Enhanced EDMs from Small Instantons

We show that models in which the strong $CP$ problem is solved by introducing an axion field with a mass enhanced by non-QCD UV dynamics at a scale $\Lambda_{\rm SI}$ exhibit enhanced sensitivity to external sources of $CP$ violation. In the presence of higher-dimensional $CP$-odd sources at a scale $\Lambda_{\rm CP}$, the same mechanisms that enhance the axion mass also modify the axion potential, shifting the potential minimum by a factor $\propto\Lambda^2_{\rm SI}/\Lambda^2_{\rm CP}$. This phenomenon of $CP$-violation enhancement, which puts stringent constraints on the scale of new physics, is explicitly demonstrated within a broad class of"small instanton"models with $CP$-odd sources arising from the dimension-six Weinberg gluonic and four-fermion operators. We find that for heavy axion masses $\gtrsim 100$MeV, arising from new dynamics at $\Lambda_{\rm SI}\lesssim 10^{10}$GeV, $CP$ violation generated up to the Planck scale can be probed by future electric dipole moment experiments.


Introduction
The Standard Model (SM) of particle physics has two sources of CP violation. The wellestablished and measured source of CP violation in the quark mixing sector, the Kobayashi-Maskawa phase [1], is responsible for a multitude of CP -violating phenomena observed in the quark flavor-changing transitions. At the same time, this phase induces electric dipole moments (EDMs) of neutrons and heavy atoms well below current experimental limits. The other source of CP violation, the nonperturbative parameter θ of quantum chromodynamics (QCD), is largely irrelevant for flavor physics, but tends to induce large EDMs. The nonobservation of EDMs that imply the smallness of theta, |θ| 10 −10 [2,3], contrasted with the naive expectation of θ ∼ O(1), poses a naturalness problem for the Standard Model, the strong CP problem.
There are two generic approaches to resolve the strong CP problem. The first approach involves promoting the θ parameter to a new dynamical field, the QCD axion [4][5][6][7][8][9][10], which symbolically can be represented as where G c µν is the gluon field strength, G cµν ≡ 1 2 ε µνρσ G c ρσ with c the adjoint index and f a is the decay constant of the axion field a. The QCD vacuum energy, which for small θ can be parametrically expressed as can be made to dynamically relax to the minimum of the potential V (a). In this expression, Λ QCD is the nonperturbative scale of the strong interactions, and m q is the light quark mass. As a result, any initial value of θ = a/f a will relax to the minimum of the axion potential. In the absence of additional sources of CP violation, this minimum is exactly at θ = 0, as in Eq. (2). Therefore, the neutron EDM that scales as is also relaxed to zero. Consider now additional sources of CP violation placed at some new physics scale Λ CP that we will assume to be larger than the electroweak scale (for example, this could be due to supersymmetric theories with large CP -violating phases). Integrating out the new physics at this scale will, in general, result in a number of generic consequences: 1. The theta parameter may receive additive corrections to its value, θ → θ + θ rad . Since G G is a dimension four operator, θ rad can depend only on the ratio of scales, and therefore has Λ 0 CP scaling. Potentially, this can be a large correction, but the axion mechanism will remove the theta term together with θ rad . whether there is a similar decoupling as for the standard QCD axion, where all observables from, for example, dimension six operators, scale as Λ 2 QCD /Λ 2 CP , or if there is an enhancement of CP violation mediated by the induced θ which is similar to models attempting to solve the strong CP problem using exact parity or CP symmetries.
To answer this question we compute the topological susceptibility and mixed correlators in heavy axion models that arise from two sources of CP violation: the dimension six Weinberg gluonic operator and a CP -odd four-fermion operator. Such CP -odd operators induce a linear term in θ (or equivalently a) in the axion potential leading to a shift θ ind in the potential minimum. Similar contributions were proposed in [36], and were estimated on dimensional grounds for fermionic and scalar operators in [37,38,43].
Instead of relying on dimensional analysis our computation employs a simple, noninteracting instanton (or anti-instanton) background that ignores strong coupling effects, where we are able to extract qualitative results which show that the induced theta, θ ind ∝ Λ 2 SI /Λ 2 CP . This induced shift is qualitatively different from the usual QCD axion scenario and solutions based on exact discrete symmetries due to the presence of the new scale Λ SI . While there is still decoupling in the Λ CP → ∞ limit, our results show that the induced θ can enhance the magnitude of observable EDMs, even to the point that if Λ 2 SI /Λ 2 CP is too large, the strong CP problem will reappear. Thus, models with a dynamically enhanced axion mass are subject to bounds depending on the amount of CP violation that is present at energy scales that may significantly exceed 100 TeV. Interestingly, the enhanced EDMs are potentially observable in future EDM experiments. This paper is organized as follows: in Section 2, we investigate vacuum correlators in an instanton (or anti-instanton) background with different sources of CP violation that shift the axion potential minimum. In Section 3, we consider different heavy QCD axion models with small instantons, deriving the resulting size of the induced θ and subsequent constraints on the CP -violating scale, Λ CP . We reach our conclusions in Section 4.

