Abstract
It is shown that in a spacetime of curvature is a natural ultraviolet completion of in the flat-spacetime Standard Model with Higgs field , scale , and loop factors and . This curvature completion rests on the fact that -mass gauge theory in flat spacetime turns, on the cut view , into a massless gauge theory in curved spacetime. It provides a symmetry reason for curved spacetime, wherein gravity and matter are both low-energy effective phenomena. Gravity arises correctly if new physics exists with at least 63 more bosons than fermions, with no need to interact with the and with dark matter as a natural harbinger. It can source various cosmological, astrophysical, and collider phenomena depending on its spectrum and couplings to the .
, spectrally completed with the discovery of its Higgs boson, is experimentally affirmed to describe physics at the Fermi scale, . Its validity in the direction comes to an end at a physical scale , at which it is environed by new physics. Integrating out all trans-Fermi high-frequency fluctuations, low-energy fields develop the effective actionin the flat spacetime of metric such that the action encodes the tree-level interactions augmented by logarithmic contributions. entrains a -sized vacuum energy along with a -sized Higgs boson mass, and adds -sized gauge boson masses so that hypercharge, isospin, and color are all explicitly broken at the [1].
In the above, is a fundamental scale of nature just as . It is a physical scale rather than a formal momentum cutoff introduced to regulate the loop integrals. In this sense, is a physical effective theory whose no part, including and , can be modified at will. If were a formal cutoff it would be possible to eradicate both and simply by switching from cutoff regularization to, say, dimensional regularization.
Quarks and leptons, whose masses vary with only logarithmically, stay put at the Fermi scale. In view of the Higgs mechanism, therefore, and stand as subversive outliers. They render the unnatural [2–5]. Its natural extensions like supersymmetry, extra dimensions, and technicolor have not even glimpsed at collider searches [6]. It must therefore be naturalized by a different mechanism. That mechanism, if exists, must be able to (i) eradicate to restore gauge invariance, (ii) ameliorate to stabilize the Higgs sector, and (iii) elucidate to reveal the physics behind it. They are approached below by restoring gauge invariance (not by using affine geometry as in [7], though the results agree).
In search for a mechanism to restore gauge symmetries at the , it proves effectual to introduce and its by-parts arrangement for gauge as two gauge kinetic functionals which convert to one another through by-parts integration. This by-parts equivalence of theirs, , can be put in use to construct as an equivalent of . Corresponding effective action is equivalent to . This dynamical equivalence is, however, quite vulnerable to spacetime geometry, indeed viewing as a specific value assigned to a curved metric and inductively unassigning it as , as is, can be carried into curved spacetime to find that simply because . The reason is that , unlike , must involve not just but also , where is the Ricci curvature of . This nonequivalence is actually a blessing in disguise for taming the gauge sector. Indeed, if in is construed as a specific value assigned to , then its inductive unassignment takes into with which and thus so that all gauge symmetries are restored at the . The unassignment of () renders the Poincare (gauge) breaking by futile. The classical curvature restores gauge invariance to naturalize the gauge sector.
This naturalization mechanism extends over entirety of the if is construed as a curved spacetime effective action evaluated at the curvature and then at the metric . Namely, is nothing but the scale cut view of : such that and both embrace , albeit with different physical meanings. Physically, must involve (a)no extra couplings not found in as no quantum fluctuations are left to induce any new coupling,(b)no extra forces except gravity as spacetime can attain required elasticity if nears gravitational scale [8],
so that it comes to take the familiar Einstein-Hilbert form though already at one loop (see [7] for a variant study). is negative for bosonic and fermionic degrees of freedom in the . This means that gravity can be induced properly only if the is extended by new fields belonging to some “new physics” () sector lying at a scale . Its effective action is of the form if it is secluded from the . If it has no scalars [2–5], so that, through (12), and add up to give as a completely -natural effective field theory governing and dynamics in curved spacetime [7] such that is Higgs-curvature direct coupling, and is Newton’s constant (see also [9, 10]). Not only these curvature couplings but also those in and are all given by couplings in the flat spacetime effective actions and .
In (18), vacuum energy, already naturalized in the with corrections , needs to be naturalized also in the as it is still far bigger than the observational value of . This is the cosmological constant problem [9, 10].
In (18), the , secluded from the , is a natural home to noninteracting dark matter. Its spectrum, modelable as gauge theory or with two fermions or anything with , is secluded enough to form a dark matter observable via only its weight [11].
In (18), there exist no higher-curvature terms because, after inducing the Einstein-Hilbert term, no suitable couplings are left in the . If needed, they can be added by hand but that reduces predictive power of . Indeed, adding, for instance, the ghost-free quadratic action does not unnaturalize (18), yet this ambiguates it by the undetermined coefficients of Weyl and Gauss-Bonnet invariants.
In (18), logarithmic contributions coming through lead to multiplicative renormalizations of and . These logarithms can actually be construed as loop integrals in a -dimensional momentum space of total volume so that the formal equivalence with small but finite enables the two logarithmic parts of (18) to be formulated in the dimensional regularization scheme with associated renormalization methods [1].
In (18), is decoupled from . Their coupling can cause new effects. One possibility is Higgs mass shifts like , which beget logarithmic unnaturalness as in supersymmetry unless [12]. The interacting dark matter under search in direct detection experiments is another possibility. The recent LHC diphoton signal [13] can well be a precursor of such .
In (18), matter and gravity meet in a physically consistent framework in that they both are sub-Fermi effective interactions. This is a crucial property because putting quantized matter into curved geometry is an aporia as quantum gravity is distant and classical gravity is inconsistent [14, 15]. They do not enhance naturalness [16, 17].
In summary, restoration of gauge invariance has led to naturalization of the via gravity. This is not surprising because, at least in macroscopic world, it is gravity that dictates what is natural and what is unnatural. It is in this sense that unnatural in flat spacetime turns into natural in curved spacetime. The mechanism makes, as were also with the previous work [7], three salient predictions:(I)Gravity arises as a large-distance effective force consistently coupled to the low-energy quantum effective action. It is the requisite physics that completes the and renders the physical.(II)New physics exists as a highly crowded sector ( for ), which does not have to couple to the matter. This secluded sector, which can in principle lie at any scale sufficiently below , can source dark matter as a noninteracting nonbaryonic matter which can be sensed via only its weight. In this setup, is all -natural.(III)New physics may interact with the partly or wholly. In this case, scalar fields in the , even its vector-like fermions, can cause the Higgs boson mass to shift by depending on coupling strengths. They destabilize the unless . The LHC diphoton signal, if real, may be stemming from such sector.
Additional Points
Future research will reveal more about the formalism.
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work is supported in part by the TÜBTAK Grant 115F212. It is dedicated to honorable memory of Professor Namık K. Pak who was a dedicated teacher, an indefatigable researcher, and a great friend.