Fast-Rolling Relaxion

We discuss new mechanisms to stop the relaxion field during inflation. They can be realized in a generic model, including the original model but in a quite different parameter region. We consider a fast-rolling relaxion field, which can go over the bumps created by QCD-like dynamics. Then, in one of the mechanisms, we stop it with a parametric resonance of the Higgs field. The mechanisms are free from a super-Planckian field excursion or a gigantic number of e-folds of inflation. The relaxion has a mass around the weak scale and mixes with the Higgs boson, which enhances the testability of our mechanisms.


I. INTRODUCTION
The succession of null results of new physics search at the Large Hadron Collider (LHC) might imply that the energy scale of new physics is higher than the TeV scale. If this is the case, the Higgs mass parameter seems to be severely fine-tuned since it is highly sensitive to the energy scale of new physics. The gap between the two scales is worth considering since a mechanism might reside behind it.
Among various possibilities, it is interesting that the electroweak (EW) scale is determined dynamically with a new field, called the relaxion [1]. It scans the Higgs mass squared during inflation, rolling down a linear potential. After the relaxion passes the critical point, where the Higgs mass squared becomes zero, the Higgs field develops a vacuum expectation value (VEV). It triggers a back-reaction and the relaxion stops around the critical point. Since the Higgs mass squared is determined independently of its initial value, we can naturally explain the gap between the two scales. The back-reaction can be implemented by using an axion-like coupling of the relaxion to strong dynamics, which generates bumps that depend on the value of the Higgs field. Although the idea is in an early stage of development, there are already a number of papers discussing its phenomenology or its variants .
In the original model, the relaxion is assumed to slow-roll so that it stops immediately after the bumps match the slope of the linear potential. Since the potential barrier is very low at that point, the first selected vacuum quickly decays or hops into the lower vacuum through quantum tunneling or the Hubble fluctuations. The relaxion continues to decay, or hop for the first bumps, to lower vacua until it finds a vacuum whose lifetime is much longer than the age of the universe. This process, however, requires a gigantic number of e-folds since the Hubble volume during inflation is very small. To be consistent with the age of the universe, the bubble nucleation rate in the final vacuum, γ, should satisfy with H 0 being the Hubble constant of the current universe. For such a vacuum to decay during the inflation era, we need a gigantic number of e-folds, N e , as with H being the Hubble constant during inflation. It easily exceeds 10 150 even with a low scale inflation, which causes problems in the inflation sector. If one considers slow-roll inflation during the relaxation, for example, the number of e-folds is expected to be lower than 10 24 to avoid fine-tuning problems [6,[39][40][41][42]. Alternatively, one may assume eternal inflation during the relaxation. It, however, induces multiverse, which obscures the virtue of the relaxion mechanism. 1 A simple solution to the above problem is to violate one of the slow-roll conditions and allow the relaxion to fly over the bumps with its kinetic energy. 2 In compensation, we need another mechanism to stop the relaxion at a desired position. Several alternative mechanisms have already been proposed. For example, [5,[7][8][9][10][11] use particle production and [4] uses potential instability to stop the relaxion. Such mechanisms can also avoid a super-Planckian excursion of the relaxion, which is often troublesome when one UV-completes the model. In addition, it also reduces the number of e-folds during the slow-rolling phase.
In this paper, we discuss two stopping mechanisms, which can be realized without extending the original model. One is to use Higgs coherent oscillation caused by a parametric resonance, and the other is rather close to the original mechanism. In this paper, we focus on the former one and provide a detailed analysis. For the latter, we only sketch the idea and give an example in the Appendix C. In both mechanisms, the relaxion mass lies around the weak scale and mixes with the Higgs boson, which enhances the testability of these mechanisms. In addition, classical rolling always dominates over quantum fluctuations during the relaxation. Thus, the Higgs VEV is determined almost identically over different Hubble patches, which cures the oddity raised in the original model [1].
This paper is organized as follows. In Section II, we briefly review the original relaxion model with QCD-like dynamics, which we use throughout this paper. An overview of our first mechanism is given in Section III. Since we use a parametric resonance to stop the relaxion, we explain it in Section IV. Then, we discuss resonant particle production in Section V.
After a short review of a vacuum decay rate in Section VI, we summarize theoretical and experimental constraints in Section VII. Then, we show an example parameter region in Section VIII. In Section IX, we discuss how an appropriate potential of the relaxion can be 1 It has been also argued that eternal inflation is generically incompatible with the (refined) de Sitter swampland conjecture [43][44][45]. The hilltop eternal inflation is marginally consistent with the refined de Sitter swampland conjecture, but the Hubble constant naturally lies around the Planck scale in such a case. 2 Another solution is make = 0 after the relaxation, which is discussed in the context of solving the strong CP problem in the QCD relaxion model [1]. obtained, identifying it with a pseudo Nambu-Goldstone boson (NGB). We also discuss the origin of the fast-rolling in the latter part of the section. The final section is devoted to our conclusions.

II. MODEL
We review the original non-QCD relaxion model [1] in this section.
The Lagrangian is defined as where and L SM is the SM Lagrangian without the Higgs potential. Here, we denote the relaxion field and the Higgs doublet as X and Φ, respectively. The Higgs quartic coupling, λ, is almost the same as that in the SM, while the Higgs mass, M 2 , is assumed to lie around a new physics scale. To relax the hierarchy between the two scales, we introduce a coupling between the Higgs boson and the relaxion, where the small coupling constant, , is technically natural (see Section IX). Since breaks the shift symmetry of the relaxion, we expect a tadpole term of the relaxion with |r| 1/16π 2 . We assume M 2 , and r are positive. We have another source of the shift symmetry breaking due to non-QCD strong dynamics, which we assume to have the form of The simplest example of the new strong sector is given in [1], where we introduce new light fermions having the same SM charges as a right handed neutrino, N and N c , and new heavy fermions having the same Standard Model (SM) charges as a lepton doublet, L and L c . 3 They are charged under a new SU (3) group, whose field strength is denoted as G a µν . The relevant part of the Lagrangian is given by 3 In this paper, all the fermions without a dagger represent left-handed Weyl fermions.
where f is a decay constant of the relaxion, y andỹ are Yukawa couplings, and L kin includes the kinetic terms. After integrating out L, the mass of N is given by at the one-loop level. If m (eff) N is smaller than the dynamical scale of the new strong dynamics, Λ c , N N c condensates and generates In this paper, we assume that Λ c is smaller than m L .