Instanton Correlation Functions
We begin with briefly reviewing QCD dynamics and the instanton solution that will be used to compute various instanton correlation functions. The pure Yang-Mills part of the QCD Lagrangian is given by where g is the QCD gauge coupling, θ is the QCD vacuum angle and a = 1, . . . , 8 labels the gauge adjoint representation. The BPST instanton solution [44] is given by where the instanton is located at x 0 and has a size ρ. The η a µν denote the group-theoretic 't Hooft η symbols [45]. The topological charge is defined to be where Q = 1 for the one instanton solution (5). We will next compute correlation functions in the instanton (or anti-instanton) background (5) that will be useful in obtaining contributions to EDM observables such as the neutron EDM.

Topological susceptibility
The vacuum-to-vacuum amplitude in QCD can be written as where the Euclidean action for (4) in an instanton background of charge Q [46] is given by The topological susceptibility is then introduced as [8,11,47] where G G is shorthand notation for G a µν G aµν . Since the amplitude in the |Q| > 1 instanton background becomes more exponentially suppressed, only the Q = ±1 configurations dominate the path integral. Henceforth, we refer to S E in (8) only for |Q| = 1. In the instanton background (5) we then obtain the two-point correlator where the running coupling g(1/ρ) encodes corrections from the quantum fluctuations. In (10) we have replaced the path integral over the fluctuation A µ with an integration in (11) over the collective coordinates (see Ref. [45]) where, assuming an SU (N ) gauge group, 1 the coefficient and C 1 , C 2 are order one constants (C 1 = 0.466, C 2 = 1.679 using Pauli-Villars regularization [49]). The gauge coupling running is given by In principle, we could consider an ensemble of instantons and anti-instantons [50][51][52] to compute correlation functions. However, the qualitative aspects of such an ensemble can be simply captured by one instanton and one anti-instanton [53,54], where the (anti-)instantons are assumed to be noninteracting with each other and can be justified in the weak coupling regime. Thus, we will compute correlation functions by adding the contribution from an instanton background to that in an anti-instanton background. The total contribution to the topological susceptibility, obtained by performing the x integration first that arises from (9), followed by the x 0 integration in (11), is then given by Assuming an asymptotically free theory, the integral in (14) is divergent for large instantons but can be evaluated with a IR cutoff ρ IR on the instanton size. Assuming N = 3 with ρ IR = 1/Λ QCD we obtain χ(0) ∝ Λ 4 QCD .

