Clockwork inflation with nonminimal coupling
Abstract
We suggest a clockwork mechanism for a Higgslike inflation with the nonminimal coupling term . The seemingly unnatural ratio of parameters, of the self quartic coupling of the inflaton, , and the nonminimal coupling, , is understood by exponential suppression of by the clockwork mechanism, instead of a large nonminimal coupling. The portal interaction between the inflaton and the Standard Model (SM) Higgs doublet is introduced as a source of reheating and the inflaton mass. Successful realization of inflation requires that the inflaton gets a mass around (sub) GeV scale, which would lead to observable consequences depending on reheating process and its lifetime.
pacs:
98.80.CqI introduction
Higgs inflation is a successful model of inflation based on the Standard Model (SM) of particle physics Bezrukov and Shaposhnikov (2008). A Jordan frame action for Higgs inflation include nonminimal coupling between the inflaton field, or ‘Higgs’ field, and the Ricci scalar ,
where is the reduced Planck mass. The potential satisfies the following condition at a large field limit, ,
(2) 
then the potential in Einstein frame has an asymptotically flat plateau and accommodates enough number of efoldings Park and Yamaguchi (2008). To fit the cosmological observation, however, the ratio is requested to be extremely small or finetuned as
(3) 
and calls for additional explanation Kim and Park (2011).
Conventionally, the small value is explained by a large nonminimal coupling with Bezrukov and Shaposhnikov (2008) but it causes unwanted low scale cutoff for a gravition interaction at around , which is well below the Planck scale Burgess et al. (2010); Bezrukov et al. (2011). It is also noticed that the small ratio could be obtained by a small quartic coupling at the inflationary scale due to the renormalization group (RG) running with , for the SM Higgs field De Simone et al. (2009); Bezrukov and Shaposhnikov (2009). However, it relies on the largish experimental uncertainty in the top quark mass measurement Patrignani et al. (2016). For updated analysis, see Hamada et al. (2014, 2015) and also Bezrukov and Shaposhnikov (2014); Bezrukov et al. (2015).
In this paper, we would suggest an alternative, simple explanation for the small ratio by clockwork mechanism. We don’t need any unnaturally small or large couplings at a defining scale but still realize a successful inflation. Especially, we only introduce a mild value of nonminimal coupling, thus the Planck scale cutoff is maintained.
The main idea of clockwork mechanism was first proposed in order to generate a transplankian period of the pseudo scalar inflaton potential Choi et al. (2014), and utilized in more general cases Choi and Im (2016); Higaki et al. (2016); Kaplan and Rattazzi (2016). It is also generalized to the fields with different spins, and recognized that the localization of the wave functions in the site space resembles that in the deconstruction of the extra dimensional model Giudice and McCullough (2017a), although the details are not exactly the same Craig et al. (2017); Giudice and McCullough (2017b). There are also interesting applications of the mechanism for various phenomenological problems such as dark matter, flavor, composite Higgs, axion of muon and seesaw mechanism Kehagias and Riotto (2017); Farina et al. (2017); Ahmed and Dillon (2016); Hambye et al. (2016); Coy et al. (2017); BenDayan (2017); Park and Shin (2018); Hong et al. (2018); Patel (2017); Kim and Mcdonald (2018). We also note that other possibilities of inflationary scenarios in the context of e.g. linear potential model, hybrid potential modal and Starobinsky’s model were considered in Kehagias and Riotto (2017); Im et al. (2018). Discussions on continuum limit and connection with linear dilaton models are in Giudice et al. (2017); Choi et al. (2017).
This paper is organized as follows: In the next section, Sec. II, we first review the basic idea of clockwork mechanism for our purpose then apply to the Higgslike inflation model in Sec. III. Finally we conclude in Sec. IV. For definiteness, in below, we consider “Higgslike inflation” driven by a SM singlet scalar and the inflation takes place by the interplay between a positive quartic coupling and the nonminimal coupling.