III. MECHANISM
In this section, we give an overview of our first mechanism. The main difference from the original model is that the relaxion does not slow-roll due to where H is the Hubble constant during inflation and the dot indicates the time derivative.
In this case, the original stopping mechanism does not work since the kinetic energy of the relaxion is not negligible. Instead, we use the following mechanism.
1. The initial Higgs mass squared is assumed to be positive at the onset of relaxation.
The relaxion rolls down the potential with its terminal velocity until the Higgs mass squared becomes smaller than Λ 2 h .
2. When the Higgs mass squared decreases down to a certain positive value, the Higgs field starts to oscillate homogeneously due to a parametric resonance, which is explained in the next section. The amplitude of the oscillation grows gradually as the Higgs mass becomes smaller.
3. The Higgs oscillation is then fed back to the relaxion roll in the following ways; (i) the height of the bumps of the relaxion potential oscillates, (ii) the relaxion slows down due to the additional Hubble friction acting on the Higgs field. Since they hinder the roll, the relaxion eventually hits a bump and bounces back.
4. Just after the bounce of the relaxion, the Higgs field finds its mass is negative due to the negative contribution from the second term in Eq. (5), and develops a VEV. The sudden development of the Higgs VEV plays a role of an anchor, which secures the relaxion in a potential well between the bumps.
5. Once the anchor bites, the Higgs field can not return to the symmetric point due to the Hubble friction, and the oscillation around the VEV dumps quickly. It finalizes the relaxation.
It should be noticed that the Higgs mass squared is typically positive when the relaxion stops unlike in the original mechanism. However, it does not matter since the Higgs mass squared has been reduced to a value smaller than Λ 2 h , and the negative Higgs mass squared is provided by Λ 2 h cos(X/f ).

IV. EDGE SOLUTION
In this section, we show the behavior of the Higgs field around the critical point, which triggers our stopping mechanism.
Before going into discussion, let us summarize the equations of motion for the homogeneous modes of the relaxion and the Higgs field, which are denoted byX(t) andh(t), respectively. They are given bÿ where we choose a basis of the Higgs field as Here, η a 's are taken so that their expectation values are always zero without loss of generality.
One might think that nothing happens during the roll of the relaxion since the simplest solution to the equations of motion is with t 0 being a constant. However, it is not always a stable solution and there appears another branch of stable solutions where the oscillation ofh grows gradually.

A. Simplified System
Let us first see that the solution of (14) and (15) is not stable using a simple differential equation. Plugging Eq. (14) into Eq. (12), one obtains a differential equation that looks likë with y ∼h and Here, ω, m 2 0 and δm 2 are the constants determined by the model parameters and we have ignored the Hubble friction acting on the Higgs field for simplicity.
where the unit of scale is arbitrary. The initial conditions of y are irrelevant if it is small enough but non-zero.
We can see that y starts to oscillate around t ∼ 2 and the amplitude grows gradually afterwards, which shows that the solution described by (14) and (15) Since m 2 decreases monotonically due to the motion of the relaxion, it enters the above resonance band and y starts to grow exponentially.
The exponential growth, however, stops immediately after the non-linear term in Eq. (16) becomes important. To understand the effect of the non-linear term, let us analyze the differential equation given byÿ assuming m 2 is constant. A relevant oscillating solution is given bỹ where sn(u, k) is the Jacobi elliptic sine function and Here, K(k) is the complete elliptic integral of the first kind, which is defined as It motivates us to approximate the original differential equation as with A being the amplitude of the oscillation. Let us describe the behavior of the solution as follows. When m 2 enters the resonance band, A starts to grow exponentially. Since it makes m 2 larger, the growth stops immediately. However, since m 2 decreases due to the motion of the relaxion, it re-enters the resonance band and the amplitude grows until m 2 gets out of the resonance band. These occur repeatedly and the amplitude is kept around the edge of the resonance band.
Let us discuss it more quantitatively and justify the above description. Assuming m 2 is a constant, one can construct a solution to Eq. (24) that behaves like where u(t) is an O(1) function having a periodicity ω. As discussed in Appendix A, ν can be approximated as around the first resonance band. When ν has an imaginary part, the amplitude grows exponentially. Assuming m 2 is kept around the edge of the resonance band, i.e. Im(ν) 0 , we can predict the amplitude as We show the amplitude, A, determined by Eq. (27) with the dashed orange line in the left panel of Fig. 1. Notice that the time dependence of A comes from m 2 (t). As we can see from the figure, the predicted amplitude agrees well with the numerical solution, supporting our intuitive description.
Another justification comes from the frequency of the amplified mode. Since the real part of ν is one, we expect that the frequency of y(t) peaks around ω/2. In the right panel of Fig. 1