Fermion contributions
The introduction of fermions modifies the path integral and the collective coordinate integration. In the massless fermion limit, the pure vacuum-to-vacuum transition amplitude is zero. Instead, the instanton now causes transitions from left-handed to right-handed fermions violating the U (1) chiral symmetry so that, for example, 0|ψ Ri ψ Li |0 = 0. Thus, instantons only contribute to correlation functions in which each fermion flavor and chirality appears at least once. The effect of massless fermions is usually formulated as an "effective" Lagrangian [45,49] where the determinant is taken over the N f fermion flavors, and ψ α L,R (x 0 ) are the fermion zero modes. The constant e 0.292N f assumes Pauli-Villars regularization and the gauge coupling running (13) now includes the fermion Note that because of the explicit appearance of the fermion zero modes ψ L,R (x 0 ) in (15), there is only a contribution to the axion potential if the external fermion zero mode legs are closed. There are two ways this can occur. The first way is to assume that the fermions have an explicit mass m f (corresponding to a nonzero Higgs vacuum expectation value (VEV), v ≈ 246 GeV) that connects left-and right-handed fermion fields. The determinant in the effective action then gives a contribution ∝ (ρ m f ) N f for N f fermion flavors. This is the case for the usual contributions from "large" instantons with ρ ∼ ρ IR = 1/Λ QCD and m f Λ QCD . However, since we are interested in "small" instantons corresponding to instanton sizes (∼ 1/Λ SI ) much smaller than the inverse of the electroweak scale, a second possibility is to close the external fermion zero-mode legs in (15) with N f /2 Higgs bosons. This contribution will be proportional to the product of Yukawa couplings (times a loop factor) and is larger than the Higgs VEV contribution that now scales as . Instead of proceeding with the 't Hooft determinant operator in the effective Lagrangian (15) we will follow the approach taken in Refs. [38,40] and directly compute the vacuum-to-vacuum amplitude by including the Higgs-fermion Yukawa interaction in the path integral.
Consider a Higgs field H which couples to N f flavors of massless fermions with the following Euclidean action where S H is the quadratic (free) part of the Higgs action and y i are the Yukawa couplings. The Yukawa couplings, or equivalently the fermion masses, have been redefined to be real with their phase included inθ = θ + Arg DetM q , where M q is the quark mass matrix. The vacuum-tovacuum amplitude now takes the form where the action S E is defined in (8) with θ →θ. The first line in (17) shows the collective coordinate integration arising from the gauge field part of the path integral and the second line contains the Higgs and massless fermion contributions to the path integral with S (0) ψ the quadratic (free) part of the fermion action. Integrating over the fermionic fields introduces the factor e 0.292N f and the running gauge coupling now contains fermionic contributions via b 0 → b 0 − 2/3N f . Finally, the path integral over the Higgs field gives a nonzero contribution to the amplitude provided all Higgs fields are contracted where (N f − 1)!! is the number of Higgs contractions and the quantity I is given by [38,40] In the second line of (18) we have substituted for the scalar Feynman propagator ∆ H (x 1 − x 2 ) and the fermions have been replaced with their respective zero mode expressions given in [49].
Note that for an instanton background we have two zero modesψ j,L in an anti-instanton background) where the subscripts L, R, which are suppressed hereon, denote leftand right-handed fields, respectively. Thus, combining (18) and (17) gives the final expression (assuming m H ρ 1) with S E defined in (8) (assuming θ →θ), and The expression (19) shows how the instanton density in the vacuum-to-vacuum amplitude is modified in the presence of massless fermions and a Higgs-fermion Yukawa interaction. As expected, the amplitude vanishes if any Yukawa coupling is zero. Thus, the topological susceptibility (14) in the presence of massless fermions is obtained by the substitutions In the case of "large" instantons associated with the scale 1/Λ QCD , the expression for the vacuum-to-vacuum amplitude differs from (19). As already mentioned, each light fermion (m f Λ QCD ) introduces an e 0.292 ρ m f factor 2 . This can be seen via the first line in (18) where ∆ H can be replaced by v 2 , which just gives I = v 2 , and hence: where the product runs only over N L light fermions and m i = y i v/ √ 2.
2 In QCD, χ(0) ∝ m f , whereas the χ(0) resulting from (21) ∝ m N L f . The difference can be understood in terms of instanton-(anti-)instanton interactions-either via mixing between the fermion zero modes of the instanton with those of the anti-instanton [55], or using 't Hooft vertices with fermion legs joined between an instanton and anti-instanton [51].

Weinberg gluonic operator
The Weinberg operator is a purely gluonic, CP odd, dimension six term given by O W = GG G [56] that leads to the Lagrangian term where Λ W is an effective UV scale. The operator (22) can induce a shift in the axion potential minimum, which can be computed by considering the mixed correlator [13,57] In the instanton background (5) we obtain where f abc are the structure constants. Note that for an SU (N ) gauge group, the SU (2) instanton solution is embedded in the top left corner of the N × N matrix of SU (N ) generators. Thus, the sum in (24) only gives nonzero contributions for a, b, c = 1, 2, 3. Furthermore, Again performing the integrals first over x and then x 0 gives where we have also included the anti-instanton contribution.
In the presence of fermions, χ W (0) is obtained by making the substitutions C[N ] → C f [N ] for small instantons (or by introducing the factor (ρ m f ) N L , as in (21) for large instantons), θ →θ and b 0 → b 0 − 2/3N f in the running gauge coupling g(1/ρ).

