YGHP17-02

Inflection-point inflation

in hyper-charge oriented U(1) model

Nobuchika Okada, Satomi Okada and Digesh Raut

Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL35487, USA

Graduate School of Science and Engineering, Yamagata University,

Yamagata 990-8560, Japan

Inflection-point inflation is an interesting possibility to realize a successful slow-roll inflation when inflation is driven by a single scalar field with its value during inflation below the Planck mass (). In order for a renormalization group (RG) improved effective potential to develop an inflection-point, the running quartic coupling must exhibit a minimum with an almost vanishing value in its RG evolution, namely and , where is the beta-function of the quartic coupling. In this paper, we consider the inflection-point inflation in the context of the minimal U(1) extended Standard Model (SM), a generalization of the minimal U(1) model, where the U(1) symmetry is realized as a linear combination of the SM U(1) and the U(1) gauge symmetries. We identify the U(1) Higgs field with the inflaton field. For a successful inflection-point inflation to be consistent with the current cosmological observations, the mass ratios among the U(1) gauge boson, the right-handed neutrinos and the U(1) Higgs boson are fixed. Focusing on the case that the U(1) gauge symmetry is mostly oriented towards the SM U(1) direction, we investigate a consistency between the inflationary predictions and the latest LHC Run-2 results on the search for a narrow resonance with the di-lepton final state. In addition, the inflection-point inflation provides a unique prediction for the running of the spectral index ( is the e-folding number), which can be tested in the near future.

## 1 Introduction

Inflationary universe is the standard paradigm in the modern cosmology [1, 2, 3, 4] which provides not only solutions to various problems in the Standard Big Bang Cosmology, such as the flatness and horizon problems, but also the primordial density fluctuations which seed the formation of large scale structure of the universe we see today. In a simple inflationary scenario known as slow-roll inflation, inflation is driven by a single scalar field (inflaton) when it slowly rolls down to its potential minimum. During the slow-roll, the energy density of the universe is dominated by the inflaton potential energy, which drives accelerated expansion of the universe, namely, cosmological inflation. After the end of inflation, the inflaton decays to Standard Model (SM) particles to reheat the universe to initiate the Standard Big Bang Cosmology.

The slow-roll inflation requires the inflaton potential to be sufficiently flat in the inflationary epoch. In a simple inflationary scenario such as chaotic inflation, a flat potential is realized by taking an initial inflaton value to be of the trans-Planckian scale. However, the field theoretical point of view, it may be more appealing to consider the small-field inflation (SFI) scenario, where the initial inflaton value is smaller than the Planck mass and possible higher-dimensional Planck suppressed operators are less important to the inflationary predictions. Hybrid inflation [5] is a well known example of the SFI [5], where a flat direction of the scalar potential is realized with multiple scalar fields. When one considers the SFI driven by a single scalar field, the so-called inflection-point inflation [6, 7, 8] is an interesting possibility. If the inflaton potential exhibits an inflection-point, the slow-roll inflation epoch can be realized with the initial inflaton value in the immediate vicinity of the inflection-point.

From a particle physics point of view, an inflation scenario seems more compelling if the inflaton field plays another important role in particle physics models, such as the Higgs inflation scenario [9, 10, 11] in which the SM Higgs field is identified with the inflaton field. When the SM is extended with some extra gauge groups or unified gauged groups, such models always include an extra Higgs field, in addition to the SM Higgs field, to spontaneously break the gauge symmetries down to the SM one. Similarly to the Higgs inflation scenario, it is interesting if we can identify the extra Higgs field with the inflaton. The extra Higgs field usually has Yukawa couplings with some fermions in addition to the gauge and the quartic couplings, just like the SM Higgs doublet. As will be discussed below, this gauge-Higgs-Yukawa system is essential to realize the inflection-point inflation with the identification of the Higgs field as the inflaton.

Let us consider a Renormalization-Group (RG) improved effective Higgs/inflaton potential [12]. During the inflation, we assume that inflaton value is much larger than its Vacuum Expectation Value (VEV) at the potential minimum, so that the inflaton potential is dominated by its quartic term of the form,

(1.1) |

where denotes the inflaton field, and is the running quartic coupling.
If the RG running of the inflaton quartic coupling first decreases towards high energy and then increases,
inflection-point is realized in the vicinity of the minimum point of the running quartic coupling,
where both the quartic coupling and its beta-function become vanishingly small
[7, 8].^{1}