B. Relaxion-Higgs System
The solution obtained in the previous subsection can not be directly used in our analysis since it does not include the Hubble friction and feedback to the relaxion dynamics.
We search for an edge solution, which has the form of where α, ω X , m 2 Φ and A h are treated as constants for |t| |m 2 Φ /( f ω X )|. The time dependence of these constants will be taken into account by shifting m 2 Φ so that it cancels the last term of Eq. (28). In Appendix B, we determine α as FIG. 2. A comparison between a numerical solution to Eqs. (11) and (12) and the analytic result given in Eqs. (31) and (32). and obtain relations among ω X , m 2 Φ and A h as ignoring O(1/ω 3 X ), O(H 2 /ω X ) and O(H 3 ) corrections. It should be noted that the Higgs oscillation does not grow due to the Hubble friction in the case of 3Hω X /Λ 2 h 1.
In Fig. 2, we compare a numerical solution to Eqs. (11) and (12) with the analytic results given in Eqs. (31) and (32). The left panel showsh (blue) and A h (orange), and the right panel showsẊ/f (blue) and ω X (orange). We take At t = 0, we set M 2 − X 4700 GeV 2 and takeẊ around the terminal velocity. To mimic quantum fluctuations ofh, we take 5h H/(2π) at t = 0 and switch off the Hubble friction when |h| is smaller than H/(2π). As we can see from the left panel, the amplitude is well approximated by Eqs. (31) and (32) FIG. 3. The same figures as in Fig. 2 with the parameter set given in Eq. (34).
right panel, we see that the predicted ω X roughly reproduces the time average ofẊ/f . In Fig. 3, we show another example where the relaxion hits a bump and gets trapped in a potential well. Here, we take the following parameters.
We set M 2 − X 13800 GeV 2 at t = 0. As the figure shows, the relaxion hits a bump at around t ∼ 1.3 GeV −1 and the Higgs field quickly develops a VEV just like an anchor, securing the relaxion. After the relaxation, we have M 2 − X 3000 GeV 2 and the negative Higgs mass squared is provided by Λ 2 h cos X /f . Finally, let us comment on other resonance bands. So far, we have used the first resonance band for the edge solution, but similar solutions with the second and higher resonance bands should also exist. One could eliminate them by taking a large enough Hubble constant since they are much weaker than the first one. However, it is not necessary because the Higgs field does not develop its VEV after the relaxion hits a bump. Since the anchor does not bite, the relaxion continues to roll.

V. SUPPRESSION OF RESONANT PARTICLE PRODUCTION
In the previous section, we have analyzed the homogeneous modes of the relaxion and the Higgs field. However, in field theory, we have to take into account the dynamics of inhomogeneous modes as well. In particular, resonant particle production can affect our mechanism in the following ways.
• Thermal bath induces thermal potential.
• Thermal bath behaves like friction for the Higgs field and the relaxion.
• Large fluctuations of the relaxion may average out the bumps.
• Increase of the temperature may cause phase transition and erase the bumps.
Although they do not always spoil our mechanism, we seek a parameter space where the particle production is suppressed to keep our analysis as simple as possible.
Let us start with classical field theory. We separate the fields into the homogeneous modes and fluctuations around them as In the Fourier space, the fluctuations are expanded as For small fluctuations, they satisfÿ with a(t) = e Ht being the scale factor. Here, we substituted the edge solution given by Eqs. (28) and (29). We have ignored inhomogeneous terms since they can be erased by using special solutions. We assume that m 2 Φ and ω X are almost constant in the timescale of 1/H.
We estimate the effect of resonant particle production in quantum field theory by comparing ϕ k and χ k with "free" solutions, ϕ free k and χ free k , which satisfy the same differential equations but without δm 2 ϕ , δm 2 χ or δm 2 mix . Notice that χ free k and ϕ free k should give the zero point fluctuations with appropriate normalization. Since |k| 2 /a 2 dominates over all the mass To suppress the resonant particle production, we require 7 for all |k| > 3H/2. Here, which we call the occupation numbers.
In the WKB approximation, χ free k and ϕ free k are calculated as where C χ and C ϕ are constants.
On the other hand, χ k and ϕ k are well approximated by where µ χ and µ ϕ are the characteristic exponents, which will be given explicitly later. Here, One can easily check these are in good agreement with numerical solutions.
At the first approximation, the occupation numbers are given by where Since the smallest |k|/a gives the strongest constraints, we check In the left panel of Fig. 4, we plot the imaginary part of the characteristic exponents evaluated at t = 0.3, when the Higgs oscillation has not started yet. We evaluate them by directly solving the recurrence formula for ∆(0) in Appendix A. The largest peaks of µ ϕ corresponds to the first resonance band and that of µ χ corresponds to the second resonance band. We can see that there are tiny peaks in the right of the largest peaks. They correspond to the second resonance band of µ ϕ and the third resonance band of µ χ . Notice that the first resonance band for N χ is what we use for the Higgs coherent oscillation, and hence it can not contribute to the particle production.
For the maximum occupation number of ϕ, we obtain N max ϕ 0.5. As for that of χ, we show its time dependence in the right panel of Fig. 4. For t −0.7, the resonance condition can not be satisfied due to a large Higgs mass. It has a peak structure and we get N max χ 1.0. Since they are small enough, we can safely ignore the effect of particle production with this parameter set.
In the subsequent sections, we use the approximations given in Appendix A. They are given by for the first resonance band of N ϕ , and for the second resonance band of N χ . Notice that the integration in Eqs. (54) and (55) can be executed analytically.

