Low energy implications of cosmological data in $U(1)_X$ Higgs inflation

A scalar field having the Coleman-Weinberg type effective potential arises in various contexts of particle physics and serves as a useful framework for discussing cosmic inflation. According to recent studies based on the Markov chain Monte Carlo analysis, the coefficients of such an effective potential are severely constrained by the cosmological data. We investigate the impact of this observation on the physics beyond the Standard Model, focusing on an inflationary model based on the $U(1)_X$-extended Standard Model as a well-motivated example. We examine the parameter region that is not excluded by the Large Hadron Collider (LHC) Run-2 at 139 fb${}^{-1}$ integrated luminosity, and show that the model parameters can be further constrained by the High-Luminosity LHC experiments in the near future. We also comment on the possible reheating mechanism and the dark matter candidates of this scenario.


I. INTRODUCTION
Cosmic inflation was originally proposed in the context of grand unified theories (GUTs) [1][2][3] and gravitational effective theories [4,5]. As the simple models based on the GUT scenario turned out to be unsuccessful, and as the quantum generation mechanism of the primordial fluctuations seeding the large scale structure of the Universe was found to be enormously successful, inflationary cosmology has become a major paradigm of modern physics in its own right, not necessarily associated with its particle physics origin. This, of course, does not mean search of the particle physics responsible for inflation is unimportant. With the rapidly growing data from cosmological precision measurements and collider experiments, the time may now be ripe for discussing a coherent picture of the early Universe based on particle physics.
The Coleman-Weinberg mechanism [6] is a natural realization of spontaneous symmetry breaking as a consequence of radiative quantum corrections. While the mechanism is not directly relevant within the electroweak symmetry breaking of the Standard Model, it can be, in principle, responsible for symmetry breaking in theories beyond the Standard Model [7,8]. It is natural to suppose that the Coleman-Weinberg mechanism may have played some role in the early Universe, as cosmic inflation may well be realized by some Higgs-like scalar field that existed in the early Universe. Indeed, the inflationary model based on the Coleman-Weinberg effective potential, where λ, b are dimensionless constants, µ is the renormalization scale and φ is a scalar field (inflaton), has * kawai@skku.edu † okadan@ua.edu ‡ satomi.okada@ua.edu been widely studied, although its original model is known to be disfavored by the recent cosmic microwave background (CMB) observations by more than 2-σ (e.g. [9]). At high energies a scalar field can nonminimally couple to the Ricci scalar. Taking the nonminimal coupling into account, it is known that the inflationary model with the Coleman-Weinberg effective potential (1) assumed in the Jordan frame gives the spectrum of the CMB consistent with the present observations within 1-σ [10]. Moreover, it has been pointed out recently [11] that the Markov chain Monte Carlo (MCMC) analysis constrains the coefficient b to be positive definite by more than 3-σ. Although the paper [11] focuses on the type I and type II seesaw mechanisms and concludes that the type II mechanism is favored, the implication of this observation is profound and extends beyond the seesaw model; if this claim is substantiated by further observational data, many interesting cosmological scenarios would be severely constrained or ruled out. In this paper we discuss the implication of nonzero b in a simple particle physics model beyond the Standard Model, namely the U (1) X -extended Standard Model. It is shown, by a straightforward renormalization group (RG) analysis, that the breaking scale of the U (1) X symmetry is obtained from the cosmological input, and the model parameters are constrained by the current and future Large Hadron Collider (LHC) experiments. term is generated by quantum effects. The U (1) gauge couplings may be organized into the form of a triangular matrix [13] and the covariant derivative with the gauge group connections is written where G α µ , W a µ , B µ , Z µ are the gauge fields, g 3 , g 2 , g 1 , g X are the gauge couplings, and T α , T a , Y , X are the corresponding generators and charges of the SU (3) c , SU (2) L , U (1) Y , U (1) X gauge groups. The coupling g 1 arises as mixing of the two U (1)'s.
