Finite modular axion and radiative moduli stabilization

We propose a simple setup which can stabilize a modulus field of the finite modular symmetry by the Coleman-Weinberg potential. Our scenario leads to a large hierarchy suppressing instanton-like corrections $e^{2\pi i\tau}$ and to a light axion identified as $\mathrm{Re} \tau$, where $\tau$ is the modulus field. This stabilization mechanism provides the axion solution to the strong CP problem. The potential has a minimum at a large $\mathrm{Im}\tau$ which suppresses explicit $U(1)_{\mathrm{PQ}}$ violation terms proportional to $e^{-2\pi {\mathrm{Im}\tau}}$, and hence the quality of the axion is ensured by the residual symmetry associated with the $T$-transformation, $\tau \to \tau +1$, around the fixed point $\tau \sim i\infty$.

In the models with finite modular flavor symmetries, the value of modulus τ plays an essential role to fit to the data and is often treated as a free parameter.Whereas in a UV theory, its value should be fixed dynamically through modulus stabilization.There are studies on the moduli stabilization with the three-form fluxes [48,49], the negative power of modular form [50,51] 1 , and the general superpotential invariant under the modular symmetry SL(2, Z) [53][54][55][56][57], utilizing the Klein j function [58].The latter two mechanisms will need non-perturbative dynamics to generate those potentials.Recently, effect of the radiative correction to the treelevel stabilization is discussed in Ref. [59].
In this work, we point out that the modulus can be stabilized only by the Coleman-Weinberg (CW) potential generated by couplings with matter fields.This mechanism provides a perturbative way to stabilize a modulus through the supersymmetry breaking.For illustration, we shall consider a simple model with vector-like quarks which transform as non-trivial singlets under the finite modular flavor symmetry.The resultant potential has a global minimum at Im τ ≫ 1, where the residual Z T N symmetry associated with the T transformation, τ → τ + 1, is unbroken.That can lead to a large hierarchy suppressing instanton-like correction, i.e., e −2πIm τ ≪ 1.An interesting feature of this mechanism is that the Re τ direction can be identified as the QCD axion due to the accidental Peccei-Quinn (PQ) symmetry originated from the Z T N residual symmetry.We shall discuss the condition that the axion solution to the strong CP problem is not spoiled due to the CW potential, i.e. the axion quality is good enough.
This paper is organized as follows.The finite modular symmetry is briefly reviewed in Sec. 2. In Sec. 3, we discuss the mechanism of the modular stabilization by utilizing the CW potential, and then we argue existence of the QCD axion in the setup in Sec. 4. Section 5 concludes.

Finite modular symmetry
We consider the series of groups where N ∈ N is called level and Γ := SL(2, Z) = Γ(1) is a group of 2 × 2 matrices with determinant unity.The modular transformation γ ∈ Γ(N ) for a complex parameter τ is defined as The modular group is generated by three generators The finite modular symmetry is defined as the quotient group Γ N := Γ/Γ(N ) where Γ : Here Z R 2 is the Z 2 symmetry generated by R.Under Γ N with N < 6, the generators satisfy (2.4) There are Abelian discrete symmetries Z S 2 , Z ST 3 and Z T N in associated with the S, ST and T generators which are unbroken at τ = i, w := e 2πi/3 and i∞, respectively.The finite modular symmetries are isomorphic to the non-Abelian discrete symmetries, e.g.Γ 3 ≃ A 4 .A modular form Y where ρ(r) is the representation matrix of Γ N .We assume that the chiral superfield Q with weight −k Q and representation r Q transforms as More detailed discussions can be found in e.g.Refs.[10,11].

Radiative moduli stabilization
We consider a simple supersymmetric model with the following Kähler potential and superpotential: where h ∈ N. The reduced Planck mass M p = 2.4 × 10 18 GeV is set to unity.The chiral superfield and is assumed to be a singlet under Γ N for simplicity.Here, Q i and Q i are vector-like pairs under the SM gauge group.The weight of the modular form k i = k Q i + k Q i − h and the representation r i are chosen so that the combination e K |W |2 in supergravity is invariant under Γ N .We introduce the vector-like mass parameter M Q i which could be replaced by vacuum expectation values of fields.Throughout this work, we assume that the SM fields are trivial singlets under the modular symmetry Γ N for simplicity and they have no effects in the following analysis.We shall show that the modulus τ can be stabilized without introducing a superpotential for the modulus τ , but with the 1-loop CW potential with Here, we consider the MS scheme with µ being the renormalization scale.The first (second) term in the parenthesis corresponds to the scalar (fermion) contribution.The supersymmetric mass m Q i is multiplied by the factor (−iτ + iτ † ) k i /2 due to the canonical normalization.We introduce the soft supersymmetry breaking mass squared m 2 i for the scalar component of Q i which is assumed to be independent of τ for simplicity.This is the case if supersymmetry is broken by a mechanism irrelevant to the modulus τ .We shall first see the simplest case analytically at large Im τ , and then see the results numerically later.Throughout this paper, we assume that the mass parameters are common for Q i 's, i.e.M Q i =: M Q , m i =: m 0 and k i =: k for simplicity.Note that the tree-level scalar potential in supergravity is irrelevant when