B. After the Higgs oscillation
After the Higgs oscillation begins, δm 2 mix is turned on and ϕ k and χ k are influenced by each other. In addition, the Higgs oscillation can cause resonant production of the W boson and the Z boson.
By solving Eqs. (39) and (40) directly, one finds that these differential equations have broad and strong instabilities. To avoid such instabilities, we restrict the number of the Higgs oscillations during the amplification to be O(1). Then, A h , m 2 Φ and ω X are different for each oscillation and we can avoid the exponential amplification of mode functions.
Another concern is that a single oscillation could produce too many particles, which might disturb the Higgs oscillation. Let us consider a generic particle, ζ, which has a mass of where g is a coupling constant. In the following, we estimate how much ζ is produced for each oscillation. The adiabaticity condition is violated for momenta satisfying [46] |k| 2 a 2 whose occupation number becomes around one for a single oscillation. The energy density of the created ζ particles is evaluated as Since it is much less than that of the Higgs oscillation, we can safely neglect the effect of particle production.
As we can see from

VI. VACUUM DECAY RATE
In this section, we review a vacuum decay rate, which will be used to evaluate the stability of a vacuum after the relaxation. As pointed out in the introduction, the lifetime of the first selected vacuum should be much longer than the age of the universe, otherwise one needs an unacceptably large number of e-folds to find a sufficiently long-lived vacuum.
We approximate the potential around the EW vacuum by a one-dimensional potential along X, which is given by where v 246 GeV is the Higgs VEV.
The origin of X is redefined so that the correct EW vacuum corresponds to X 0, which we label as X F . In the following, we evaluate the tunneling rate into the deeper vacuum around X 2πf , which we label as X T . The decay rate is given by the bubble nucleation rate [47,48], which is expressed as Here, and X B is the bounce solution. The prefactor A is assumed to be (100 GeV) 4 in our analysis 8 .
For the evaluation of B, we use the thin wall approximation [47], which is given by where Here, X * is a constant determined by Since it gives a lower bound on B, we get an upper bound on the bubble nucleation rate The actual constraint of vacuum stability will be discussed in the following section together with other constraints.

VII. VIABLE PARAMETER SPACE FOR THE RELAXION MECHANISM
In this section, we summarize constraints on the parameters. We divide them into four categories; (i) those for the successful relaxation, (ii) those from consistency of our analysis, (iii) those from the explicit model of the strong sector, and (iv) those from experiments. The constraints of (i) are essential and independent of what are behind the relaxion potential, while those of (iii) depend on the detail of the strong sector. Some of the constraints of (ii) may be removed, but it is beyond the scope of this paper.
A. Successful Relaxation

Roll down
We first discuss the conditions for the relaxion to roll down the potential.
The terminal velocity of the relaxion should be large enough so that the relaxion can go over the bumps, which gives To violate the slow-roll condition, we require The classical rolling should dominate over quantum fluctuations, which gives

Stopping mechanism
Next, we discuss the conditions to stop the relaxion at the desired position.
The existence condition of the edge solution can be read off from Eqs. (30) and (32) with The Higgs oscillation should start before the relaxion scans the correct Higgs mass. It where When the relaxion hits a bump, the Higgs boson should acquire a large enough VEV to anchor the relaxion. We require the amplitude of the Higgs oscillation be smaller than the

EW vacua
Lastly, we discuss the conditions related to the EW vacuum.
There must be a stationary point of the relaxion potential around the EW vacuum, which To avoid fine-tuning, there should be a sufficient number of vacua that realize the Higgs mass around the EW scale, which gives To make the lifetime of the EW vacuum longer than the age of the Universe, we require with H 0 being the current value of the Hubble constant. It is equivalent to B thin 400.

B. Consistency of Analysis
Since we assume that the inflaton potential dominates the total energy density during the relaxation, we require with M Pl being the reduced Planck mass.
For the approximation used in Eq. (76) to be reliable, we need To avoid the resonant particle production before the Higgs oscillation starts, we require To avoid the resonant particle production after the Higgs oscillation starts, we require Notice that not all of them are responsible for the particle production since the amplitude is small for the first several oscillations.

C. Model of Strong Sector
The parameters coming from the new strong sector, Λ 0 and Λ h , should be obtained naturally from an explicit model. In our analysis, we adopt the simplest model described in Section II.
Since we need fermions that condensate, we require To restrict effects of the new doublet fermions on the strong dynamics, we require The Hubble expansion should not disturb the condensation, which gives To keep the new Yukawa couplings perturbative, we require max(|y|, |ỹ|) 4π .
Since there are tree and quantum contributions in Λ 4 0 , there can be an implicit cancellation. To avoid a fine-tuning, we define fine-tuning parameter ξ as and require |ξ| 0.1 .
Notice that, once m L and Λ c are fixed, the other parameters are determined as D. Experiments

Strong sector
We first discuss experimental constraints on the new strong sector.
From Eq. (94), we expect that m L is not much larger than Λ 0 to avoid a fine-tuning.
The constraints on the new doublet fermions have been discussed in [42] in Then, they effectively decouple from Higgs phenomenology.
Lastly, we comment on the dynamics that gives the relaxion decay constant, f . Such a small decay constant is obtained, for example, from a scalar clockwork mechanism discussed in Appendix D. In such a case, we expect new particles around f . Alternatively, one could use fermion condensations for the building block of the clockwork [49]. Then, we expect a rich spectrum around 4πf ∼ 1 TeV and thus we can safely ignore their effects. Since it is highly model dependent, we do not go into details in this paper.