Ii clockwork mechanism
A clockwork (CW) mechanism can be described by the clockwork diagram in Fig. 1 where a set of heavy fields , and are linked by vertical and diagonal mass parameters, and , respectively. The mass parameters are considered to be spurions of symmetries under which the spurions are bicharged as of and of respectively. Under , and , . For a scalar potential, such a schematic picture can be discussed more transparently in the context of supersymmetry by constructing following clockwork superpotential:
(4) 
We assume that the mass parameters are essentially similar in values so that and for below. The ratio is . Then the term scalar potential is calculated as
(5)  
All s and combinations of are heavy with masses of . To figure out the zero mode of the theory, we can use the EulerLagrange equations of motion for : ^{1}^{1}1 It is noted that the charge assignment itself would not forbid terms composed of a power of in scalar potential and, in principle, they could additionally contribute to the CW decomposition of the mass eigenstates after supersymmetry is broken. To avoid this complication, we would regard the potential in Eq. 5 as our CW model. After integrating out heavy fields , whose dynamics is essentially irrelevant in our discussion, we get the effective potential of the form of . . The solutions are iteratively obtained as
(6) 
In order to find the zero mode scalar field, we see the kinetic terms after inserting the solution of the equations of motion:
(7)  
where is the canonically normalized CW zero mode scalar field as
(8) 
where the conveniently defined numerical factor, . Approximately, a gear field , , whose value is determined by equations of motion, is related with the zero mode as
(9) 
One can notice that the zero mode is close to (indeed when ) as we have depicted in Fig. 1 and is said to be ‘localized at site’. The other end point is for so that the effective coupling of the zero mode to the other sector of the model, which is described by an operator of dimension , , is highly suppressed as
(10) 
with a positive power, . The effective coupling is now read to be and its size is naturally small with . This explains how the CW mechanism would address hierarchy problems for seemingly unnatural small parameters. We would suggest a model of inflation based on the CW mechanism in the next section.
Iii Clockwork Higgslike inflation
iii.1 Clockwork mechanism for inflation
The action for ‘clockwork Higgslike inflation’ is introduced with nonminimal coupling terms and the potential terms with the CW potential :
(11)  
where the nonminimal coupling term and the CW potential are given as
(12) 
with positive and . Here we consider the CW gears as real scalar fields. The quartic potential, which is responsible for inflation, is introduced only for as
(13) 
which breaks the CW shift symmetry.^{2}^{2}2In fact, the nonminimal coupling term also breaks the CW shift symmetry, which might cause the setup radiatively unstable especially when is sizable. The loop corrections to the zero mode scalar potential mediated by heavy modes can be estimated as
Taking the masses of the CW heavy modes greater than the inflation scale (i.e., ), we can safely integrate out the heavy fields, and get the effective action for the CW zero mode. We will come back to the effect of heavy modes later. In the Einstein frame,
(15)  
The last line is obtained by inserting the clockwork solution for the gear fields (9), which yields and as
(16) 
and
(17) 
where is defined in (8). Because the nonminimal coupling term is universally contributed by all gear fields with , the effective coupling is not suppressed as , while the quartic coupling is dominated by the first gear field, . Therefore .
Having the effective theory for the zero mode fields, we get the effective coupling
(18) 
which explains the small value taking with .^{3}^{3}3Higgslike inflation with a very small quartic coupling was also discussed in different contexts such as Kaiser (1995); Komatsu and Futamase (1999); Tenkanen (2016); Alanne et al. (2017) and many others.
The field value of during inflation for the CMB scale, , is determined by the required efolding number as
(19) 
The initial value of is of the similar size of , , so we would carefully check if the heavy fields would spoil the inflation dynamics because of the mixing from the quartic potential. Let us discuss it with two field decomposition of the scalar fields: and as the eigenstates of the clockwork potential, where represents a heavy mode, which would potentially affect the inflationary dynamics closely. Then,
(20) 
In the scalar potential, , the dominant tadpole contribution for the heavy modes is coming from the quartic potential (), which gives the shift of the heavy field as
(21) 
for during inflation. The CW heavy modes are still heavier than the inflaton scale, so we can integrate them out and get the effective potential of the zero mode. For the large field value of (during inflation), the effective potential is corrected as
(22) 
For the initial value of , , the heavy field contributions for the inflation dynamics are suppressed as , compared to the leading contribution to the slow roll parameters. In short, our treatment of inflaton potential is robust and the corrections from the heavy gear fields are small.
iii.2 Higgs portal with Clockwork
It is an intriguing possibility that the standard model Higgs doublet field, , has a portal interaction with other sector of scalar field(s), . This is particularly interesting because the current measurement of top quark mass may imply metastable electroweak vacuum Isidori et al. (2001); Degrassi et al. (2012); Alekhin et al. (2012) (also see Chigusa et al. (2017); Andreassen et al. (2018) for the stateoftheart calculation of the decay rate) and the Higgs portal interactions would remedy the problem Kim et al. (2017). From the RG equation of , , and , we can obtain the positive values of and for all scales Falkowski et al. (2015). In our setup, we introduce a coupling only between the Higgs and the first gear field, Kim and McDonald (2017), in order not to disturb the inflation dynamics through the radiative corrections from the Higgs loop, but still yield the meaningful coupling between the Higgs and the inflaton field Lerner and McDonald (2009).