Besides the Standard Model particles, three singlet fermions (right-handed neutrinos) N i R and one complex scalar Φ are introduced; the former are necessary for the theory to be free from gauge and gravitational anomalies, and the latter is responsible for breaking the U (1) X symmetry and generating Majorana masses of the singlet fermions. The particle contents of the U (1) X -extended Standard Model are summarized in TABLE I. Requiring the absence of gauge and gravitational anomalies, the U (1) X charges are determined up to two real parameters x H and x Φ . As the overall scale of x Φ may be absorbed into redefinition of g X , we shall set x Φ = 1.
The remaining x H is a real constant parameter of the model. The special case x H = 0 corresponds to the U (1) B−L -extended Standard Model, whereas x H = −4/5 and x H = −2 correspond to the standard and flipped SU (5) × U (1) X that may be embedded in the SO(10) GUT. In this paper we leave x H as a free parameter.
The Yukawa terms are where H ≡ iσ 2 H * , and we work in the basis of diagonal Majorana Yukawa matrix y M . This model is a wellmotivated extension of the Standard Model, as the small nonzero neutrino masses are generated by the seesaw mechanism, the mechanism of baryogenesis via leptogenesis is readily incorporated, and Φ is a natural candidate of the inflaton field, as we now explain.
In the Higgs sector we assume the classically conformal potential where λ Φ and λ H are the dimensionless self couplings of the singlet Higgs Φ and the doublet Higgs H. For simplicity the mixed coupling λ will be assumed to be small (| λ| 1) below so that the Φ and H sectors may be studied separately (this can be done consistently, see e.g. [7]). We will be interested in symmetry breaking by the Coleman-Weinberg mechanism. Let us decompose Φ into the real and imaginary parts, and identify φ as the inflaton. Including the 1-loop corrections, the potential for φ is where µ is the renormalization scale and β Φ is the 1loop beta function given in (A9). This is the Coleman-Weinberg effective potential (1), with λ = λ Φ and b = β Φ /λ Φ . The action of the scalar sector pertinent to the inflationary dynamics, including the nonminimal coupling to the background curvature, is Here, ξ is the dimensionless coupling and M P is the mass scale which we choose to be the reduced Planck mass M P = 2.24 × 10 18 GeV. This action in the Jordan frame can be brought to the Einstein frame 1 by Weyl transformation. Then the cosmological observables can be evaluated by using the standard slow roll approximation. The  effect of positive nonminimal coupling ξ is to flatten the potential in the Einstein frame, reducing the tensor amplitude relative to the scalar amplitude, thereby shifting the model prediction comfortably within the 1-σ parameter range of the recent observational constraints [17,18]. The MCMC analysis of [11] shows that, for the fixed nonminimal coupling ξ = 100, the Planck 2018 [17,18] and BICEP2-Keck Array experiments [19,20] constrain the effective potential at the renormalization scale µ = M P as In gauge-extended Standard Models, these constraints impose conditions on the beta function β Φ at the energy scale of inflation, which then govern the physics at lower energy scales via the RG flow. An immediate consequence of identifying φ as the U (1) X Higgs field is that the conditions (8) and (9) control the breaking scale of the U (1) X gauge symmetry. Neglecting the running of λ and b for simplicity, we find from the stationarity condition dV eff /dφ = 0 that the potential (1) takes a minimum at The conditions (8), (9) then indicate that the U (1) X symmetry breaking by the Coleman-Weinberg mechanism takes place in the range The right-handed neutrino masses are given by Thus, interestingly, this rough estimate shows that the cosmological data fitting leads to the natural seesaw scale expected from the seesaw mechanism for not too small Majorana Yukawa couplings y i M . In the next section we present a refined analysis, including the RG flow of parameters.  (9) on the coefficient b of the effective potential obtained from the cosmological MCMC analysis [11]. The red curve is the threshold above which the U (1)X symmetry breaking by the Coleman-Weinberg mechanism does not occur.