Relaxion and Higgs boson
Let us move on to the collider constraints on the relaxion and the Higgs boson.
Since we assume a rather small f , the relaxion easily mixes with the Higgs boson. We define the mixing angles as where X is the expectation value of the relaxion at the EW vacuum. 9 Using these, the mass matrix of the scalars are expressed as in the basis of (χ, ϕ). 10 When θ is small, the mixing angle of χ and ϕ can be calculated as Here, the mixing angle is defined by where h 125 is the 125GeV Higgs boson and S is the relaxion-like boson. The masses of h 125 and S are approximately given by respectively. Since the Higgs mass has been measured, we tune λ so that it reproduces Here, we ignore the uncertainty since it has very little effect on our results.
Since S has couplings to the SM particles through the mixing, it can be produced at colliders. We assume that S decays only into the SM particles. The partial decay widths of the relaxion are given by where and Γ(h 125 → SM)| m 125 →m S is the Higgs partial decay width with the Higgs mass being replaced by m S . 11 The experiments at LEP, Tevatron, and LHC have extensively searched for neutral Higgs bosons. In our analysis, we use HiggsBounds [50][51][52][53][54] to obtain the 95% CL excluded region.
Finally, we discuss the Higgs phenomenology. To avoid the constraints on the Higgs signal strengths, we require [42,55] as a reference 12 . In particular, the branching ratio of the Higgs boson into two relaxions 11 We neglect possible decay of the relaxion into two photons through the strong sector. 12 By using the P-value of HiggsSignals [55,56], the number is about 0.7 at 95% CL. The ATLAS and the CMS collaborations provide global fits of the signal strengths, which give about 0.95 at 95%CL since the central values are rather high [57,58].
easily dominates over the others due to a large coupling, Thus, we forbid it kinematically by requiring Notice that all the collider constraints can be avoided if the relaxion mass and the Higgs mass are degenerated. However, we do not consider such a region to avoid additional finetuning.

VIII. PARAMETER REGION
In this section, we present an example parameter region. To reduce the number of parameters, we fix the following parameters in this section; can see, the allowed region is not small, so that we do not need to fine-tune the parameters.
Let us move on to the constraints from the consistency of our analysis. In Fig. 6, we show where our analysis becomes unreliable. The blue region is the same as in is safe.
The right panel of Fig. 6 shows the maximum occupation numbers of ϕ and χ before the Higgs oscillation starts. These numbers should be small enough so that the assumption of homogeneous fields is a good approximation. In this analysis, we assume are safe.
Next, we discuss the constraints on the strong sector. To illustrate them, let us take Λ c = 400 GeV , λ = 0.52 .
In Fig. 7, we plot |yỹ|, ξ, and m 500 GeV, we can avoid the constraints from the EW precision and the resonance searches.
Finally, we show the collider constraints on the relaxion and the Higgs signal strengths.
In Fig. 8, we shade the excluded region with green, which comes mainly from the searches for the relaxion. The blue region is the same as in Fig. 7 and the orange region violates the We pick up some sample points within the allowed region, which are shown with stars in the left panel of Fig. 9. In the right panel, we plot the evolution of the Higgs homogeneous mode for each sample point. The Higgs oscillations are expected to start at t = 1 GeV −1 . As we can see, the relaxation is successful for these sample points. We summarize phenomenology for each example point in Table I. At Point 3, the Higgs oscillation does not start. It is simply because the parameter set is around the boundary of Eq. (74). Even so, the relaxation is successful and it corresponds to the other mechanism explained in Appendix C.

IX. THE RELAXION POTENTIAL AND COSMOLOGICAL HISTORY
In the preceding sections, we have discussed the dynamics of the relaxion and the Higgs field assuming that their potential is given by Eq. (4). The most important feature of the  potential is the shift symmetry of X that is broken by two distinct sources. In this section, we give a sketch of the model in which the relaxion is identified with a pseudo NGB. The details of the model is discussed in Appendix D. We also discuss cosmological history that is compatible with the fast-rolling relaxion.

A. Pseudo Nambu-Goldstone Relaxion
When the relaxion appears as a pseudo NGB, we expect that its potential has a form of as the NGB resides in a compact field space. Here, we introduced two energy scales, F H and F L , which are associated with spontaneous breaking of a global U (1) symmetry. We take F L to be f in Eq. (4), while F H to be very large so that it reproduces the linear potential of the relaxion. The X-dependence of the potential arises from explicit breaking of the global U (1) symmetry. An arbitrary phase, δ, is irrelevant in the following discussion.
Before discussing an explicit model that generates Eq. (113), let us first clarify the relation between Eq. (113) and Eq. (4).
To identify the first term in Eq. (113) as the linear potential in Eq. (4), we need a large enough F H . Since the relaxion needs to scan its field range of O(M 2 / ) to find the EW scale, we require The coefficient of the linear potential in Eq. (4) is obtained by expanding X around an arbitrary point, X 0 , as Then, we have 13 The first term in Eq. (113) also contributes to the Higgs mass squared by For example, if we take F H 5 × 10 8 GeV and κ H 10 −9 , we have 0.8 GeV and M 20 TeV. 14 In Fig. 10, we show an illustration of the relaxion potential with F H = 40F L . The larger structure of the potential is governed by the first term in Eq. (113), while the small wiggle comes from the second term. We close up the potential around X 0 /F H = π/3 for a small interval, δX ∈ [−F H /2, F H /2], to illustrate that it is well approximated by a linear potential with a small modulation.