Now the scalar potential is extended for the SM Higgs and the singlet fields,
(23) 
and the nonminimal coupling is also extended as .
It is known that during the inflation, the Higgs gets a effective mass squared from the nonminimal coupling as , therefore the term makes the Higgs unstable for a positivie Espinosa et al. (2008). However from the Higgs portal coupling, there is the another source that makes the Higgs stable during inflation from the Higgs portal interactions Lebedev and Westphal (2013) as
(24) 
which is much bigger than the contribution from the Higgs nonminimal coupling. For the positive , the Higgs can be stable during inflation even if the the Higgs quartic is negative at high energy scale.
After the end of inflation, the inflaton will start to oscillate, and the Higgs particles could be produced through parametric resonance in the preheating stage Kofman et al. (1994, 1997). There are several studies about the bound on in order not to destabilize the Higgs field after inflation with the assumption that the inflaton field is oscillating with a quadratic potential around its minimum Herranen et al. (2015); Ema et al. (2016); Kohri and Matsui (2016); Enqvist et al. (2016); Ema et al. (2017). For the Higgslike inflation with a large nonminimal coupling constant (), such a quadratic approximation is valid for a long time until the oscillating amplitude of the canonically normalized inflaton field becomes of , so that most of preheating history is the same as that of the literatures. In our case the situation is a little bit different, because and there is no source of constant mass terms for except the Higgs VEV. It starts to roll dominantely with a quartic potential, , which means that we cannot simply take the quadratic approximation for the motion of . If thermalization arises much quicker than the case with a quadratic potential, the Higgs could be trapped at the origin due to its thermal potential. Therefore, it needs further studies for the Higgs stability with after inflation. On one hand, if is negative the Higgs is destabilized during inflation, and spoils the previous discussion. In this sense, we naturally take .
At present Universe, the clockwork gears are very heavy so that we cannot produce it. For the zero mode, the Higgs portal provides the mass term as
(25) 
The mass is and the couplings between the Higgs particles and the zero mode particles are
(26) 
The Higgs can decay as with the coupling , which is compatible with the current LHC bound on the Higgs invisible decay, Tanabashi et al. (2018). The numerical value of leads to a light mass for :
(27) 
Our inflationary scenario thus predicts the GeV scale light particles which are coupled to the Higgs weakly, whose experimental search would be extremely interesting and deserves further study Park and Shin .
Iv Conclusion
Higgs(like) inflation is an attractive model of inflation which explains the cosmological data with the collaborative helps from the nonminimal coupling and the inflaton potential. On the other hand, the required ratio of the selfcoupling constant () and the nonminimal coupling () is unnaturally small, , which thus requires additional explanation. Clockwork mechanism provides an interesting answer. By construction, the effective coupling of the inflaton potential is efficiently suppressed by the factor .
The constructed clockwork framework leads to interesting implications to the rest of cosmological history and observational consequences:

During inflation: Having and in our setup, the unitarity problem of conventional Higgs inflation Burgess et al. (2010); Bezrukov et al. (2011) would be relieved. The stochastic quantum fluctuations of the scalar fields coupled to the inflaton are quite suppressed because they are all heavy () and their effects on the inflaton potential is subleading. The SM Higgs also can be stable thanks to the large positive mass squared from the Higgsinflaton coupling.

Reheating: Just after the inflation, the dominant potential of the inflaton is quartic, . Because all other CW gear fields are heavy enough, we can safely focus on dynamics of the inflaton and the Higgs fields with quartic potentials and the initial conditions as , and . Since we cannot take a quadratic approximation for the potential of , the dynamics for the production of the Higgs and other SM particles are all involved. More detailed study about the (p)reheating in this kind of system (with a scale invariant scalar potential for the Higgsinflaton) would be quite interesting Park and Shin .

Late time dynamcis of the inflaton: If the clockwork mechanism for the inflation works, the inflaton mass at its minimum () is given by the Higgs portal interaction, and around (sub) GeV. In our minimal example, it has a symmetry, so the inflaton (i.e., the quanta of the inflaton field) could contribute to the measured amount of dark matter Aghanim et al. (2018). The relic density of the inflaton depends on the reheating procedure, and we could give further constraints on the the size of the coupling or the breaking scale of , which can predict the observations in experiments searching for ALPs. We remain it as a future work.
Acknowledgements.
This work is supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2016R1A2B2016112) and (NRF2018R1A4A1025334) (SCP), and by IBS under the project code, IBSR018D1 (CSS).References
 Bezrukov and Shaposhnikov (2008) F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B659, 703 (2008), arXiv:0710.3755 [hepth] .