(1.2) |

where and are the gauge and Yukawa couplings, respectively, and and are positive coefficients whose actual values are calculable once the particle content of the model is defined. Here, we have neglected terms proportional to ( term and the anomalous dimension term) because the SFI requires the quartic coupling , as will be shown later. Hence the quantum corrections to the effective Higgs potential are dominated by the gauge and Yukawa interactions. Realization of the inflection-point requires a vanishingly small beta-function at the initial inflaton value, namely . This condition leads to a relation between and , or in other words, the mass ratio of gauge boson to the fermion in the Higgs model is fixed. Since the Higgs quartic coupling at low energy is evaluated by solving the RG equation, the resultant Higgs mass also has a unique relation to the gauge and the fermion masses. Therefore, in the inflection-point inflation scenario with the Higgs field as the inflaton, there is a correlation between the very high energy physics of inflation and the low energy particle phenomenology.

Recently, two of the authors of this paper (N.O. and D.R.) have investigated the inflection-point inflation in the minimal gauged (baryon number minus lepton number) extension of the SM [15], where the Higgs field as the inflaton field [16]. In order to realize the successful inflection-point inflation, we have obtained the predictions for the mass spectrum for the gauge boson ( boson), the right-handed neutrinos, and the Higgs boson as a function of the initial inflaton value () and the inflaton/Higgs VEV (). Considering the reheating after inflation with the fixed particle mass spectrum, we have identified the allowed parameter regions to satisfy the Big Bang Nucleosynthesis constraint. We have found that the entire parameter region for GeV can be tested by the future collider experiments such as the High-Luminosity Large Hadron Collider (LHC) [17] and the SHiP [18] experiments.

In this paper, we generalize the minimal model to the so-called non-exotic U(1) extension of the SM [19]. The non-exotic U(1) model is the most general extension of the SM with an extra anomaly-free U(1) gauge symmetry, which is described as a linear combination of the SM U(1) and the U(1) gauge groups. The particle content of the model is the same as the one in the minimal model except for the generalization of the U(1) charge assignment for particles. The orientation of the U(1) gauge group is characterized by a U(1) charge of the SM Higgs doublet (). For example, is the U(1) limit, while the U(1) gauge group is oriented to the SM U(1) direction for . In this context, we investigate the inflection-point inflation with the identification of the U(1) Higgs field as the inflaton. As we will discuss in the following, the inflation analysis weakly depends on and hence our results are similar to those in Ref. [16]. However, there is a sharp contract in low energy phenomenologies, in particular, the U(1) gauge boson phenomenology at the LHC. An upper bound on the U(1) gauge coupling, , has been obtained from theoretical consistencies in Ref. [16]. The boson with such a small coupling can be explored in future collider experiments only for a small mass region such as GeV. For this case in the inflection-point inflation scenario, the reheating temperature is estimated to be GeV [16]. Such a low reheating temperature may not be desirable in terms of thermal dark matter physics and baryogenesis scenario. On the other hand, in the U(1) generalization, the coupling of the boson with the SM fermions is controlled by for with being the U(1) gauge coupling, and therefore the boson coupling becomes sizable for . In this paper, we will investigate the inflection-point inflation for this hyper-charge oriented extension of the SM, which opens up a possibility to explore the mass region of TeV at the LHC Run-2 while successfully realizing the inflection-point inflation.

The paper is organized as follows. In the next section, we give a brief review of the slow-roll inflation. In Sec. 3, we present the inflationary predictions for the scenario, where the inflaton potential exhibits an inflection-point-like behavior during the slow-roll. In Sec. 4, we consider the minimally gauged extension of the SM, where the Higgs field is identified with the inflaton field. To realize the inflection-point in a Higgs/inflaton potential, we consider the RG improved effective Higgs/inflaton potential. In Sec. 5, we consider the constraints on the model parameters from the Big Bang Nucleosynthesis and the current collider experiments. We also discuss the prospects of testing the scenario in the future collider experiments, such as the High-Luminosity LHC and SHiP experiments. Sec. 6 is devoted to conclusions.

## 2 Basics of Inflection-Point Inflation

The inflationary slow-roll parameters for the inflaton field () are given by

(2.1) |

where GeV is the reduced Planck mass, is the inflaton potential, and the prime denotes the derivative with respect to . The amplitude of the curvature perturbation is given by

(2.2) |

which should satisfy from the Planck 2015 results [20] with the pivot scale chosen at Mpc. The number of e-folds is defined as

(2.3) |

where is the inflaton value at a horizon exit corresponding to the scale , and is the inflaton value at the end of inflation, which is defined by . The value of depends logarithmically on the energy scale during inflation as well as on the reheating temperature, and it is typically taken to be .

