# Higgs Chaotic Inflation in Standard Model and NMSSM

###### Abstract

We construct a chaotic inflation model in which the Higgs fields play the role of the inflaton in the standard model as well as in the singlet extension of the supersymmetric standard model. The key idea is to allow a non-canonical kinetic term for the Higgs field. The model is a realization of the recently proposed running kinetic inflation, in which the coefficient of the kinetic term grows as the inflaton field. The inflaton potential depends on the structure of the Higgs kinetic term. In the simplest cases, the inflaton potential is proportional to and in the standard model and NMSSM, respectively. It is also possible to have a flatter inflaton potential.

###### pacs:

98.80.Cq^{†}

^{†}preprint: IPMU-10-0145

^{†}

^{†}preprint: KEK-TH-1394

The inflation is strongly motivated by the recent WMAP results Komatsu:2010fb . It is a non-trivial task to construct a successful inflation model, partly because the properties of the inflaton are poorly known. The inflaton may be only weakly coupled to the standard model (SM) sector. In this case, since the number of cosmological observables are limited, it might be difficult to pin down the inflation model even with the Planck data :2006uk . Alternatively, the inflaton may be a part of the SM or its extensions Murayama:1992ua ; Kasuya:2003iv ; Allahverdi:2006iq , in which case we may be able to study the properties of the inflaton at collider experiments such as the LHC. The latter idea has recently attracted much attention since the proposal of the SM Higgs inflation Bezrukov:2007ep . In the model of Ref. Bezrukov:2007ep , the flat potential is achieved by introducing a non-minimal coupling to the gravity Salopek:1988qh (see also Refs. Einhorn:2009bh ; Lee:2010hj ; Ferrara:2010yw ; Ferrara:2010in ; Kallosh:2010ug ; BenDayan:2010yz for the inflation with non-minimal coupling to gravity in supergravity). In this letter we pursue another approach to the inflation in the SM and its extensions: we construct a Higgs chaotic inflation model by allowing a non-canonical kinetic term. As we shall see below, the model is a realization of the running kinetic inflation Takahashi:2010ky ; Nakayama:2010kt .

Recently, a new class of inflation models was proposed by one of the authors (FT) Takahashi:2010ky , in which the kinetic term grows as the inflaton field, making the effective potential flat Dimopoulos:2003iy ; Izawa:2007qa . This model naturally fits with a high-scale inflation model such as chaotic inflation Linde:1983gd , in which the inflaton moves over a Planck scale or even larger within the last or e-foldings Lyth:1996im . This is because the precise form of the kinetic term may well change after the inflaton travels such a long distance. In some cases, the change could be so rapid, that it significantly affects the inflaton dynamics. We named such model as running kinetic inflation. Interestingly, the power of the inflaton potential generically changes in this class of inflation models. The phenomenological aspects of the running kinetic inflation was studied in detail in Ref. Nakayama:2010kt .

First let us consider the Higgs inflation in the SM. In order to identify the Higgs with the inflaton, there are two issues. First, if the potential were valid up to large field values, the chaotic inflation with a quartic potential would occur. However, the quartic chaotic inflation is strongly disfavored by observation Komatsu:2010fb . Secondly, in order to satisfy the WMAP normalization, a quartic coupling must be as small as which would result in an unacceptably light Higgs mass. These issues can be avoided if the potential becomes flatter at large field values. There are two ways. One is to introduce a non-minimal coupling to gravity Salopek:1988qh and the other is to make use of the running kinetic term Takahashi:2010ky . We will focus on the latter possibility in this letter.

The key idea is to add the following interaction,

(1) |

where is the Higgs doublet, is a numerical coefficient, and denotes a gauge covariant derivative ^{1}^{1}1
In Ref. Germani:2010gm a different kind of non-canonical kinetic term was considered.
.
Here and in what follows we adopt the Planck unit, .
In the unitary gauge, we can write down the Lagrangian for the Higgs :

(2) |

For small , the effect of non-canonical kinetic term is irrelevant, while, for large , the kinetic term grows, that is why the name “running kinetic inflation.” The canonically normalized field in this regime is given by

(3) |

and the effective potential becomes

(4) |

Thus, the quadratic chaotic inflation occurs. We emphasize here that the potential changes from to because of the running kinetic term. A large kinetic term makes the effective potential flatter, and it is straightforward to obtain a flatter potential by increasing the power of in the coefficient of the kinetic term. The WMAP normalization gives , and so, if is sufficiently large, , the quartic coupling can be of . Such a large coupling is analogous to the non-minimal coupling to gravity in Ref. Bezrukov:2007ep , and it may be obtained by tuning or some UV dynamics Dimopoulos:2003iy ; Izawa:2007qa . Note that the inflation takes place for sub-Planckian values of , while the value of exceeds the Planck scale.

