Weyl Symmetry Inspired Inflation and Dark Matter

Yong Tang and Yue-Liang Wu Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan International Centre for Theoretical Physics Asia-Pacific, Beijing, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Abstract Motivated by the original Weyl scaling gauge symmetry, we present a theoretical framework to explain cosmic inflation and dark matter simultaneously. This symmetry has been resurrected in recent attempts to formulate the gauge theory of gravity. We show the inspired inflation can be well consistent with observations and probed in future. Furthermore, Weyl gauge boson can be a dark matter candidate and decay due to its novel couplings to standard model Higgs field, which can lead to testable signatures in dark matter indirect searches, Higgs invisible decay and production rate at colliders.

Introduction: The accumulated compelling evidence for dark matter (DM) has been challenging the standard model (SM) of fundamental physics for decades. The supporting observations, such as cosmic mircowave background (CMB), large-scale structure, rotation curves, scope from cosmological to galactic scales [1,2]. For the intrinsic nature of DM, however, we are still lacking sufficient information since the robust evidence is only able to suggest that DM must have gravitational interaction. Nevertheless, explanations of DM would require extensions of SM, either in the sector of particle physics or gravity.
Gauge symmetry has played a guiding principle for constructing fundamental laws of nature since last century when Weyl first proposed [3] the scaling symmetry and tried to unify the electromagnetic interaction with Einstein's general relativity. The original scale factor has to be modified as a phase to account for the gauge U (1) theory for electromagnetic interaction [4]. U (1) is later generalized to non-abelian theory by Yang and Mills [5], which describes the interactions of all known fundamental particles in SM by incorporating the Higgs mechanism [6][7][8]. Variants of Weyl/scaling symmetry, however, still stimulate explorations of theoretical and phenomenological studies, see Refs.  for examples in cosmology and particle physics. Recently, Refs. [36,37] has shown the original Weyl symmetry can play a crucial role in formulating the gauge theory of gravity.
In this paper we propose that the original Weyl symmetry can provide a framework to explain the cosmic inflation [38][39][40][41] and DM simultaneously [42]. The starting inflationary Lagrangian can be Weyl invariant and responsible for the generation of Planck scale. Theoretical predictions of observables in this scenario are consistent with currect experiments and testable in future. The gauge boson associated with local Weyl symmetry may be identified as a DM candidate if its mass and coupling are in the right region. Thanks to the novel connection with SM Higgs field, Higgs's invisible decay and single-or double-production at colliders can be used to probe this scenario. Additionally, the intrinsic decay of Weyl boson into SM particles would induce signatures at DM indirect searches in cosmic-ray, gamma-ray and neutrino spectra.
Framework: To illustrate the main physical points, we start with the following general bosonic Lagrangian with two scalars ϕ and φ, where R is the Ricci scalar, W µ the Weyl field, F µν = ∂ µ W ν −∂ ν W µ and the covariant derivative D µ = ∂ µ − g W W µ for scalars. More complete Lagrangian can be found in Refs. [36,37] where gravity is formulated as a gauge theory of the fundamental field χ a µ with its connection to metric, χ a µ χ b ν η ab = g µν . Throughout our paper, we use the sign convention, η ab = (1, −1, −1, −1). The potential V has a general form of 4 i=0 c i φ i ϕ 4−i . The parameters in the front of scalar kinetic terms can have two possible signs, ζ i = ±1. The negative sign can appear in theories with compactified Kaluza-Klein extra-dimension and is not necessary associated with theoretical issues, as long as the total energy of the system is positive [43]. We shall take ζ i = 1 unless otherwise stated. Note that W µ does not couple to fermions directly, which can be understood from the Lagrangian for a massless fermion Since there is no i in the covariant derivative for W µ , W µ -dependent terms will cancel in the parentheses.
The above general Lagrangian has a local symmetry and is invariant under the following local Weyl or scaling transformation where the scale factor λ (x) acts as a gauge parameter that may be taken in the domain λ > 0 without losing generality. When we fix φ 2 = v 2 , Einstein-Hilbert term R is recovered.
In such an Einstein basis, Weyl boson W µ gets a mass M W = g W v due to the kinetic term of φ. Afterwards, the theory describes Einstein's gravity with a non-minimally coupled scalar ϕ and the massive gauge boson W µ . We shall show that ϕ can be responsible for cosmic inflation and W µ can be a dark matter candidate if its mass light and coupling constant are in the proper range.
Inflation: To demonstrate how the above framework can provide a mechanism for cosmic inflation, we shall illustrate with a concrete example with β = 0 and V = c(ϕ 2 − φ 2 ) 2 .