B. Relaxion Potential from Clockwork Mechanism
Here, we discuss a model that achieves the potential in Eq. (113). As we have mentioned, the X-dependence of the potential originates from explicit breaking of the global symmetry.
A difficulty in model building is that we need two hierarchical decay constants, F H and F L .
Let us first consider a global U (1) symmetry that is broken by two condensation operators; one is at O(F H ) and the other is at O(F L ). We assume that the U (1) charges of these operators are the same. We consider an effective scalar potential given by where π H and π L are the phase components of the condensation operators. It has a similar structure to that of Eq. (113), but we also have the third term, which is consistent with the U (1) symmetry realized by the shifts of π H and π L , We assume that the coefficient g is order one and thus the third term has the largest coefficient. Since one of the linear combinations of π H and π L becomes very heavy because of the third term, we identify the other lighter combination as the relaxion.
The potential of Eq. (119), however, does not provide the desirable relaxion potential of Eq. (113). In the limit of vanishing explicit breaking, the mass eigenstates of π H and π L are given by Thus, we find that π H and π L include X component as This difficulty can be circumvented by introducing a hierarchical U (1) charge between the two condensation operators. When the operator corresponding to π H has charge Q H 1, the third term is modified as Here, the U (1) symmetry is realized by In this case, the relaxion component of π H and π L are given by Substituting Eq. (126) to the potential in Eq. (123), we obtain the desirable potential of It should be noted that an exponentially small Q H can be achieved by the clockwork mechanism [59][60][61] (see also [62]). For example, let us consider N + 1 sectors containing condensation operators. We assume the charges of these operators decrease in geometric progression with ratio q −1 (q > 1). Identifying the phase component of the operator in the first sector as π L and that in the (N + 1)-th sector as π H , we effectively get an exponentially small charge of Q H = q −N . For example, the hierarchy between F H 5 × 10 8 GeV and F L 10 2 GeV is realized with q = 3 and N = 14. 16 There are two caveats in the construction of an explicit clockwork model. First, the mass scales of the explicit breaking of the U (1) symmetry should not exceed those of the spontaneous breaking since otherwise the existence of the NGB is invalidated. Second, the F H and F L should be stabilized in a way that the larger mass scale does not interfere the lower mass scale. In Appendix D, we elaborate on these points and consider a clockwork mechanism with progressively increasing VEVs to overcome these difficulties.
Finally, let us comment on the origin of the explicit breaking terms (see Appendix D for details). The second term in the potential in Eq. (123) can be obtained by using the model in Section II with X/f → π L /F L . Similarly, we can construct a model that gives the first term in the potential in Eq. (123). Let us consider a new strong sector that couples to π H , whose Lagrangian is given by Here Section II, respectively. Assuming that L H is heavier than the dynamical scale, Λ H , and N H is lighter, we find With these relations, we find, for example, that M

C. Cosmological History
In our mechanism, we assume that the initial velocity of the relaxion is large enough so that it can go over the bumps created by the strong dynamics. In this section, we discuss cosmological history where the relaxion starts to roll before or during inflation.
A simple possibility is that there is a radiation dominated era before the relaxion mechanism takes place. We assume that the maximal temperature is high enough so that all the terms in Eq. (113) vanish. The succeeding inflation era starts typically before these terms are recreated. In the inflation era, the temperature decreases very quickly and the first dynamics confines at T Λ H , which generates the first term in Eq. (113). 18 It accelerates the relaxion unless the relaxion accidentally sits around an extremum. The relaxion reaches the terminal velocity within about one e-fold after the first confinement. Subsequently, the second dynamics confines at T Λ, which generates the second term in Eq. (113). If the relaxion has been accelerated enough, it can go over the created bumps, which explains the origin of the fast-rolling.
In Fig. 11, we show the time evolution ofẊ during inflation (blue line), ignoring the Higgs field. Here, we assume that the inflation era starts at t ∼ O(0.1) GeV −1 with H = 10 GeV.
The parameters are taken to be = 0.  17 The mass squared, M N , is required to be at least of O(ỹ H y H /16π 2 M L ) to be technically natural. 18 Here, we assume that the global U (1) symmetry has been broken well before the inflation starts. 11. The evolution ofẊ during inflation with |Φ| = 0 (blue line). Low scale inflation with the Hubble constant H = 10 GeV is assumed to take place after radiation domination. (The time variable t can be regarded as an e-folding number, N e = Ht.) During radiation domination, the strong dynamics responsible for the relaxion potential in Eq. (113) does not exhibit confinement, and hence, the relaxion does not have any potential. The first dynamics confines at T Λ H , which generates the first term in Eq. (113), and the relaxion is accelerated. The second dynamics confines at T Λ c , which modulate the relaxion potential. The dashed cyan line shows the terminal velocity expected for a purely linear potential. The dashed orange line shows the evolution ofẊ without the second term in Eq. (113).
for a purely linear potential with the dashed cyan line.
As we can see from the figure, the relaxion reaches the terminal velocity around t 1 GeV −1 . Soon after, the second term in Eq. (113) is turned on and the velocity of the relaxion starts to oscillate. Our stopping mechanism is expected to work around t ∼ 10 3 GeV −1 and the linear potential approximation is good in this timescale. Note that the relaxion continues to roll in the figure since we do not take into account the Higgs field.
We give another possibility for the origin of the fast-rolling, which does not require a radiation dominated era before inflation. Suppose that the field value of the relaxion at the onset of the inflation is randomly distributed in space. Here, we assume that the potential of Eq. (113) has already been developed. Then, in some of the Hubble patches, we find a small region that takes the value of the relaxion in a different potential well. If the vacuum energy of the interior is higher than that of the exterior, such a region starts to contract and eventually collapses. At the point of collapse, the released energy is converted to the kinetic energy of the relaxion and the relaxion starts to roll down the potential.
Let us move on to cosmological history after the relaxation. As is discussed in [1], we need to reproduce the observed CMB fluctuations and reheat the SM particles safely. In our mechanism, the e-folds during the relaxation is estimated as which is about 10 5 with the parameter sets in the previous section. The observed CMB fluctuations can be compatible with such a number of e-folds if we consider, for example, new inflation. After the inflation, the maximal temperature should not be raised above ∼ 100 GeV, otherwise the relaxion would start to roll again. Thus, we need another wellseparated sector, into which the potential energy of the inflaton is dumped. We do not go further in details since it falls out of the scope of this paper.
Finally, let us comment on baryogenesis. Although the upper bound on the maximal temperature is similar as in [1], the situation is worse because we can not use the sphalerons to generate the baryon asymmetry. There are several ways to generate it with a very low reheating temperature by using the decay of inflaton or heavy particles [63][64][65][66], or by using the oscillations of mesons or baryons [67][68][69].