Next we apply the same idea to the Higgs inflation in supergravity. As is well known, it is difficult to implement the chaotic inflation in supergravity because of the exponential pre-factor in the scalar potential, where is the Kähler potential. In order to construct a chaotic inflation model in supergravity, there must be flat directions in the field space along which the Kähler potential does not grow. The flat direction can be realized by either symmetry or tuning. In the latter case we can assume that a certain interaction in the Kähler potential is enhanced, which results in an approximate flat direction Izawa:2007qa ; NT . Instead, we here adopt the symmetry to ensure the flatness, following the construction in Refs. Takahashi:2010ky ; Nakayama:2010kt . In both cases, the inflation dynamics is essentially the same. In the pioneering paper by Kawasaki, Yamaguchi and Yanagida Kawasaki:2000yn , a shift symmetry on the inflaton, ( is a real transformation parameter), was introduced so that the Kähler potential depends only on , not on . In Ref. Takahashi:2010ky , the shift symmetry is generalized to , based on the idea that the form of the kinetic term may change after the inflaton traverses more than the Planck scale.

Let us now construct a Higgs inflation model in the singlet extension of MSSM. We introduce a chiral superfield, , to represent the gauge invariant :

(5) |

where and are the up- and down-type Higgs superfields. In the scalar components, we can express

(6) |

We require that the Kähler potential for the Higgs fields is invariant under the following transformation;

(7) |

where is a real transformation parameter. This corresponds to the above-mentioned shift symmetry with . We will discuss the case of another value of later. The symmetry (7) means that the composite field transforms under a Nambu-Goldstone like shift symmetry.

The Kähler potential satisfying the shift symmetry (7) must be a function of :

(8) |

where is a numerical coefficient of and we normalize ; is real (imaginary) for even(odd) . Note that the and terms are absent. Instead, the kinetic term for arises from the terms of , whose contribution is proportional to . One can show that remains constant along the inflationary trajectory by noting that appears explicitly in the Kähler potential and therefore acquires a large mass during inflation Takahashi:2010ky . So, we drop terms with because it does not change the form of the kinetic term.

We can impose a discrete symmetry that is consistent with the shift symmetry (7). Requiring , an invariant under the shift symmetry, be also invariant under the discrete symmetry up to a phase factor, we find that must be either or . If , the would flip its sign, and so, with any odd should vanish. If , there is no such constraint, since itself is invariant under .

In order to have a successful inflation, we introduce explicit symmetry breaking terms in both the Kähler and super-potentials:

(9) | |||||

(10) |

where the - and -terms are the symmetry breaking terms, and we assume . There could be other symmetry breaking terms, but we assume that they are soft in a sense that the shift symmetry remains a good symmetry at least up to the inflaton field value of . Here is a singlet superfield. can be stabilized at the origin during and after inflation if we add in the Kähler potential with . The presence of not only simplifies the inflaton potential, but also helps to avoid a situation that the inflaton potential becomes negative due to in the scalar potential Kawasaki:2000yn . Here and in what follows we impose symmetry under which and flip the sign, in order to suppress dangerous couplings such as . The charge assignment of and are shown in Table 1.

It may be instructive to write down explicitly the Kähler and super-potentials in terms of and :

(11) | |||||

(12) |

with . We note that the form of the superpotential (12) is equivalent to the part of the interactions in NMSSM.

The scalar potential in supergravity is given by

(13) |

Since we have imposed the symmetry, along the inflationary trajectory. The relevant Lagrangian for the inflation is then given by

(14) | |||||

(15) |

Since we explicitly break the shift symmetry (7) by the term, there appears a non-vanishing exponential prefactor. However, for , the exponential prefactor is close to unity, and therefore can be dropped. Note that the inflaton does slow-roll even if the exponential pre-factor gives a main contribution to the tilt of the potential, as long as is much smaller than unity. Except for the exponential factor, one can see that and in Eq. (2) are related to and as and .

U(1) | ||||
---|---|---|---|---|

For , the Lagrangian can be approximated by

(16) | |||||

(17) |

where we have defined . The inflationary trajectory is given by , and so, the imaginary component of vanishes. Let us rewrite the inflaton as

(18) |

where is a real scalar. The Lagrangian for the canonically normalized inflaton is therefore given by

(19) |

for . Thus, thanks to the shift symmetry, the inflaton can take a value greater than the Planck scale, and the chaotic inflation takes place.

The inflaton field durning inflation is related to the e-folding number as

(20) |

and the inflation ends at . The power spectrum of the density perturbation is given by

(21) |

where we have used in the second equality the WMAP result Komatsu:2010fb . The coupling is therefore determined as

(22) |

In order for the inflation driven by (19) to last for e-foldings, the following inequality must be met;

(23) |

The spectral index and the tensor-to-scalar ratio are respectively given by

(24) | |||||

(25) |

For , they vary as and .

