Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry

We discuss a scheme to implement the relaxion solution to the hierarchy problem with multiple axions, and present a UV-completed model realizing the scheme. All of the $N$ axions in our model are periodic with a similar decay constant $f$ well below the Planck scale. In the limit $N\gg 1$, the relaxion $\phi$ corresponds to an exponentially long multi-helical flat direction which is shaped by a series of mass mixing between nearby axions in the compact field space of $N$ axions. With the length of flat direction given by $\Delta \phi =2\pi f_{\rm eff} \sim e^{\xi N} f$ for $\xi={\cal O}(1)$, both the scalar potential driving the evolution of $\phi$ during the inflationary epoch and the $\phi$-dependent Higgs boson mass vary with an exponentially large periodicity of ${\cal O}(f_{\rm eff})$, while the back reaction potential stabilizing the relaxion has a periodicity of ${\cal O}( f)$. A natural UV completion of our scheme can be found in high scale or (mini) split supersymmetry (SUSY) scenario with the axion scales generated by SUSY breaking as $f\sim \sqrt{m_{\rm SUSY}M_*}$, where the soft SUSY breaking scalar mass $m_{\rm SUSY}$ can be well above the weak scale, and the fundamental scale $M_*$ can be identified as the Planck scale or the GUT scale.


I. INTRODUCTION
Recently a new approach to address the hierarchy problem has been proposed in [1].
The scheme introduces a scalar degree of freedom, the relaxion φ, making the Higgs boson mass a dynamical field depending on φ. During the inflationary epoch, the Higgs boson mass-square µ 2 h (φ) is scanned by the rolling φ from a large positive initial value to zero. Right after the relaxion crosses the point µ h (φ) = 0, so that µ 2 h (φ) becomes negative, a nonzero Higgs vacuum expectation value (VEV) is developed and a Higgs-dependent back reaction potential begins to operate to stabilize the relaxion 1 . One can then arrange the model parameters in a technically natural way to result in the relaxion stabilized at a point where the corresponding Higgs VEV is much smaller than the initial Higgs boson mass.
An intriguing feature of the relaxion mechanism is that the relaxion potential involves two very different scales. One is the period of the back reaction potential, and the other is the excursion range of the relaxion necessary to scan µ h (φ) from a large initial value to zero. To see this, let us consider the relaxion potential given by where V 0 is the potential driving the rolling of φ during the inflationary epoch and V br is the periodic back reaction potential stabilizing φ right after it crosses µ h (φ) = 0. In fact, the key feature of the mechanism can be read off from the following form of potential: where M h denotes the initial Higgs boson mass, 0 and h are small dimensionless parameters describing the explicit breaking of the relaxion shift symmetry in V 0 and µ 2 h , respectively, and finally f is the relaxion decay constant in the back reaction potential.
In non-supersymmetric theory, the Higgs mass parameter M h is naturally of the order of 1 A mechanism to cosmologically relax the Higgs boson mass down to a small value through a nucleation of domain wall bubbles has been discussed in [2]. the cutoff scale of the model. On the other hand, in supersymmetric theory, it corresponds to the scale of soft supersymmetry (SUSY) breaking mass which can be well below the cutoff scale of the model. In any case, we are interested in the case that M h is much larger than the weak scale: which might be explained by the relaxion mechanism.
Let us now list the conditions for the relaxion mechanism to work. First of all, in order for the rolling relaxion to cross µ h (φ) = 0 without a fine tuning of the initial condition, it should experience a field excursion In order for the scalar potential to be technically natural under radiative corrections, the symmetry breaking parameters 0 and h should obey On the other hand, from the stability condition ∂ φ V = 0, one finds and therefore ∆φ f As for the back reaction potential, generically Λ br (h = 0) may not be vanishing, and then one needs Also, in order not to destabilize the weak scale size of the Higgs VEV, its magnitude should be bounded as An immediate consequence of the above conditions is that the relaxion should experience a field excursion much bigger than f in the limit M h v: The required excursion is huge in the case that the back reaction potential is generated by the QCD anomaly, in which Λ 4 br ∼ f 2 π m 2 π and therefore Even when the scale of the back reaction potential saturates the bound (9), the required relaxion excursion is still much larger than f as long as M h is higher than the weak scale by more than a few orders of magnitudes.  [3][4][5][6][7][8][9][10] for recent discussions of the related issues. In this paper, we discuss an alternative scenario in which the relaxion corresponds to an exponentially long multi-helical flat direction in the compact field space spanned by N sub-Planckian periodic axions: Such a long flat direction is formed by a series of mass mixing between nearby axions, producing a multiplicative sequence of helical windings of flat direction, which results in ∆φ for ξ = O(1). Our scenario is inspired by the recent generalization of the axion alignment mechanism for natural inflation [11] to the case of N axions [12]. Although it requires a rather specific form of axion mass mixings, our scheme does not involve any fine tuning of continuous parameters, nor an unreasonably large discrete parameter.