Simplified analysis
We argue the model of radiative stabilization of the modulus τ where there is only one pair of (Q, Q), so we omit the index i.The singlet modular form of Γ N can be expanded as where r = 1 t is a singlet whose charge is 0 ≤ t < N under the Z T N symmetry 2 .Assuming |q| = e −2πIm τ ≪ 1 and m 2 0 ≪ M 2 Q , the CW potential is given by where x := 2Im τ , p := 2πt/N and MQ := c 0 M Q .At the leading order, there is no potential for Re τ which will be important for the axion interpretation discussed in the next section.The first derivative of the potential is given by The point x = k/p is a maximum for sufficiently large k/p, and hence this potential can have a minimum at where the second parenthesis of Eq. (3.6) is vanishing.Here, W(z) is the Lambert function satisfying W(z)e W(z) = z which has two real values for −e −1 < z < 0, and the larger one can be a solution where our approximation |q| ≪ 1 is valid.Taking µ = MQ and p = 2π/3 , the location of the minimum is x 0 = 3.9, 7.9, 12 and 16 for k = 6, 8, 10 and 12, respectively.x 0 increases slightly by decreasing µ.Thus we find that there is the minimum at Im τ ≫ 1 in the CW potential.Note that this minimum exists only for t ̸ = 0 where the modular form 1t is a non-trivial singlet.For the CW potential from the trivial-singlet modular form, the potential increases as Im τ increases in contrast to that from the non-trivial one, and thus there is no minimum at Im τ ≫ 13 .See also the potential in Ref. [52].Therefore the dominant CW potential should be generated from the non-trivial singlet one for the existence of the minimum at Im τ ≫ 1, so that the potential tends to approach to zero for a sufficiently large Imτ .

Numerical analysis
We shall study the potential Eq. (3.2) numerically without relying on the q-expansion in Eq. (3.4).For illustration, we consider Γ 3 ≃ A 4 as the finite modular group, and two pairs of vector-like quarks whose superpotential is given by We can easily choose the representations and modular weights such that only those two terms are allowed but mixing terms like Q1 Q 2 are forbidden.An explicit assignment of representations and modular weights will be shown in Eq. (4.13) when we shall discuss the vector-like quark decays.The modular forms of weight 12 are given by where the functions Y 1,2,3 are defined as [1] Y with η(τ ) being the Dedekind eta function.The q-expansion of the modular forms are given by4 The modular weights are chosen to k = 12 so that the strong CP problem is solved with keeping the modulus mass heavy, as discussed in the next section 5 .Here, we introduce the two vector-like pairs, so that the mixed anomaly of A 4 with the QCD is canceled [61,62].The singlets are trivial under the S-transformation, and transform as 1t under the Ttransformation.Hence, in our setup, det(ρ(T )) = w 1+2 = 1 and the anomaly is canceled.The anomaly associated with the modular weight can be canceled by the Green-Schwarz mechanism with an another gauge coupling modulus or by adding log Y (k) 1 0 (τ ) to the gauge kinetic function with a certain coefficient, where Y (k) 1 0 (τ ) is a trivial singlet modular form and could be explained by threshold corrections from heavy modes [63].
Figure 1 shows the shape of the potential along the Im τ direction.Here, we take m 2 /M 2 Q = 10 −8 and µ/M Q = 10 −2 so that the minimum resides at Im τ ≃ 13.The red solid (blue dashed) line corresponds to Re τ = 0 (−0.5).Note that the fundamental domain is Im τ ≥ 1 ( √ 3/2) at Re τ = 0 (−0.5).In the left panel, the origin of the horizontal axis is √ 3/2 and the global picture of the potential is shown.The right panel is the shape of the potential around the global minimum at large Im τ .For Im τ ≲ 1, where the q-expansion is not efficient, there is a local minimum at τ ∼ w, but V CW (τ = w) = 0, since Y (12) 1 1,2 (w) = 0, is shallower than the global minimum at Im τ ∼ 12 where the potential value is negative.The potential clearly depends on Re τ at small Im τ , while it is independent of Re τ within numerical precision.Thus, the CW potential in the Re τ direction is approximately flat near the global minimum at Im τ ≫ 1.As a result, our scenario of modulus stabilization can lead a light axion as well as a large hierarchy by |q| = e −2πIm τ ∼ 10 −36 .We will confirm this flatness analytically in the next section to discuss the quality of the axion solution to the strong CP problem.