X. CONCLUSION
In the original relaxion mechanism, quantum tunneling is essential for finding a sufficiently long-lived vacuum. However, it requires an unacceptably large number of e-folds and the inflation sector seems to be extremely fine-tuned.
We have shown that a fast-rolling relaxion can easily avoid the above problem without extending the original relaxion model. In our mechanism, we allow the relaxion to fly over the bumps and trigger the Higgs oscillation around the critical point. Then, the relaxion accidentally hits a bump and stops due to the immediate development of the Higgs VEV.
After the EW scale is relaxed, the first selected vacuum need not decay if its lifetime is much longer than the age of the universe.
In our mechanism, the edge solution plays an important role. We have examined it in detail and obtained the relations among the amplitude of the Higgs oscillation, the Higgs mass, and the velocity of the relaxion. They offer not only a physical explanation of the solution but also a very effective way to search for viable parameters.
To keep our mechanism viable in field theory, we have discussed possible resonant particle production. In particular, the Higgs oscillation could produce too many particles and hence is dangerous. We discussed that it can be avoided by limiting the number of the Higgs oscillations to be O(10).
We We have also discussed a more realistic model and cosmological history, where the relaxion resides in a compact field space and the origin of the fast-roll is explained naturally. Importantly, the model is realized without introducing a super-Planckian field space.
We think a higher cut-off is an interesting direction to explore. Unfortunately, such a region typically suffers from a large number of Higgs oscillations and the analysis becomes much more complicated. However, our mechanism is possibly compatible with particle production and such a region may be explored by using lattice calculations. In this appendix, we give estimates of the Floquet exponents of the Mathieu equation following [70].
The Mathieu equation is defined as with real A and q.
From the Floquet's theorem, there exist solutions that satisfy u(z + π) = e iπν u(z) .

(A2)
Here, ν is a complex constant called the Floquet exponent. Such a function can be expanded with functions with periodicity π as where c n 's are expansion coefficients.
In the following, we evaluate using the Whittaker-Hill formula.
We first obtain relations among the expansion coefficients by substituting Eq. (A3) to Eq. (A1). We get where These relations can be expressed as where As shown in [70], Next, we see the analytic structure of ∆(ν) ≡ det Ξ. As can be seen from Ξ, ∆(ν) has simple poles 19 at and is analytic for other ν. Since ∆(ν + 2) = ∆(ν) and ∆(ν) = ∆(−ν), it is enough to study strip Re(ν) ∈ [0, 1], where we have only one pole 20 at We define a function, which has a pole at the same position in the strip and satisfies D(ν + 2) = D(ν) and D(ν) = D(−ν). Thus, for an appropriate constant, C = 0, has no singularities for any ν ∈ C. Since Θ(ν) is a bounded entire function, it must be a constant from the Liouville's theorem. Taking the limit of ν → i∞, we have since Ξ goes to the identity matrix. Thus, we get Θ(ν) = 1 .
(A17) 19 For a special case, A ∈ Z, two simple poles merges into a pole of order 2. The following discussion is also applicable to such a case. 20 When A < 0, we slightly tilt the strip so that only one pole is in one strip.
Using ν = 0 as a reference point, we have for A = 4 2 with ∈ Z. For Eq. (A7) to have a non-trivial solution, we need ∆(ν) = 0, which gives which is called the Whittaker-Hill formula. Notice that we can extend it to A ∈ R.
Next, we evaluate ∆(0). We define finite truncated determinants, ∆ p 's, as whereξ n = ξ n (ν = 0) = q 4n 2 −A . Then, we have In the following, we obtain a recursion formula for ∆ p . The Laplace expansions for the first row give where and ∆ (−n) p is ∆ p without the first n columns and the first n rows. In the second term, we further developed the determinant along the first column.
Similarly, we have After erasing ∆ From these, we obtain The first three terms are given by Here, we defined ∆ −1 so that it reproduces ∆ 2 .
The leading terms in q 2 can be calculated as It shows that instability of Eq. (A19) appears around A = n 2 , n ∈ Z for a small q. The most important resonance bands are A = 1 and A = 4, which we call the first and the second resonance bands, respectively.
For a more precise determination of the Floquet exponents around these resonance bands, we solve Eq. (A28) up to a sufficiently large p. We get Here, the rational coefficients are determined by rationalizing numerical results with p = 1000.

Appendix B: Edge solution in the relaxion-Higgs system
In this appendix, we derive Eqs. (31) and (32).
We put an ansatz for the edge solution as where ϕ 0 and χ 0 are functions and α, A h , m 2 Φ and ω X are constants. Here, we assume |t| |m 2 Φ /( f ω X )|.
Appendix D: More on the Relaxion Potential As we have mentioned in Subsection IX B, a naive clockwork mechanism is not suitable for our relaxion potential. In this appendix, we explain the difficulties in model building and discuss how to overcome them by using clockwork with progressively increasing VEVs.