As we will see, our scheme finds a natural UV completion in high scale or (mini) split supersymmetry (SUSY) scenario with soft SUSY breaking scalar mass m SUSY v. In the UV completed model, the axion scales are generated by SUSY breaking [13][14][15] as where M * can be identified as the Planck scale or the GUT scale. With the (N − 1) hidden Yang-Mills gauge sectors which confine at scales below f i to generate the desired axion mass mixings, the canonically normalized relaxion has a field range where n j > 1 corresponds to the number of flavors of the gauge-charged fermions in the j-th hidden sector. One can then arrange the microscopic parameters in a technically natural way to make the resulting relaxion potential V 0 (φ) and the φ-dependent Higgs boson mass µ 2 h (φ) vary with an exponentially large periodicity of O(f eff ), while the back reaction potential V br (h, φ) has a periodicity of O(f i ). An interesting feature of our model is that the desired V 0 (φ) and µ 2 h (φ) arise as a natural consequence of the solution of the MSSM µ-problem advocated in [13][14][15].
The outline of the paper is as follows. In the next section, we describe the basic idea with a simple toy model and discuss the scheme within the framework of an effective theory of N axions. In section 3, we present a UV model with high scale SUSY, realizing our scheme in the low energy limit. Section 4 is the conclusion.

II. EXPONENTIALLY LONG RELAXION FROM MULTIPLE AXIONS
To illustrate the basic idea, let us begin with a simple two axion model. The lagrangian density of the model is given by where h is the SM Higgs doublet and φ i are the periodic axions: with a scalar potentialṼ where Here M h is an axion-independent mass parameter which is comparable to the cutoff scale of the above effective lagrangian, and n > 1 is an integer which will be determined by the underlying UV completion. We assume and therefore the model is stable against the radiative corrections which replace the Higgs As for the back reaction potential, one can consider two different possibilities. One option is to generate it by the coupling of φ 1 to the QCD anomaly, yielding where y u denotes the up quark Yukawa coupling to the SM Higgs field h, and Λ QCD is the QCD scale. This option corresponds to the minimal model, however generically is in conflict with the axion solution to the strong CP problem. Alternative option is to introduce a new hidden gauge interaction which confines around the weak scale and generates a back reaction potential given by [1,16] In order for the model to be technically natural, the underlying dynamics to generate the back reaction potential should be arranged to make sure that the above conditions on m 1 and m 2 are stable against radiative corrections.
The above two axion model involves the shift symmetries which are broken byṼ 0 down to the relaxion shift symmetry The flat direction associated with U (1) φ has a helical winding structure in the compact 2-dim field space of φ i as depicted in Fig. (1). Then the periodicity of the flat direction is enlarged as which is larger than the original axion periodicities 2πf 1 ∼ 2πf 2 by the winding number n.