 Park and Yamaguchi (2008) S. C. Park and S. Yamaguchi, JCAP 0808, 009 (2008), arXiv:0801.1722 [hepph] .
 Kim and Park (2011) Y. Kim and S. C. Park, Phys. Rev. D83, 066009 (2011), arXiv:1010.6021 [hepph] .
 Burgess et al. (2010) C. P. Burgess, H. M. Lee, and M. Trott, JHEP 07, 007 (2010), arXiv:1002.2730 [hepph] .
 Bezrukov et al. (2011) F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. Sibiryakov, JHEP 01, 016 (2011), arXiv:1008.5157 [hepph] .
 De Simone et al. (2009) A. De Simone, M. P. Hertzberg, and F. Wilczek, Phys. Lett. B678, 1 (2009), arXiv:0812.4946 [hepph] .
 Bezrukov and Shaposhnikov (2009) F. Bezrukov and M. Shaposhnikov, JHEP 07, 089 (2009), arXiv:0904.1537 [hepph] .
 Patrignani et al. (2016) C. Patrignani et al. (Particle Data Group), Chin. Phys. C40, 100001 (2016).
 Hamada et al. (2014) Y. Hamada, H. Kawai, K.y. Oda, and S. C. Park, Phys. Rev. Lett. 112, 241301 (2014), arXiv:1403.5043 [hepph] .
 Hamada et al. (2015) Y. Hamada, H. Kawai, K.y. Oda, and S. C. Park, Phys. Rev. D91, 053008 (2015), arXiv:1408.4864 [hepph] .
 Bezrukov and Shaposhnikov (2014) F. Bezrukov and M. Shaposhnikov, Phys. Lett. B734, 249 (2014), arXiv:1403.6078 [hepph] .
 Bezrukov et al. (2015) F. Bezrukov, J. Rubio, and M. Shaposhnikov, Phys. Rev. D92, 083512 (2015), arXiv:1412.3811 [hepph] .
 Choi et al. (2014) K. Choi, H. Kim, and S. Yun, Phys. Rev. D90, 023545 (2014), arXiv:1404.6209 [hepth] .
 Choi and Im (2016) K. Choi and S. H. Im, JHEP 01, 149 (2016), arXiv:1511.00132 [hepph] .
 Higaki et al. (2016) T. Higaki, K. S. Jeong, N. Kitajima, and F. Takahashi, Phys. Lett. B755, 13 (2016), arXiv:1512.05295 [hepph] .
 Kaplan and Rattazzi (2016) D. E. Kaplan and R. Rattazzi, Phys. Rev. D93, 085007 (2016), arXiv:1511.01827 [hepph] .
 Giudice and McCullough (2017a) G. F. Giudice and M. McCullough, JHEP 02, 036 (2017a), arXiv:1610.07962 [hepph] .
 Craig et al. (2017) N. Craig, I. Garcia Garcia, and D. Sutherland, (2017), arXiv:1704.07831 [hepph] .
 Giudice and McCullough (2017b) G. F. Giudice and M. McCullough, (2017b), arXiv:1705.10162 [hepph] .
 Kehagias and Riotto (2017) A. Kehagias and A. Riotto, Phys. Lett. B767, 73 (2017), arXiv:1611.03316 [hepph] .
 Farina et al. (2017) M. Farina, D. Pappadopulo, F. Rompineve, and A. Tesi, JHEP 01, 095 (2017), arXiv:1611.09855 [hepph] .
 Ahmed and Dillon (2016) A. Ahmed and B. M. Dillon, (2016), arXiv:1612.04011 [hepph] .
 Hambye et al. (2016) T. Hambye, D. Teresi, and M. H. G. Tytgat, (2016), arXiv:1612.06411 [hepph] .
 Coy et al. (2017) R. Coy, M. Frigerio, and M. Ibe, (2017), arXiv:1706.04529 [hepph] .
 BenDayan (2017) I. BenDayan, (2017), arXiv:1706.05308 [hepph] .
 Park and Shin (2018) S. C. Park and C. S. Shin, Phys. Lett. B776, 222 (2018), arXiv:1707.07364 [hepph] .
 Hong et al. (2018) D. K. Hong, D. H. Kim, and C. S. Shin, Phys. Rev. D97, 035014 (2018), arXiv:1706.09376 [hepph] .
 Patel (2017) K. M. Patel, Phys. Rev. D96, 115013 (2017), arXiv:1711.05393 [hepph] .
 Kim and Mcdonald (2018) J. Kim and J. Mcdonald, (2018), arXiv:1804.02661 [hepph] .