The clockwork potential
To demonstrate the clockwork mechanism, we consider a model with N complex scalars, φ k (k = 0, · · · N − 1) [59][60][61] (see also [62]). We assume a global U (1) symmetry where the charge of φ k is 3 −k . Then, the complex scalars are connected via where κ k 's denote small coupling constants. We also assume that each complex scalar obtains a VEV via a scalar potential, where we omit the coefficients for brevity. Notice that we dropped |φ k | 2 |φ l | 2 with k = l by hand and ignored the other Planck-suppressed terms allowed by the global U (1) symmetry.
For a while, we assume that f k = O(f ) = O(10 2 ) GeV for all k. Then, if κ k 1, the VEVs are given by φ k After all the φ k 's obtain VEVs, the connecting terms in Eq. (D1) lead to the potential of the phase components of φ k as where φ k = f k e iπ k /f k / √ 2. Due to this potential, all of the phase components obtain masses of O κ 1/2 f except for the pseudo Nambu-Goldstone boson (NGB), which we denote as X.
Here, the pseudo NGB corresponds to the mode that keeps the following relation; where F is the effective decay constant. By substituting φ k = e iX/(3 k F ) into the kinetic terms, we find that X is canonically normalized if Notice that Eq. (D4) shows that the relative contribution of each phase component π k to the pseudo NGB is suppressed by 3 −(k−1) f k /F . This property is valid even for κ i = O(1).
An intriguing feature of the clockwork mechanism is the response to the explicit breaking of the U (1) symmetry. For example, let us consider a situation where the U (1) symmetry is broken by two tiny parameters with mass dimension 3, E 0 and E N −1 , which have the U (1) charges 1 and 3 −(N −1) , respectively. In this case, φ 0 and φ N −1 couple to E's via which leads to a non-trivial potential of X as Here, we ignored the phases of E 0 and E N −1 since they are irrelevant for the relaxion mechanism. For N 1, the periodicity of the second term, ∆X = 3 N −1 ×2πF , is much longer than that of the first term, ∆X = 2πF . In this way, the clockwork mechanism accommodates cosine potentials of the pseudo NGB with hierarchically different periodicity.
A caveat here is that the mass scales of the explicit breaking should be much smaller than f k 's, i.e. E 0 , E N −1 f 3 . Otherwise, the explicit breaking terms in Eq. (D6) disturb the U (1) symmetric potential of Eq. (D2), which invalidates the existence of the pseudo NGB.
In subsection IX B, however, we utilized the explicit breaking term much larger than O(f ).
To allow such large explicit breaking, we need to extend the model and will discuss it in the next subsection.

Clockwork with progressively increasing VEVs
To allow explicit breaking with a mass scale larger than O(f ), we consider f k 's that progressively increase by a factor of a (3 ≥ a > 1), i.e. f k = a k f 0 . 23 In order to protect the mass scale of the k-th site against that of the (i + 1)-th site, we assume that the connecting Furthermore, by requiring that the explicit breaking satisfies we find that the pseudo NGB mode is lighter than all the other modes. With a large enough N , the desired scalar potential of Eq. (113) can be realized.
Before closing this subsection, let us comment on the relation between the relaxion potential discussed above and the that given in subsection IX B. For that purpose, we define 23 The mass scales f k do not need to be an exact geometric series. The upper bound on a comes from the requirement that Eq. (D5) remains of O(f 2 ). π L and π H by respectively. Then, the potential of π L and π H is given where the third term comes from the connecting terms in Eq. (D1). This potential reproduces the one in Eq. (123).
respectively. Then, the connecting superpotential is given by where κ k andκ k are coefficients of O(a −2 ). The VEV of each sector is obtained from the superpotential of where Y k 's are chiral superfields and f k 's are progressively increasing constants; f k+1 = af k (2 ≥ a > 1). Due to SUSY, f k 's are stable against quantum corrections. Here, the absence of the terms such as Y i f 2 j and Y i Φ jΦj (i = j) is technically natural. 24 In this model, the NG mode corresponds to the massless flat direction of the superpotential,X where Φ's depend on Π's via Here, the imaginary parts of the scalar components of Π k andX correspond to π k and X in the previous section, respectively. The decay constant for the canonically normalized X is given by Similarly as in the previous section, other supermultiplets than the relaxion direction become heavy. Furthermore, due to the gravity mediated SUSY breaking, the fermionic and scalar partners of the relaxion eventually obtain the masses of O m 3/2 . Thus, the model is reduced 24 One may also assume a discrete symmetry in each site under which the combination of X k andΦ k are neutral. Then, by assuming that each discrete symmetry is broken by f 2 k , we can forbid Y i f 2 j and Y i Φ jΦj (i = j). In this case,κ k is highly suppressed, though it does not affect the clockwork mechanism.
to the clockwork mechanism in the non-supersymmetric model below the scale of m 3/2 .

Explicit breaking in supersymmetric model
The explicit breaking of the U (1) symmetry can be implemented in the following way.
First, let us consider the explicit breaking at the N -th site by a breaking parameter E H , which has the U (1) charge of −2 −(N −1) and mass dimension 2. Then, the explicit breaking term is given by This superpotential leads to a runaway potential of the real part of the scalar component of X, which we denote as X s , as Here, the first term comes from the F -term potential ofX, while the second term comes from the gravity mediated SUSY breaking mass squared. We take E N −1 to be real for simplicity and F H denotes 2 N −1 F . In order to keep the VEVs of Φ 0 andΦ 0 around O(F ), we require The cosine potential of the relaxion is given by the SUSY breaking A-term contribution as The mass scale of this coefficient is O(10) TeV for E Similarly, the explicit breaking at the first site is implemented by where the U (1) charge of E 0 is 1 and its mass dimension is 2. To validate the notion of spontaneous U (1) breaking, E 0 should satisfy Under this condition, the scalar potential of X s is dominated by the SUSY breaking effects.
The SUSY breaking A-term contributions leads to the cosine potential as The relaxion also obtains a potential from the F -term contributions, They are, however, subdominant for which we assume in our analysis. where W α denotes the chiral gauge field strength superfield.
The dynamical scale of the SU (N c ) gauge theory, Λ H , depends on Φ N −1 viã where Λ H is the dynamical scale for m L = m N = 0 at H =H = 0 and Φ N −1 = f N −1 / √ 2. For m L , m N Λ H , the non-perturbative superpotential [71][72][73] and the mass terms of L HLH and N HNH lead to This term corresponds to the explicit breaking of the N -th site, By plugging into Eq. (D21), we obtain which leads to . 25 Thus, we successfully obtain the relaxion potential 25 More precisely, we need to consider one of the linear combinations of the two Higgs doublets, though it does not alter the correspondence of the parameters significantly.