The relaxion shift symmetry U (1) φ is slightly broken by small nonzero values of , and Λ br . Note that this particular form of U (1) φ breaking is technically natural as long as the first condition of (16) is satisfied. To find the effective potential of the flat relaxion direction, one can rewrite the model in terms of the canonically normalized heavy and light axions [11,12]:  which can be identified as the inflaton direction. One easily finds that the length of this periodic flat direction is given by where gcd (n 1 , n 2 ) denotes the greatest common divisor of n 1 and n 2 . This shows that a super-Planckian flat direction with ℓ flat ≫ M P l ≫ f i can be developed on the twodimensional sub-Planckian domain if In Fig. 1, we depict the flat direction in the fundamental domain of axion fields, which has a length given by (12). Since the axionic inflaton of natural inflation rolls down along this periodic flat direction, its effective decay constant is bounded as which means that at least one of n i should be as large as gcd (n 1 , Turning on the second axion potential a nontrivial potential is developed along the periodic flat direction having a length (12). for which where f H = f 1 f 2 /f eff . In the limit Λ 4 f 4 2 Λ 4 br , it is straightforward to integrate out the heavy axion φ H to derive the low energy effective lagrangian of the light axion φ. The resulting effective potential of the canonically normalized φ is given by where We can now generalize the above two axion model to the case of N > 2 axions to enlarge the effective axion scale further [12]. The lagrangian density is given by with Λ 4 i f 4 N Λ 4 br . The model involves the N axionic shift symmetries: which are broken byṼ 0 down to the relaxion shift symmetry: with the corresponding flat direction given by Turing on small nonzero values of , and Λ br , the relaxion shift symmetry (30) is slightly broken and nontrivial potential of φ is developed. Although our way to break U (1) φ is rather specific, it is technically natural as the model involves many continuous or discrete axionic shift symmetries which are distinguishing our particular way of symmetry breaking from other possibilities. It is again straightforward to integrate out the (N − 1) heavy axions which receive a large mass fromṼ 0 [12]. For the canonically normalized φ, the resulting effective potential is given by where For simplicity here we assumed that all f i are comparable to each other, or f 1 is the biggest among {f i }.
Obviously, in the limit N where the ellipsis stands for the (N − 1) heavy axions receiving a large mass fromṼ 0 . For an illustration of this feature, we depict in Fig. (2) the relaxion field direction for the case of N = 3, n 1 = 2, n 2 = 4.
The effective potential (32) can easily realize the relaxion mechanism under several consistency conditions. First of all, like (16) of the two axion model, we need in order for the model to be stable against radiative corrections, while allowing µ h = 0 for certain value of φ. Without invoking any fine tuning, there is always a certain range of with the back reaction potential stabilizing the relaxion at the value giving h = v. The stabilization condition leads to From (35), this then yields a lower bound on f eff : where v 4 /Λ 4 br (h = v) ∼ 10 12 when V br is generated by the QCD anomaly, or v 4 /Λ 4 br (h = v) has a model-dependent value not exceeding O(1) when V br is generated by the hidden color dynamics which confines around the weak scale.
To summarize, in our scheme for the relaxion mechanism, v M h can be technically natural with an exponential hierarchy between the two effective axion scales: which is arising as a consequence of a series of mass mixing between nearby axions in the compact field space of N axions. Although it relies on a rather specific form of axion mass mixings, the scheme does not involve any fine tuning of continuous parameters, nor an unreasonably large discrete parameter.

III. A UV MODEL WITH HIGH SCALE SUPERSYMMETRY
In this section, we construct an explicit UV completion of the N axion model discussed in the previous section. We first note that our scheme requires that the axion potential On the other hand, we wish to have an explicit UV model providing the full part of the axion potential in (28), as well as a mechanism to generate the axion scales f i . This implies that our UV model should allow the natural size of the Higgs boson mass, i.e.
M h , to be much lower than its cutoff scale. As SUSY provides a natural framework for this purpose, in the following we present a supersymmetric UV completion of the low energy effective potential (32).
