum rule in (51), 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.

V 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.

V.1 CCB is suppressed

Nullification of the gauge part in (49) ensures that color and electromagnetism are restored and electroweak breaking is rendered spontaneous up to doubly Planck-suppressed terms involving $\phi$ and ${\mathcal{V}}_{\mu}$. This is important for assuring neutralization of the CCB in accordance with the symmergence principle. To that end, exact form of the affine connection in (44), which is never derivative-free, ensures that gauge boson masses are impossible to arise at any order in $1/M^{2}$. Indeed, iteration of (50) up to next order

[TABLE]

exemplifies that what arise at each order involve derivatives of $\phi$ and ${\mathcal{V}}_{\mu}$.

V.2 The UV boundary might have a say in CCP

The total vacuum energy at one loop

[TABLE]

is composed of the tree-level potential plus the loop corrections involving the UV scale $\Lambda_{U}$. It does not have to be small. In fact, it can readily exceed the current observational bounds lam-cdm to give cause to the CCP ccp . Needless to say, the CCP arises from not the $\Delta^{4}$ 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 (53), can conspire to yield the observed value $V_{obs}=\left(2.57\times 10^{-3}\ {\rm eV}\right)^{4}$ lam-cdm is what the CCP is all about ccp . Indeed, the empirical equality

[TABLE]

gives a fix on the UV boundary in that the solution $\Lambda_{U}=\Lambda_{U}^{0}$ expresses $\Lambda_{U}$ 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 $8\pi M_{Pl}$ susy-ccp , can prevent fine-tuning and render $\Lambda_{U}^{0}$ physical.

Symmergent gravity and the NP it necessitates agree with all the existing bounds thanks to their prediction that gravity is Einstein (as in lam-cdm ), missing matter can be ebony (as in cdm-current ), and non-SM interactions are not a necessity (as in exotica ). In case of experimental discoveries, however, they can be probed with conclusive tests via the dark matter and SM-NP couplings.

V.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 not $f(R)$ gravity fR but just the GR has an answer in symmergence. And current observations lam-cdm seem to have already confirmed the symmergence.

V.4 Higgs-curvature coupling is predicted

The non-minimal Higgs-curvature coupling non-min-curve , symmergement of the Higgs quadratic UV sensitivity in (10), is a model-dependent loop factor. It has the numerical value

[TABLE]

as follows from (3). This prediction is specific to the SM spectrum. It changes by an amount $\propto\lambda^{2}_{H\digamma^{\prime}}$ in the presence of the NP. This change, as ensured by the bound (59), 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 $c_{H}$.

Obviously, $c_{H}$ is too small to facilitate the Higgs inflation Higgs-inflation . 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 non-min-cinli .

The non-minimal $H^{\prime}$-curvature coupling non-min-curve is expected to have similar features. Indeed, $c_{H^{\prime}}$ in (50) can be directly computed with a concrete NP model. It may differ significantly from $c_{H}$. In fact, what drives cosmic inflation could well be $H^{\prime}$ not $H$ demir2 ; talk-NP .

V.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 cutoff-dimreg ; demir2 ; talk-NP ; talk-qdis

[TABLE]

expresses all amplitudes in terms of the renormalization scale $\mu$ after subtracting $1/\epsilon$ terms in ${\overline{\text{MS}}}$ scheme. Their independence from $\mu$ 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 $\Lambda_{W}$. It can be directly tested at present coupling-LHC and future coupling-FCC ; coupling-ILC colliders.

V.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:

Ebony NP. This kind of NP has no non-gravitatinal couplings to the SM ebony . It forms an insular sector (composed of, for instance, high-rank non-Abelian gauge fields and fermions demir2 ; talk-NP ). In view of the present searches, which seem all negative, this ebony, pitch-dark NP shows good agreement with all the available data cdm-current . 2. 2.

Dark NP. This type of NP is neutral under the SM but couples to it via Higgs, hypercharge and lepton portals demir2 ; portals ; neutrino-portal . Indeed, at the renormalizable level, scalars $H^{\prime}$, Abelian vectors $V^{\prime}_{\mu}$ and fermions $N^{\prime}$ can couple to the SM via

[TABLE]

in which the couplings $\lambda^{2}_{HH^{\prime}}$, $\lambda_{BZ^{\prime}}$, $\lambda_{HN^{\prime}}$ can take, none to significant, a wide range of values with characteristic experimental signals cdm-current ; dm-LHC . 3. 3.

Visible NP. In this case, the NP is partly or wholly charged under the SM (e.g., $H^{\prime}$ in (57) can be an SU(2) doublet) demir2 . It couples to the SM with SM coupling strengths, and it seems to have already been sidelined by the current bounds exotica ; visible .

The NP sector, whose subsectors are depicted in Fig. 2, can be of any composition (ebony to dark), can have any couplings (none to significant) and can lie anywhere (from $\Lambda_{{W}}$ to $\Lambda_{{U}}$) as long as the gravitational scale in (36) comes out right. Non-necessity of any sizable NP-SM couplings is what distinguishes the NP of the symmergence from SUSY, extra dimensions, composites models and others beyond 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 exotica .