The slow-roll approximation is valid as long as the conditions , , and hold. In this case, the inflationary predictions are given by

(2.4) |

where , and are the scalar spectral index, the tensor-to-scalar ratio and the running of the spectral index, respectively, which are evaluated at . The Planck 2015 results [20] set an upper bound on the tensor-to-scalar ratio as , while the best fit value for the spectral index () and the running of spectral index () are and , respectively, at CL.

In the SFI scenario, to realize the slow-roll inflation the inflaton potential must exhibit an inflection-point-like behavior,
where the potential is very flat.^{2}^{3}

(2.5) |

where is a constant and , and are the first, second and third derivatives of the inflaton potential evaluated at . When and are vanishingly small, the inflaton potential exhibits the inflection-point-like behavior. From Eqs. (2.1) and (2.5), the slow-roll parameters are then given by

(2.6) |

where we have used the approximation . Similarly, the power-spectrum is expressed as

(2.7) |

Using the constraint from the Planck 2015 results, we can express and as

(2.8) |

where we have used and as will be verified later. For the following analysis we set at the center value from the Planck 2015 results [20].

We define the inflaton value () at the end of inflation by . Using Eq. (2.3), the e-folding number () is given by

(2.9) |

where we have parametrized as with . The inflection-point-like behavior of the inflaton potential requires and , so that we can approximate . This approximation is justified later. As we will also show later, and hence the e-folding number is approximated as

(2.10) |

Using Eq. (2.8), is then given by

(2.11) |

From Eqs. (2.8) and (2.11), we find , and is a good approximation for .

Using Eqs. (2.4), (2.6), (2.8), and (2.11), we now express all inflationary predictions in terms , and . From Eqs. (2.6) and (2.8), the tensor-to-scalar ratio () is given by

(2.12) |

The running of the spectral index () is found to be

(2.13) |

Note that this value is a unique prediction of the inflection-point inflation independently of and . This prediction is consistent with the current experimental bound, [20]. It is expected that the future experiments can reduce the error to [21], and therefore the prediction of the inflection-point inflation scenario can be tested in the future.

## 3 The Inflection-point U(1) Higgs Inflation

SU(3) | SU(2) | U(1) | U(1) | |

3 | 2 | |||

3 | 1 | |||

3 | 1 | |||

1 | 2 | |||

1 | 1 | |||

1 | 2 | |||

1 | 1 | |||

1 | 1 |

The model we will investigate is the anomaly-free U(1) extension of the SM, which is based on the gauge group SU(3)SU(2)U(1)U(1). The particle content of the model is listed in Table 1. The covariant derivative relevant to U(1) U(1) is given by

(3.1) |

where () are U(1) (U(1)) charge of a particle, and the gauge coupling
is introduced in association with a kinetic mixing between the two U(1) gauge bosons.
Although we set zero at the symmetry breaking scale, it is generated
through the RG evolutions.
The particle content includes three generations of right-hand neutrinos
and a U(1) Higgs field , in addition to the SM particle content.
The U(1) gauge group is identified with a linear combination of
the SM U(1) and the U(1) gauge groups,
and hence the U(1) charges of fields are determined by
two real parameters, and .
Note that in the model the charge always appears as a product
with the U(1) gauge coupling and it is not an independent free parameter.
Hence, we fix throughout this paper.
In this way, we reproduce the minimal model with the conventional charge assignment
as the limit of .^{4}

The Yukawa sector of the SM is extended to have

(3.2) |

where the first and second terms are the neutrino Dirac Yukawa coupling and the Majorana Yukawa coupling, respectively. Without loss of generality, the Majorana Yukawa couplings are already diagonalized in our basis. Once the U(1) Higgs field develops non-zero VEV, the U(1) gauge symmetry is broken and the Majorana masses for the right-handed neutrinos are generated. Then, the seesaw mechanism [22] is automatically implemented in the model after the electroweak symmetry breaking.

The renormalizable scalar potential for the SM Higgs doublet () and the U(1) Higgs () fields is given by

(3.3) |

where all quartic couplings are chosen to be positive. At the potential minimum, the Higgs fields develop their VEVs as

(3.4) |

Associated with the U(1) symmetry breaking (as well as the electroweak symmetry breaking), the U(1) gauge boson ( boson), the right-handed Majorana neutrinos, and the U(1) Higgs boson () acquire their masses as

(3.5) |

where GeV is the SM Higgs VEV, and we have used the LEP constraint [23] . Because of the LEP constraint, the mass mixing of the boson with the SM boson is very small, and we neglect it in our analysis in this paper.