 Im et al. (2018) S. H. Im, H. P. Nilles, and A. Trautner, JHEP 03, 004 (2018), arXiv:1707.03830 [hepph] .
 Giudice et al. (2017) G. F. Giudice, Y. Kats, M. McCullough, R. Torre, and A. Urbano, (2017), arXiv:1711.08437 [hepph] .
 Choi et al. (2017) K. Choi, S. H. Im, and C. S. Shin, (2017), arXiv:1711.06228 [hepph] .
 Kaiser (1995) D. I. Kaiser, Phys. Rev. D52, 4295 (1995), arXiv:astroph/9408044 [astroph] .
 Komatsu and Futamase (1999) E. Komatsu and T. Futamase, Phys. Rev. D59, 064029 (1999), arXiv:astroph/9901127 [astroph] .
 Tenkanen (2016) T. Tenkanen, JHEP 09, 049 (2016), arXiv:1607.01379 [hepph] .
 Alanne et al. (2017) T. Alanne, F. Sannino, T. Tenkanen, and K. Tuominen, Phys. Rev. D95, 035004 (2017), arXiv:1611.04932 [hepph] .
 Isidori et al. (2001) G. Isidori, G. Ridolfi, and A. Strumia, Nucl. Phys. B609, 387 (2001), arXiv:hepph/0104016 [hepph] .
 Degrassi et al. (2012) G. Degrassi, S. Di Vita, J. EliasMiro, J. R. Espinosa, G. F. Giudice, G. Isidori, and A. Strumia, JHEP 08, 098 (2012), arXiv:1205.6497 [hepph] .
 Alekhin et al. (2012) S. Alekhin, A. Djouadi, and S. Moch, Phys. Lett. B716, 214 (2012), arXiv:1207.0980 [hepph] .
 Chigusa et al. (2017) S. Chigusa, T. Moroi, and Y. Shoji, Phys. Rev. Lett. 119, 211801 (2017), arXiv:1707.09301 [hepph] .
 Andreassen et al. (2018) A. Andreassen, W. Frost, and M. D. Schwartz, Phys. Rev. D97, 056006 (2018), arXiv:1707.08124 [hepph] .
 Kim et al. (2017) J. Kim, P. Ko, and W.I. Park, JCAP 1702, 003 (2017), arXiv:1405.1635 [hepph] .
 Falkowski et al. (2015) A. Falkowski, C. Gross, and O. Lebedev, JHEP 05, 057 (2015), arXiv:1502.01361 [hepph] .
 Kim and McDonald (2017) J. Kim and J. McDonald, (2017), arXiv:1709.04105 [hepph] .
 Lerner and McDonald (2009) R. N. Lerner and J. McDonald, Phys. Rev. D80, 123507 (2009), arXiv:0909.0520 [hepph] .
 Espinosa et al. (2008) J. R. Espinosa, G. F. Giudice, and A. Riotto, JCAP 0805, 002 (2008), arXiv:0710.2484 [hepph] .
 Lebedev and Westphal (2013) O. Lebedev and A. Westphal, Phys. Lett. B719, 415 (2013), arXiv:1210.6987 [hepph] .
 Kofman et al. (1994) L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994), arXiv:hepth/9405187 [hepth] .
 Kofman et al. (1997) L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. D56, 3258 (1997), arXiv:hepph/9704452 [hepph] .
 Herranen et al. (2015) M. Herranen, T. Markkanen, S. Nurmi, and A. Rajantie, Phys. Rev. Lett. 115, 241301 (2015), arXiv:1506.04065 [hepph] .
 Ema et al. (2016) Y. Ema, K. Mukaida, and K. Nakayama, JCAP 1610, 043 (2016), arXiv:1602.00483 [hepph] .
 Kohri and Matsui (2016) K. Kohri and H. Matsui, Phys. Rev. D94, 103509 (2016), arXiv:1602.02100 [hepph] .
 Enqvist et al. (2016) K. Enqvist, M. Karciauskas, O. Lebedev, S. Rusak, and M. Zatta, JCAP 1611, 025 (2016), arXiv:1608.08848 [hepph] .
 Ema et al. (2017) Y. Ema, M. Karciauskas, O. Lebedev, and M. Zatta, JCAP 1706, 054 (2017), arXiv:1703.04681 [hepph] .
 Tanabashi et al. (2018) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 030001 (2018).
 (56) S. C. Park and C. S. Shin, in prepration .
 Aghanim et al. (2018) N. Aghanim et al. (Planck), (2018), arXiv:1807.06209 [astroph.CO] .