III. IMPLICATIONS ON COLLIDER PHYSICS
In order to simplify the analysis we neglect the mixed Higgs coupling λ. This can be done safely as the corresponding beta function (A10) stays negligibly small along the RG flow. Up to 1-loop, only the U (1) X Higgs self coupling λ = λ Φ , the U (1) X gauge coupling g X and the Majorana Yukawa coupling y i M concern the dynamics of inflation. We also assume for simplicity that the three diagonal components of the Majorana Yukawa coupling are degenerate, The boundary conditions are set by the cosmological data fitting at the scale of inflation µ = M P . The value of λ is fixed by (8), and the condition (9) constrains β Φ , which relates the values of y M and g X at µ = M P . We shall regard the value of y M at µ = M P as an input parameter and g X to be determined within the likelihood of 68% confidence level. The condition (8) suggests λ g 2 X , y 2 M . When 96g 2 X 3y 4 M , from the beta function (12) one can see that the constraints (9) give conditions on the gauge coupling at µ = M P , with the center value g X = 0.0509. The RG-improved effective potential is and a potential minimum is characterized by the stationarity condition dV (φ)/dφ = 0, which gives If the U (1) X symmetry is broken by the Coleman-Weinberg mechanism, λ becomes negative at the potential minimum. The breaking scale is the vacuum expectation value v φ of the U (1) X Higgs field φ at the minimum, which is found by solving the condition (17) for φ. The inflaton (U (1) X Higgs boson) mass m 2 φ = d 2 V (φ)/dφ 2 | φ=v φ and the Z boson mass M Z = 2v φ g X at the breaking scale are found by solving the RG equation from µ = M P down to µ = v φ .

A. RG flow and the Coleman-Weinberg mechanism
The solutions to the RG equations (12), (13), (14), with the boundary conditions set at µ = M P , are shown in FIG. 1 and FIG. 2. At the boundary λ is fixed by (8) and the gauge coupling g X is given by the 1-σ constraints (9) through the beta function (12). The Yukawa coupling at µ = M P and the x H parameter are treated as input parameters. x H = 10. For small y M (M P ), λ is seen to become negative at lower energy, indicating that the symmetry breaking takes place by the Coleman-Weinberg mechanism.
For larger values of y M (M P ), λ stays positive all the way and thus there is no symmetry breaking by the Coleman-Weinberg mechanism. This implies that instead of the classically conformal Higgs potential (4) one needs to consider the renormalizable potential implementing the Higgs mechanism, where v H = 246 GeV is the Standard Model Higgs vacuum expectation value and v Φ is a symmetry breaking scale of U (1) X ; the potential involves extra parameters and the scenario becomes less predictive. We thus focus on the more interesting case in which the U (1) X symmetry breaking is realized by the Coleman-Weinberg mechanism. FIG. 2 shows the behavior of g X (the left panel) and λ (the right panel), as y M (M P ) is fixed at 0.1 and x H is varied as 0, 10, and 20. One can see a similar tendency as in FIG. 1, that is, when x H is small the symmetry breaking takes place at higher energy scale, and as x H is increased the breaking scale v φ becomes smaller, and then for x H larger than some critical value the Coleman-Weinberg symmetry breaking cease to occur. Both FIG. 1 and FIG. 2 show that the U (1) X breaking scale v φ can be as low as TeV. It is thus interesting to discuss possible signals that may be found in colliders.

B. Z boson mass
When the U (1) X symmetry is broken, the Z boson becomes massive and its mass is given by M Z = 2g X v φ . FIG. 3 shows the Z boson mass on the x H -y M (M P ) plane, computed at the U (1) X breaking scale φ = v φ . The left (right) panel uses the 1-σ lower (upper) bound of λb/4 from the cosmological data (9) as the boundary condition of the renormalization group equations at φ = M P . As one can see from FIG. 1 and 2, the Coleman-Weinberg mechanism ceases to operate for large values of y M (M P ) and for large values of x H . The threshold is indicated by the red curves, above which the U (1) X symmetry cannot be broken by the Coleman-Weinberg mechanism. The plots clearly indicate that the Z boson can be very light when y M (M P ) is small and x H is moderately large.