V.7 Electroweak stability restricts SM-NP coupling

The Higgs part of (34) makes it clear that the quadratic UV sensitvity in flat spacetime changes to Higgs-curvature coupling in metric-affine spacetime. This solves the BHP veltman . In the presence of NP, however, the Higgs sector is destabilized by not only the BHP but also the SM-NP coupling lhp2 ; demir2 ; talk-NP . Indeed, the SM-NP interactions in (57), for instance, lead to a new Higgs mass correction (in addition to $\delta S_{H}$ in (10))

[TABLE]

with a loop factor ${\tilde{c}}_{H}$. This is sensitive to not ${\Lambda_{{U}}}$ but ${{m_{\digamma^{\prime}}}}$. The problem is that heavier the NP larger the shift in the Higgs mass and 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 beyond ; lhp2 . In symmergence, however, SM-NP coupling is not a necessity demir2 ; talk-NP ; talk-hepac ; talk-qdis . It works with any coupling strengths such as the seesawic ones

[TABLE]

with which the Higgs mass shift in (58) falls below the Higgs mass to ensure stability of the electroweak scale. This seesawic structure, explicated in Fig. 3, implies that heavier the NP weaker its coupling to the SM.

V.8 The NP can be probed in cosmic rays

The Planck-suppressed $\phi$ and ${\mathcal{V}}_{\mu}$ composites in (52) do not produce any mass terms. They can produce, however, sizable signals if the NP scalars $H^{\prime}$ develop large VEVs (compared to $M_{Pl}$). This effect, as depicted in Fig. 4, leads to observable effects. The $H^{\prime}$ decays into two gluons of GZK energy GZK , for instance, occurs at a rate

[TABLE]

with $\langle H^{\prime}\rangle\simeq m_{H^{\prime}}$. These gluons can partake in ultra high energy cosmic rays cosmic-ray upon hadronization. The diphotons, produced similarly, can contribute to diffuse gamma-ray background gamma-ray . These events, which weaken at low $\langle H^{\prime}\rangle$, are cosmic probes of symmergence.

V.9 The NP can be probed in colliders

One immediate implication of the bound (59) is that heavier the NP larger the luminosity needed to probe them at colliders. This can prove important in designing accelerators and detectors, as implied by the exemplary analyses below.

V.9.1 NP Scalars and Vectors

One way to see if the underlying model is symmergence or not is to measure scalar and vector masses and then determine if their production cross sections comply with the seesawic couplings in (59). For instance, $m_{H^{\prime}}$ can be measured from either of $HH\rightarrow H^{\prime}\rightarrow HH$ in Fig. 5 (a) or $HH\rightarrow H^{\prime}H^{\prime}$ in Fig. 5 (b) such that the ratio of their cross sections

[TABLE]

acts as an efficient probe of symmergence. This procedure works also for the $Z^{\prime}$ scatterings in Fig. 5 (c) and Fig. 5 (d). These $2\rightarrow 2$ scatterings (and various other channels) can be directly tested at present coupling-LHC and future coupling-FCC ; coupling-ILC colliders.

V.9.2 Right-Handed Neutrinos

Symmergence has testable implications also for right-handed neutrinos. Indeed, if the bound (59) is to be respected and if the active neutrinos are to have correct masses ($m_{\nu}\lesssim 1\ {\rm eV}$) then the right-handed neutrinos must have masses neutrino-portal ; vissani

[TABLE]

which can be probed at future experiments (based presumably on Higgs factories and accelerator neutrinos biz ) if not indirectly at the near-future SHiP experiment SHiP . For $m_{\nu}\lesssim 0.1\ {\rm eV}$ sunny , as revealed with recent cosmological data, the bound in (62) increases by an order of magnitude.

V.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 seagull ).

The dark NP can be probed directly cdm-current ; dm-LHC . The scalar field $H^{\prime}$ in (57), for instance, develops a vanishing VEV, $\langle H^{\prime}\rangle=0$, and becomes stable if it is real and has odd ${\mathbb{Z}}_{2}$ 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 $H^{\prime}H^{\prime}\rightarrow W^{+}W^{-},ZZ,\bar{t}t,\cdots$. The coannihilations into $W/Z$ are Goldstone-equivalent to $H^{\prime}H^{\prime}\rightarrow HH$ in Fig. 5 (b) so that $H^{\prime}$ scalars attain the observed relic density cdm-current ; scalar-dm if

[TABLE]

or, impliedly,

[TABLE]

after using the seesawic structure in (59) for $\lambda_{HH^{\prime}}$. This bound on $m_{H^{\prime}}$, which makes sense barely as it lies beneath $\Lambda_{W}$, implies that thermal dark matter cannot weigh above the electroweak scale. (Non-thermal 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 cdm-current ; dm-LHC and contrasting it with (64) or, more safely, with $\Lambda_{W}$.

VI 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 Fig. 6 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.

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 talk-qdis can shed light on strong gravity regime (like black holes). Third, $Q_{\mu\nu}$ in the MAG action (38) is suggestive of bimetric gravity bimetric , whose investigation may provide alternative views of symmergence. Fourth, $Q_{\mu\nu}$ in the MAG action (38) is reminiscent of the disformal curvature couplings disformal , 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.

This work is supported in part by TÜBİTAK grants 115F212 and 118F387. The author is grateful to Hemza Azri, Dieter van den Bleeken and Tekin Dereli for fruitful discussions on especially the emergent MAG. He thanks S. Vagnozzi and H. Terazawa for useful e-mail and mail exchanges.