The inflation ends when the slow-roll condition is violated at , and the inflaton starts to oscillate about the origin. The dynamics of the inflaton is then described by a complex scalar field rather than the real scalar . As the amplitude of the inflaton decreases, the term becomes more important. For , the Lagrangian becomes

(26) |

where we have defined a canonically normalized field at low scales, . Note that the power of the scalar potential changes from to after inflation. When the amplitude becomes of the order of the weak scale, the description by the D-flat direction is no longer valid, and we should consider the dynamics of and separately as usual. In the end, they should develop vacuum expectation values (VEVs), leading to the electroweak phase transition.

In order to have a successful electroweak phase transition, the -term with a right magnitude must be generated. We may add small explicit breaking of the discrete symmetry to produce a tadpole of , which makes to develop a VEV, generating the -term. Alternatively, we may identify the field as the singlet field in the NMSSM, in which the superpotential takes the following form,

(27) |

where is a coupling constant.
Note that the presence of in the superpotential does not destabilize the
inflation dynamics. In order to have a chaotic inflation in NMSSM, we need to consider a different shift symmetry.
Instead of the symmetry, let us assign symmetry
on and the Higgs field.^{2}^{2}2If the is exact, domain walls will be produced. To avoid the domain-wall problem
we need to introduce a small breaking. See Table 2. The simplest shift symmetry consistent
with the symmetry is given by^{3}^{3}3
We can also consider a shift symmetry with .
The potential would be proportional to where .

(28) |

Along the same line, we can realize a chaotic inflation with the Higgs fields as the inflaton. The Kähler potential is given by

(29) |

where is in general non-zero. The potential is given by

(30) |

where is the canonically normalized field. The spectral index and the tensor-to-scalar ratio, respectively, are and for . The WMAP normalization gives .

Note that, while is determined by the WMAP normalization, the low-energy effective coupling between the singlet and the Higgs is given by

(31) |

So, if , the effective coupling can be . Similarly, the SM Yukawa interactions break the shift symmetry:

(32) |

where are the Yukawa couplings, and we suppressed the generations. The physical Yukawa couplings at the low energy are similarly scaled as

(33) |

Therefore the top Yukawa coupling can be close to , if . The coefficients of the breaking terms are suppressed by a factor of wherever either or appears: , and . This structure might be related with the UV theory behind the shift symmetry.

We emphasize here that the presence of is essential for constructing a chaotic inflation model in supergravity. It is stabilized at the origin and its dynamics is not relevant for the inflation, and so, may be considered as a spectator field. Interestingly, however, in the Higgs chaotic inflation model, the same plays an important role in low-energy phenomenology. For instance, in the NMSSM, the fermionic superpartner of can be dark matter.

So far we have considered the possibility that the Higgs fields play the role of the inflation. It is straightforward to apply the above idea to the other flat directions in MSSM. In this case we need to adopt a flat direction which is lifted by the superpotential of the form Gherghetta:1995dv . (For instance the direction could be lifted by , and is identified with ). In particular, if the flat direction has a non-zero baryon/lepton number, the baryon/lepton numbers would be explicitly violated by the interactions in the Kähler potential, and so, the baryogenesis a la Affleck-Dine Affleck:1984fy ; Dine:1995kz is possible Takahashi:2010ky ; Nakayama:2010kt . There would be no baryonic isocurvature perturbation Kasuya:2008xp because the degree of the freedom orthogonal to the inflaton is heavy during inflation.

Let us briefly mention the reheating in the Higgs inflation model. The SM particles are naturally created by the inflaton decay in the Higgs inflation, but the process could be complicated by the non-perturbative decay. In the SM Higgs inflation and the NMSSM Higgs inflation with the symmetry, the inflaton passes near the origin after inflation, and so, the preheating is likely to occur Bezrukov:2008ut ; GarciaBellido:2008ab . On the other hand, in the last example, the inflaton acquires a non-zero angular momentum due to the non-zero . Then the preheating may not be efficient. If the non-perturbative decay is efficient in the former case, the resultant reheating temperature would be very high, and too many gravitinos may be produced from thermal scattering, while the non-thermal gravitino production Kawasaki:2006gs ; Endo:2007ih is generically suppressed in the Higgs inflation.

If the Higgs chaotic inflation is realized in nature, we will be able to study the properties of the inflaton, namely the Higgs fields, at the collider experiments as well as the CMB observation.

###### Acknowledgements.

This work was supported by the Grant-in-Aid for Scientific Research on Innovative Areas (No. 21111006) [KN and FT] and Scientific Research (A) (No. 22244030) [FT], and JSPS Grant-in-Aid for Young Scientists (B) (No. 21740160) [FT]. This work was supported by World Premier International Center Initiative (WPI Program), MEXT, Japan.## References

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