We identify the physical U(1) Higgs field with the inflaton, which is defined as in the unitary gauge, and consider the inflation trajectory and in the scalar potential in Eq. (3.3). In our analysis, we employ the RG improved effective potential along this inflation trajectory, which is dominated by the inflaton quartic term and given by

(3.6) |

where is the solution to the RG equation listed in Appendix. With the RG improved effective potential, we express the coefficients in the expansion of Eq. (2.5) as

(3.7) |

where the prime denotes , and is the beta function of the quartic coupling given by

(3.8) | |||||

Using and to realize the inflection-point like behavior of the inflaton effective potential, we obtain

(3.9) |

For the couplings being in the perturbative regime, we evaluate as a good approximation. Hence the last equation in Eq. (3.7) is simplified to be and Eq. (2.11) leads to

(3.10) |

where we have used . For , is found to be very small.

In evaluating , we simply assume is negligibly small. Although we have set , non-vanishing is generated through its RG evolution through its beta function consisting of two terms: one is proportional to and the other is proportional to . However, as we will see later, the inflection-point inflation requires , and hence stays negligibly small at any scales. As a result, Eq. (3.9) with Eq. (3.10) leads to , and we find

(3.11) |

where we have taken the degenerate Yukawa couplings for three right-handed neutrinos , for simplicity. Therefore, the mass ratio between the right-handed neutrinos and the gauge boson is fixed to realize a successful inflection-point inflation.

Now RG equations for , and at the 1-loop level are approximately give by

(3.12) |

Using the second equation in Eq. (3.9) and Eqs. (3.10)-(3.12), we express the U(1) gauge coupling at as

(3.13) |

Finally, from Eqs. (2.12) and (3.10), the tensor-to-scalar ratio () is given by

(3.14) |

which is very small, as expected for the SFI scenario.

At the end of inflation, is explicitly given by

(3.15) |

We evaluate from . If we assume that the first term dominates in the parenthesis of the final expression above, we find by using Eqs. (2.8), (2.11), and (3.10), which is inconsistent with . Therefore, the second term must dominate, and hence we obtain

(3.16) |

Before presenting our numerical results, let us check the consistency of our analysis. In the previous section we have approximated the inflaton potential by Eq. (2.5), neglecting the higher order terms. For consistency, we need to check if the contribution of higher order terms can actually be neglected in our model. Consider the following expansion of inflaton potential at ,

(3.17) |

where is the -th derivative of the potential evaluated at . As has been discussed in the previous section, and are fixed by the experimental values of the scalar power-spectrum () and the spectral index (), respectively. For the consistency of our previous analysis, we require that the terms and higher contribute negligibly in determination of compared to at the end of inflation. Using Eqs. (2.1) and (3.17), is expressed as

where we have used . This leads to constraint

(3.19) |

where . To proceed further we need to evaluate Eq. (3.19) explicitly for the minimal U(1) model. As has been shown previously in this section, all the higher order derivatives of the potential can be approximately given by , where is a constant. For example, and . We find that the most severe bound for both cases is from term. Using Eqs. (3.16) we obtain an upper bound on .

Let us now present the numerical results of the inflection-point U(1) Higgs inflation scenario. For the rest of the paper, we employ the e-folding number . We set and choose to be , and hence all our results presented in the rest of this section are controlled by only three parameters, , and .

In Fig. 1, we show the U(1) gauge coupling (left) and the inflaton quartic coupling (right) at as a function of (see Eqs. (3.10) and (3.11)). Here, we have fixed , which is motivated by the LHC phenomenology to be discussed in the next section.

In Fig. 2, we plot the running quartic coupling (left) and the RG improved effective inflaton/Higgs potential (right). Here we have fixed and , which corresponds to , , and . In the left panel, the dashed line indicates . In the right panel, we see the inflection-point-like behavior of the inflaton potential around , marked with a dashed-dotted vertical line.

Here let us look at the inflationary predictions of our scenario. The prediction for the tensor-to-scalar ratio () is given by Eq. (3.14). For the upper bound on , the resultant tensor-to-scalar ratio is given by , which is too small to test in the future experiments. However, as discussed in the previous section, the inflection-point inflation predicts the running of the spectral index to be , independently of . This predicted value is within the reach of future precision measurements [21].