Axion solution and its quality
We have seen that the CW potential Eq. (3.2) induced by the superpotential Eq. (3.8) is very flat in the Re τ direction.The stabilization mechanism can realize the axion solution to the strong CP problem by assuming that the matter fields are vector-like quarks as in the KSVZ axion model [64,65].The effective θ-angle θ in the QCD is given by where θ 0 is a constant and ϕ := 2πRe τ .The ϕ dependence appears as a result of the chiral anomaly of the QCD with the approximate U (1) PQ , which originates from the residual Z T 3 symmetry as shown below.For Im τ ≫ 1, the Yukawa coupling is given by so we can find an accidental U (1) PQ symmetry as where α is a real transformation parameter.Hence, the PQ charge of Q t Q t is −t/3, and the U (1) PQ is anomalous.This symmetry is ensured from the residual Z T 3 symmetry corresponding to α = 2π, and the explicit breaking effects of the U (1) PQ are from q = e 2πiτ which is invariant under Z T 3 but violates U (1) PQ 6 .Therefore, Re τ can be identified as the QCD axion.Since the kinetic term of the modulus is given by7 the axion decay constant is f a = √ hM p /(2πx 0 ) ∼ O (10 16 ) GeV which requires a fine-tuning of the initial condition or early matter domination [66], to avoid the overabundance of the axion.
The axion potential is given by Here, ∆V := V CW | x=x 0 is the axion-dependent part of the CW potential which can spoil the axion solution.The shift of the θ-angle due to ∆V is estimated as with where ϵ := e −πx 0 /N .In our model, k = 12 and N = 3.Here, m 2 Q 2 is the mass squared of Q 2 defined in Eq. (3.3) with taking x = x 0 and dropping the negligible corrections at O (q). β t is the ratio c 1 /c 0 in the expansion of the modular forms Y 1 2 , respectively.Note that the O (q) correction proportional to m 2 Q 1 is canceled at the minimum x = x 0 and thus the leading term is This shift of the angle should be less than O (10 −10 ) to be consistent with the measurement of the neutron electric dipole moment [67].A possible modulus τ -dependence of Λ QCD from the gauge kinetic function will also be suppressed by O (q), and thus is sub-dominant for the stabilization.Similarly the ϕ dependence of ∂ ϕ θ is also negligible.In the A 4 model discussed in the previous section, we obtain Thus the axion quality is ensured if ϵ ≲ 10 −12 , where Im τ ∼ 13 as shown in Fig. 1.The required tuning becomes mild if sin θ 0 ≪ 1 which might happen when the (generalized) CP is conserved at some level as discussed in e.g.Ref. [56] 8 .In such a case, the strong CP problem is partially solved by the modular axion ϕ ∼ Re τ .Before closing, we discuss the masses of the modulus X ∝ Im τ and the vector-like quarks Q 1,2 .The modulus mass may need to be heavier than O (10 TeV), so that the modulus decay occurs before the big-bang nucleosynthesis and the moduli problem is avoided [70][71][72][73].After canonically normalizing the kinetic term Eq. (4.4), the modulus mass is given by Hence the modulus mass is related to ∆θ as , in the A 4 model.Thus the modulus mass decreases as the axion quality is higher by ϵ → 0.
For ∆θ/ sin θ 0 ∼ 10 −10 , ϵ ≲ 10 −12 is required to avoid the moduli problem 9 .The mass of the lighter vector-like quark Q 2 , whose charge under Z T 3 is 2, is given by thus M Q ∼ M p is necessary for m Q 2 ≳ O (TeV) to be consistent with the LHC constraints [82][83][84][85][86]. Conversely, if we require m Q 2 ≳ TeV and MQ ≲ M p , the hierarchy parameter is bounded from below as ϵ ≳ 10 −8 × x −k/4 0 , and thus larger weight is preferred to allow ϵ ∼ O (10 −12 ) without having the too light vector-like quark.Figure 2 shows the masses of the modulus X ∝ Im τ and the vector-like quarks.As Im τ increases, the axion quality becomes better, whereas the modulus and vector-like quark masses decrease, and hence there is the lower bound ∆θ ≳ 10 −13 for sufficiently heavy vector-like quark Q 2 .
The vector-like quarks should be able to decay into SM particles.For illustration, if the vector-like quarks have the same quantum number as the down-type quarks, denoted by d, we can write down the superpotential where H d and q are respectively the down-type Higgs doublet and the doublet SM quark.Here, the representations and weights of quarks are set to and those of H d is the trivial-singlet with weight 0. We omit the flavor indices of the SM quarks.With this assignment, the interactions in Eqs.(3.8) and (4.12), as well as the SM Yukawa coupling yH d qd with y being a constant, are realized, while the Q i d is vanishing because of the negative weight.The size of mixing s Q i is estimated as where i = 2 in the second line, and we used Y ∼ ϵ t .Thus the vector-like quarks decay promptly at the collider scale.It is noted that the CW potential induced by the mixing term is negligible because ⟨H d ⟩ ≪ M Q .