C. Constraints by LHC
The strongest constraints on the model parameters come from the exotic gauge boson search in hadron colliders, by the s-channel process p + p → Z → e + e − /µ + µ − . Since g X is already constrained to be small by the LHC experiments, for computing the cross section σ(pp → Z → e + e − /µ + µ − ) σ(pp → Z )BR(Z → e + e − /µ + µ − ) one may use the narrow-width approximation, Here, f q and f q are the parton distribution functions for a quark and an anti-quark,ŝ ≡ xys is the invariant squared mass of the colliding quarks. For the LHC Run-2, √ s = 13 TeV. In our LHC analysis we follow [21]. In

D. Other aspects of the model
Let us conclude this section by commenting on the mechanism of reheating, the scenario of baryogenesis, and dark matter candidates that are most plausible in this model.

Reheating temperature
The mass of the Z gauge boson is M Z = 2g X v φ and that of the right-handed neutrinos is The square of the inflaton mass is, using (16), As the 96g 4 X term dominates the beta function β Φ in Eq. (12), the inflaton mass is evaluated as M φ 3/2g X M Z /π M Z . The inflaton cannot decay into the heavier Z boson. Being a Standard Model gauge singlet, it cannot decay into the Standard Model gauge bosons either. Thus the dominant decay channel of the inflaton into the Standard Model particles is through the mixing with the Standard Model Higgs field [24].
The mixing arises from the last term of the classically conformal potential (4) which we thus far have ignored. Using the unitary gauge H = (0, h/ √ 2) the potential is written in which the running of λ Φ (φ) is indicated explicitly as it is crucial for the Coleman-Weinberg symmetry breaking. The quantum corrections for λ H ∼ O(0.1) and for λ are negligible. From the stationarity conditions ∂V /∂h = 0 and ∂V /∂φ = 0 we have As a benchmark, let us consider M Z = 6 TeV and g X = 0.03, giving v φ = 10 5 GeV. The Standard Model Higgs vacuum expectation value is The last term of (24) is thus negligibly small, justifying our treatment of the mixed coupling term in our analysis above. Redefining the h and φ fields around the expectation values as h → h + v h and φ → φ + v φ , the mass term of the potential is written with the mass matrix elements The mass matrix is diagonalized by rotating the fields, with the angle given by tan 2θ = 2 m 2 /(m 2 φ − m 2 h ). Let us, from FIG. 4, choose 0.022 ≤ g X ≤ 0.031 (30) as the benchmark values of the low energy U (1) X gauge coupling compatible with the 1-σ cosmological data. For M Z = 6 GeV, the inflaton mass is given by (28) as The Standard Model Higgs mass is m h = 125 GeV. Thus for our parameter choice the mixing angle is very small, and consequently the field h is almost h, and φ is almost φ.
The 'inflaton' φ dominantly decay to f f = bb, cc and τ τ with the coupling y f sin θ/ √ 2, where y f are the corresponding Standard Model Yukawa couplings. The total decay width of φ is where m b = 4.2 GeV, m c = 1.3 GeV and m τ = 1.8 GeV.
The reheating temperature may be evaluated 3 by comparing the decay width with the Hubble expansion rate, Using the number of relativistic degrees of freedom from the Standard Model g * = 106.75, the reheating temperature for our benchmark parameter range (30), (31), (32) is found to be 83.3 TeV ≤ T rh ≤ 174 TeV.
The observed baryon asymmetry of the Universe is naturally explained by the scenario of baryogenesis via leptogenesis [28]. In our model, the right-handed neutrinos are thermally created during reheating, and the baryon number can be generated by the sphaleron processes during the electroweak phase transition. Although the temperature (35) is lower than ∼ 10 9 GeV that is necessary in generic scenarios [29], it is sufficiently higher than the lower bound for successful leptogenesis 4 when the resonant enhancement [31][32][33] takes place. Since nearly degenerate right-handed neutrino mass is necessary for the resonance, our assumption of the degenerate Majorana Yukawa couplings is not only for the sake of simplicity but is indeed necessary for the successful baryogenesis scenario.