First of all, to have N axions with the decay constants f i M Planck , we introduce N global U (1) symmetries under which where X i and Y i are gauge-singlet chiral superfields with the U (1) i -invariant superpotential where M * corresponds to the cutoff scale of the model, which might be identified as the GUT scale or the Planck scale. Here and in the following, we ignore the dimensionless coefficients of order unity in the lagrangian. We assume that SUSY is softly broken with SUSY breaking soft masses In particular, the model involves the soft SUSY breaking terms of X i and Y i , given by where To achieve the N axions in the low energy limit, we need all m 2 X i are tachyonic, which results in Then the canonically normalized axion components φ i can be identified as Now we need a dynamics to generate the axion potentialṼ 0 in (28), developing an exponentially long flat direction as described in the previous section. For this purpose, we introduce (N − 1) hidden Yang-Mills sectors associated with the gauge group G = including also the charged matter fields where Ψ i and Φ ia are the fundamental representation of SU (k i ), while Ψ c i and Φ c ia are antifundamentals. These gauged charged matter fields couple to the U (1) i -breaking fields X i through the superpotential Note that X i couples to a single flavor of the SU (k i )-charged hidden quark Ψ i + Ψ c i , while X i+1 couples to n i flavors of the SU (k i )-charged hidden quarks Φ ia + Φ c ia . With this form of hidden Yang-Mills sectors, the N global U (1) symmetries are explicitly broken down to a single U (1) by the U (1) i × SU (k j ) × SU (k j ) anomalies. The charged matter fields Ψ i + Ψ c i and Φ ia + Φ ia get a heavy mass of O(f i ), so can be integrated out at scales below f i . This yields an axion-dependent threshold correction to the holomorphic gauge kinetic function τ i of SU (k i ) at scales below f i : where we ignored the dependence on |X i |, while including the soft SUSY breaking by the gaugino masses M λ i ∼ m SUSY . As a consequence, at scales below f i , the global symmetry breaking by the U (1) i × SU (k j ) × SU (k j ) anomalies is described by the following axion effective interactions: where (F ) SU (k i ) denotes the gauge field strength of SU (k i ) and (F ) SU (k i ) is its dual. As we wish to generate the axion potential |Ṽ 0 | M 4 h ∼ m 4 SUSY from the above axion couplings, we assumeΛ whereΛ i denotes the confining scale of the hidden gauge group SU (k i ). In such case, the resulting axion potential is determined by the non-perturbative effective superpotential describing the formation of the SU (k i ) gaugino condensation [17]: Our next mission is to generate the axion potential V 0 and the axion-dependent Higgs mass-square µ 2 h in (28), driving the evolution of the relaxion during the inflationary epoch, while scanning the Higgs mass-square from an initial value of O(m 2 SUSY ) to zero. This can be done by introducing a superpotential term given by together with the associated Kähler potential term: Here we ignore the irrelevant terms such as |X N | 4 or |X N −1 | 4 in the Kähler potential. Note that the couplings in W 3 leads to a logarithmically divergent radiative correction to ∆K [18], and our model is stable against such radiative correction as long as the coefficient of ∆K is of order unity. Note also that W 3 provides a solution to the MSSM µ problem as it yields naturally the Higgsino mass µ eff ∼ m SUSY [13][14][15].
After integrating out the (N − 1) axions which receive a heavy mass fromṼ 0 , while leaving the light relaxion φ as described in the previous section, the Kähler potential term ∆K gives rise to where and δ is a phase angle which is generically of order unity. In our scheme, the MSSM Higgsino mass µ eff originates from W 3 , and therefore is naturally of the order of m SUSY [13][14][15]. Again, after integrating out the (N − 1) heavy axions, we find the MSSM Higgs parameters µ eff and Bµ eff depend on the relaxion φ as where Then the determinant of the MSSM Higgs mass matrix also depends on φ via To complete the model, we need to generate the back reaction potential V br . In regard to this, we simply adopt the schemes suggested in [1]. One option is to generate V br through the QCD anomaly. For this, one can introduce where Q + Q c is an exotic quark which receive a heavy mass by X 1 ∼ f 1 . Once this heavy quark is integrated out, the axion φ 1 couples to the gluons as After the (N −1) heavy axions are integrated out, this leads to the relaxion-gluon coupling where Then the resulting back reaction potential is obtained to be where y u is the up quark Yukawa coupling to the SM Higgs field h, and δ br is a phase angle of order unity.
Alternatively, we can consider a back reaction potential generated by an SU (n HC ) hidden color gauge interaction which confines at scales around the weak scale [1,16]. For this, one can introduce the hidden colored matter superfields with the superpotential couplings where L is an SU (n HC )-fundamental and SU (2) L -doublet with the U (1) Y charge 1/2, L c is its conjugate representation, N is an SU (n HC )-fundamental but SU (2) L × U (1) Y singlet, and N c is its conjugate representation. At scales below m SUSY , all superpartners can be integrated out, leaving the following Yukawa interactions between the relevant light degrees of freedom: where L + L c and N + N c denote the fermion components of the original superfields, tan β = H u / H d , and is presumed to be lighter than m SUSY . Note that a nonzero Dirac mass of N + N c is induced by radiative corrections below m SUSY , giving Now this effective theory at scales below m SUSY corresponds to the non-QCD model proposed in [1,16], yielding a back reaction potential of the form V br = m 2 1 hh † cos 2 where we have expressed the axion component φ 1 in terms of the light relaxion field φ,

IV. CONCLUSION
In this paper, we have addressed the problem of huge scale hierarchy in the relaxion mechanism, i.e. a relaxion excursion ∆φ ∼ 2πf eff which is bigger than the period 2πf of the back reaction potential by many orders of magnitudes. We proposed a scheme to yield an exponentially long relaxion direction within the compact field space of N