Summary and discussions
In this work, we point out that the modulus of the finite modular symmetry can be stabilized by the Coleman-Weinberg (CW) potential.For illustration, we studied the model with Γ 3 ≃ A 4 symmetry and two vector-like quark pairs which transform as non-trivial singlets, namely 1 1 and 1 2 .This is the minimal possibility to cancel the mixed QCD anomaly of the finite modular symmetry, but to induce that of the U (1) PQ symmetry.The CW potential has the global minimum at Im τ ≫ 1 if the corresponding modular form is non-trivial singlet under A 4 , where the residual symmetry Z T 3 remains unbroken.Since the potential along the Re τ direction is extremely flat due to this residual symmetry, we can regard Re τ as the QCD axion to solve the strong CP problem.The accidental U (1) PQ arises due to the Z T 3 symmetry and hence the PQ breaking effects are controlled by Γ 3 which is a different situation argued in Ref. [87].We examined the condition to ensure the quality of this finite modular axion, and correlate with the masses of the modulus X ∼ Im τ and vector-like quarks.Interestingly, the modulus X and the lighter quark Q 2 are expected to be O (TeV) scale for ∆θ < 10 −10 and thus could be probed by cosmology and collider experiments.This mechanism can be generalized to other modular forms as long as the CW potential is dominated by that with non-zero Z T 3 -charge and the potential converges to zero for Im τ ≫ 1.
The modular A 4 symmetry in our scenario can not be used to explain the hierarchical structure of the SM fermions, as studied in Refs.[36][37][38][39][40][41][42][43][44][45][46][47], since the hierarchy parameter ϵ is O (10 −12 ) and is too small.Such a tiny ϵ is required due to the correlation of the axion quality ∆θ with the modulus mass in Eq. (4.10).This relation can be relaxed if the τ -independent angle θ 0 is small or the level N is large.In the latter case, there exists a small hierarchy |e 2πiτ /N | in the SM sector and a large one |e 2πiτ | in the vector-like quark sector. 10 .Thus, we could construct a model which explains the SM fermion hierarchies by the modulus stabilized by the mechanism proposed in this paper.An explicit model building is subject of our future work.
We have proposed a new scenario to lead to a large hierarchy e −2πIm τ ∼ O (10 −36 ) and a light axion by stabilizing the modulus at Im τ ≫ 1.Although we have applied it to solve the strong CP problem by assuming Q and Q as vector-like matter fields with the QCD colors, our scenario would be useful to explain other large hierarchies, which would be required for the proton stability, tiny neutrino masses, CP/flavor violations, quintessence and so on, together with light axions by assuming Q and Q as visible or hidden matter fields.We would study it elsewhere.
In this work, we assume that the soft scalar masses of the vector-like quarks are independent of the modulus τ and supersymmetry is dominantly broken by other sector, so that the supersymmetry breaking sector does not contribute to the stabilization of the modulus τ .In addition, there would be contributions from gravitational instantons [88] which would change the potential structure.Those effects and other dynamical effects including stabilization mechanisms of other moduli, and uplifting of the vacuum energy if necessary, may become important in complete models such as superstring, but beyond the scope of this paper.

r
(τ ) with a modular weight k and representation r transforms under Γ N as