We now consider the particle mass spectrum of the model at low energies. As we have discussed, the condition of the almost vanishing leads to the relation of . The low energy mass spectrum of the boson and the right-handed neutrinos are obtained by extrapolating the couplings to low energies. From Eq. (3.11) with the upper bound on , we can see and hence the RG running effects of the gauge and Yukawa couplings are negligible and remains almost the same at low energies. On the other hand, the RG evolution of the inflaton quartic coupling significantly changes its value at low energies (as shown in the left panel of Fig. 2), since its beta function is controlled by the gauge and Yukawa couplings. Let us approximately solve the RG equations in Eq. (3.12). Since , , the solutions to their RG equations are approximately given by

(3.20) |

where and are their beta-functions evaluated at . Hence, the beta-function of the quartic coupling is approximately described as

(3.21) | |||||

where we have used in Eq. (3.9). Then we obtain the approximate solution to the RG equation as

(3.22) | |||||

where we have used Eq. (3.10) and . Using , the mass ratio of the U(1) Higgs/inflaton to the boson is given by

(3.23) |

## 4 LHC Run-2 Constraints

If kinematically allowed, the boson in the minimal U(1) model can be produced at the LHC. The ATLAS and the CMS collaborations have been searching for a narrow resonance with di-lepton final states at the LHC Run-2 and set the upper limits of a boson production cross section [24, 25]. In the analysis by the ATLAS and the CMS collaborations, the so-called sequential SM () model [26] has been considered as a reference model, where the boson has the same couplings with the SM fermions as the SM boson. In this section, we interpret the current LHC constraints on the sequential boson into the U(1) boson to identify an allowed parameter region. Then, we examine a consistency of the inflection-point inflation scenario with the LHC Run-2 constraints.

We first calculate the cross section for the process . The differential cross section with respect to the invariant mass of the final state di-lepton is given by

(4.1) |

where is the parton distribution function for a parton (quark) “”, and TeV is the center-of-mass energy of the LHC Run-2. In our numerical analysis, we employ CTEQ6L [27] for the parton distribution functions with the factorization scale . Here, the cross section for the colliding partons is given by

(4.2) |

where , the function are

(4.3) |

for being the up-type () and down-type () quarks, respectively, and the total decay width of boson is given by

(4.4) |

with . By integrating the differential cross section over a range of set by the ATLAS and the CMS analysis, respectively, we obtain the cross section to be compared with the upper bounds obtained by the ATLAS and the CMS collaborations.

In interpreting the 2016 results by the ATLAS and the CMS collaborations into the U(1) boson case, we follow the strategy in [28] (see also [29] for the minimal model). In the paper, the cross section for the process is calculated and the resultant cross sections are scaled by a -factor to match with the theoretical predictions presented in the ATLAS and the CMS papers. With the -factor determined in this way, the cross section for the process is calculated to identify an allowed region for the model parameters of , and . See Ref. [28] for the details of the strategy and the -factors. Our analysis in this section is exactly the same as that in this reference.

For TeV, we show in Fig. 3 the upper bounds on as a function of from the CMS results on search for a narrow resonance from the combined di-electron and di-muon channels [25]. The lower and upper dashed lines correspond to and , respectively. The upper bounds from the ATLAS results [24] are found to be very similar to but slightly weaker than those from the CMS results, and we have shown only the CMS results. As we can see from the cross section formula, the dashed lines approach each other for a large . The diagonal solid lines depict Eq. (3.10) for (left) and (right), along which the successful inflection-point inflation is achieved. For the diagonal solid lines with a fixed , the results for and are indistinguishable. From this figure, we find an upper bound on and , respectively, for (left) and (right). Note that even though the successful inflection-point inflation requires the U(1) gauge coupling to be very small, this scenario can still be tested at the LHC when , in other words, the U(1) gauge symmetry is oriented towards the SM hyper-charge direction.

## 5 Constraints from the Big Bang Nucleosynthesis

Let us now consider a reheating scenario after inflation to connect our inflation scenario with the Standard Big Bang Cosmology. This occurs via inflaton decay into the SM particles during the inflaton oscillates around its potential minimum. We estimate the reheating temperature () as

(5.1) |

where is the inflaton decay width into the SM particles. For the successful Big Bang Nucleosynthesis (BBN), we impose a model-independent lower bound on the reheating temperature as MeV.

In the Higgs potential of Eq. (3.3), a mass matrix between the inflaton () and the SM Higgs boson () is generated after the U(1) symmetry and the electroweak symmetry breaking:

(5.2) |

where , and GeV. We diagonalize the mass matrix by