Dark matter candidate
Although it is known that one of the right-handed neutrinos in the U (1) X -extended Standard Model can play the role of dark matter [34], for the benchmark parameter values used above, the abundance of the dark matter right-handed neutrino created in thermal processes is too small to account for the dark matter abundance of the present Universe (for a review, see [35]). We thus have to look for a candidate of dark matter outside the particle contents of TABLE I.
The simplest candidate is a Dirac fermion, which is singlet under the Standard Model gauge group and has a generic U (1) X charge Q such that |Q| = 1, 3 (TABLE II). Let us call it ζ = (ζ L , ζ R ). Unless Q = ±1 (in this case ζ has the same interaction as the right-handed neutrinos) or Q = ±3 (gauge invariant Φ * N c R ζ R , ΦN R ζ L interactions can exist), ζ is protected by the U (1) X symmetry and thus is stable to be dark matter. Adding the Dirac fermion does not spoil the anomaly cancellation of the U (1) X -extended Standard Model. The ζ field communicates with the Standard Model particles only through the U (1) X gauge interactions.
The ζ dark matter can be produced thermally or nonthermally. Let us consider the thermal production scenario first. In this case, ζ is assumed to be in thermal equilibrium with the Standard Model particles through the U (1) X interactions ζζ ↔ Z ↔ f SM f SM , and then freezes out [36]. When the mass of ζ is half of M Z , the production Z → ζζ is enhanced by the resonance process for which the cross section is In the freeze-out scenario, it is well known that the present dark matter abundance Ω DM h 2 = 0.12 is obtained 5 when the cross section is σv rel ∼ 1 pb. For our benchmark parameter value M Z = 6 TeV, the dark matter mass is m ζ = 3 TeV and one can see from (36) that the dark matter abundance Ω DM h 2 = 0.12 is reproduced if Q 10.
When |Q| O(1), the thermal dark matter scenario is not applicable as the U (1) X interaction is not strong enough to keep the Dirac fermion ζ in thermal equilibrium. In this case, it is appropriate to consider nonthermal production of dark matter by the freeze-in mechanism. After reheating, the dark matter ζ is produced from the Standard Model thermal plasma by the out-ofequilibrium processes, 5 Only in this subsection h denotes the normalized Hubble parameter.
By numerically solving the Boltzmann equations for these processes, it is known [37] that the dark matter abundance Ω DM h 2 = 0.12 is reproduced if (Qg X ) 2 (x H g X ) 2 + 0.82 1.2 (Qg X ) 4 8.2 × 10 −24 . (38) When |Q| 1, the first term of (38) (the first line of (37)) dominates. For our benchmark values x H = O (10) and g X (a few) × 10 −2 , one can see that |Q| ∼ 10 −9 gives the present dark matter abundance Ω DM h 2 = 0.12. The computation of (38) assumes M 2 Z m 2 ζ . This condition is easily satisfied by, for example, choosing m ζ = 20 TeV for M Z = 6 TeV. Also, the reheating temperature of T rh ∼ 100 TeV is sufficient for producing the thermal plasma.

IV. FINAL REMARKS
We have discussed the particle physics implications of the cosmological data-fitting on the Coleman-Weinberg type effective potential, based on the recent MCMC analysis made in [11]. We focused on the particular inflationary scenario based on the U (1) X -extended Standard Model, and analyzed the RG flow from the inflationary (Planck) scale. By the RG analysis we identified the parameter region for which the U (1) X symmetry breaking due to the Coleman-Weinberg mechanism is operative. Moreover, we found the parameter region that is already excluded by the LHC Run-2, and the region that is covered by the future HL-LHC.
Data fitting by the cosmological MCMC combined with the RG analysis opens up a new direction of research in particle cosmology. Although we focused on just one example of inflationary model here, a wide range of inflationary models based on gauge-extended Standard Models may be analyzed in a similar manner. This approach is, in a sense, amalgamation of top-down (Planck scale physics) and bottom-up (collier physics) approaches. The link between the two is provided by the RG flow.