Detailed analysis with general β and V will be presented elsewhere. After fixing for the scalar and gravity part we have Now the potential has a minimum at ϕ = v, resulting in an effective Planck scale M 2 p = 2αv 2 . To make the formalism more familiar, we can change to Einstein frame [44] by redefining the metricḡ µν = 2αϕ 2 /M 2 p · g µν and scalar field σ = M p / √ 2α · ln(ϕ/M p ), and obtain Now the formalism describes the Einstein's gravity with a minimally coupled scalar σ. Using ϕ = M p exp √ 2ασ/M p , the standard inflationary slow-roll parameters [45] can be calculated and expressed concisely as which are related with the observables, spectral index n s = 1 − 6 + 2η and tensor-to-scalar ratio r = 16 . To solve the flatness and horizon problem, the Universe has to inflate at least by e N (the typical e-folding number N 50 ∼ 60) before inflation ends at ϕ 2 = To give the correct amplitude of scalar power spectrum, parameter c should be around 5α × 10 −11 , a typical value in large-field inflation models. Thus, only α is an effectively free parameter. This minimal model gives an one-parameter inflationary scenario, in which the Hubble scale H is at order of ∼ 10 13 GeV.
We numerically solve the inflationary dynamics and present in Fig. 1 [49]. The important connection between Weyl boson and the SM of particle physics is through the Higgs field.
To be Weyl-symmetric, the SM Higgs doublet H can minimally couple to Weyl boson as where the interaction part is given by Here W µ = W µ − ∂ µ h · g W v H /m 2 W is the redefined Weyl boson in order to diagonalize the kinetic terms due to the mixing between W µ and h. Other parameters are defined as Here M W is the mass of Weyl boson before electroweak symmetry breaking, which can be different from its value at high energy scale due to the renormalization-group running or some other Higgs-like field's contributions. Note that the redefinition of Weyl boson induces an additional kinetic term for Higgs field, resulting in C H for normalization. The physical mass forh in SM will be identified as mh = 125GeV.
From the above Lagrangian, besides the novel interactions with derivative couplings, we observe immediately that there is no symmetry to guarantee the stability of Weyl boson, though the interaction W µh ∂ µh alone can not directly cause the decay W µ → 2h due to the property for on-shell Weyl boson, ∂ µ W µ = 0. However, if this interaction appears for an off-shell W µ in a decay process, channels like W µ →h + W * µ → 3h at the tree-level can happen thanks tohW µ W µ interaction. For m W mh, W µ → 2h + W * µ → 4h would be the dominant process, which has a decay width as Note that in general m W and g W can be independent. In the minimal model presented above we have the simple relation, m 2 However, in extended models with more Higgs-like fields that can have negative ζ for the kinetic term, m W and g W would be connected more complicatedly. For phenomenological studies, we shall treat m W and g W as independent from now on.
Gravitational decay is also possible, but usually subdominant unless the mass is very large. Evidently, to be a possible DM candidate, W µ would need to have a very tiny g W to be long-lived enough, namely, at least longer than the age of our Universe, t U 4 × 10 17 s.
As a matter of fact, indirect DM searches in cosmic-ray, gamma-ray and neutrino spectra have already given stronger limits, Γ W 10 −26 s −1 . For lighter W µ , the dominant decay channels are those with three fermion-antifermion pairs, such as W µ → 3b + 3b (bottom quark) or W µ → 3τ + + 3τ − (tau lepton). We show the approximate constraints in Fig. 2  and XENON1T [50]. However, the relevant high-mass region has already been excluded by indirect searches, see the red dotted curve in the right-top of Fig. 2 by XENON1T.  Due to the normalization of Higgs field, all the couplings in operators like h n O would get rescaled by a factor of 1/C n H . The decay width would be enhanced by 1/C 2 H , compared with SM value Γ SM 4.3MeV. The current upper bound from LHC measurement is Γ SM < 13MeV at 95% confidence level [52], which can give C 2 The rate for single Higgs production at the LHC will also be enhanced by 1/C 2 H , which would modify the signal strength in the observed decay channel. The combined analysis of ATLAS and CMS [53] gives the favored signal strength µ = 1.09 ± 0.11, constraining where x W = 4m 2 W /m 2 h . Since Weyl boson has a lifetime at least comparable to the age of Universe, the above channel would show itself as an invisible decay for Higgs at colliders.
Current data from ATLAS [54] and CMS [55] give a limit on the invisible branching ratio,

Br inv
26%, which is shown as the purple long-dashed line for m W < 62.5GeV in Fig. 2 and provides the strongest bound in the low mass region.
The white area in Fig. 2 is not constrained by any terrestrial experiment searches yet, therefore is still allowed, except that it is subjected to the relic abundance requirement that depends on the details of cosmological history. We consider the production of Weyl boson in the early universe from two sources, vacuum fluctuation and thermal production. Vacuum fluctuation [56,57] gives the relic abundance Ω W , where Ω DM 0.25. We show three vertical dashed lines which from left to right correspond to Ω W = (10 −2 , 10 −1 , 1)Ω DM for H = 10 13 GeV. Thermal annihilation from Higgs particles contributes to where T h is the highest temperature of Higgs particles in the thermal bath after inflation.
Note that T h is not necessarily equal to the reheating temperature, because Higgs particle may not be in thermal equilibrium with the reheated sector just after inflation. As a demonstration, the blue and black bands give the correct DM abundance for T h = 10 3 GeV and 10 15 GeV, respectively.
Finally, the possible kinetic mixing between Weyl boson and photon γ through κF µν γ µν would open up an additional portal to probe this scenario. Search strategies for general dark photons [58] due to the mixing also apply to the case of Weyl boson.