Four-fermion operators
Another class of dimension six operators which can affect the axion solution are the four-fermion operators. Such operators are suppressed by an effective mass scale Λ F and given by where λ ijkl are complex coefficients with flavor indices i, j, k, l. Note that the spinor and electroweak structure has been suppressed in (27), although it is straightforward to incorporate these details. Of particular interest is the spinor structure of (27) resulting in CP violation. These are operators of the type O F,ijkl =ψ i iγ 5 ψ jψk ψ l which are anti-Hermitian with the corresponding λ ijkl purely imaginary.
The CP -violating effect arising from (27) can be obtained by including the four-fermion interactions in the path integral (17). These operators allow for new ways to close the fermion legs in the 't Hooft vertex, as depicted in Figure 2. The largest contribution arises from just one insertion of O F , as shown in Figure 2(a), while more insertions of the four-fermion operator, such as in Figure 2(b) are suppressed by powers of Λ F . Similar to the definition (23) for χ W (0) we can define a fermion mixed correlator The only operators contributing to the fermion path integral are those with two pairs of flavor indices (i = j = k = l or i = l = k = j), i.e. O F,iijj , and O F,ijji , both of which are hereon generically referred to as O F,ij with the corresponding coupling constant λ ij ≡ λ iijj (or λ ijij ). The explicit expression for such a generic operator O F,ij can be computed as where we have also included the effect of the anti-instanton. The part of O F,ij contributing to the path integral in the instanton background is iψ † L,i ψ R,i ψ † L,j ψ R,j , while in the anti-instanton background (where G G → −G G) it is −iψ † R,i ψ L,i ψ † R,j ψ L,j . These two contributions add up 3 to give the factor of 2i in (29).
The result (29) can also be understood in terms of the results (17) and (18) from the fermionic path integral, up to the overall ratio of couplings. If we assume that O F is generated by a heavy scalar of mass Λ F , interacting with Standard Model quarks via Yukawa interactions, (18) implies a factor of 12π 2 ρ 2 /5π 2 Λ 2 F ρ 4 = 12/5ρ 2 Λ 2 F relative to the expression (19), which matches the factor inside the integral. The factor 1/(N f − 1) arises from having a fewer number of contractions-(N f − 3)!! compared to (17), assuming only one insertion of the operator O F,ij .
Furthermore, notice that y i and y j have been explicitly factored out of (29) to write the result in terms of C f [N ] defined in (20). For −iλ ij ∼ 1, this shows that the effect of the fourfermion operator, being ∝ 1/y i y j , is most enhanced for the up and down quarks compared to that from the Weinberg gluonic operator or the second and third generation quarks. However, the four-fermion operator coefficient λ ij can be chirally suppressed by Yukawa couplings [43]. For example, such four-fermion operators with a chiral suppression can arise from the overlap of fermion profiles in extra dimension models [58]. Thus, we will henceforth assume that −iλ ij ∝ y i y j so that the effect of the four-fermion operator is similar to that of the Weinberg gluonic operator as well as the contributions from the other generations of quarks.
Assuming −iλ ij = y i y j /2, we then have N f (N f −1) contributions of the fermion susceptibility (29) for both types of operators O F,iijj , and O F,ijji , each. Thus, for N f = 6 we obtain Using (30) we will place limits on a generic scale Λ F that represents all of these fermion effects. Finally, note that in supersymmetric theories the operator O F can arise from a dimensionfour term in the superpotential [59]. After integrating out the scalar superpartners this leads to a four-fermion term with 1 where Λ UV is the UV scale of the superpotential term and m SUSY is the supersymmetry-breaking scale of the scalar superpartners. The bounds on Λ F can thus be interpreted as bounds on the scalar superpartner masses.

Induced Theta
Using the results in Section 2 we can now obtain an estimate for the shift in the axion potential minimum due to CP -odd operators. In the presence of the Weinberg operator the axion potential is modified by a linear term in the axion field where we have promoted the theta angle to the axion field,θ → a/f a . This leads to a shift in the potential minimum by an amount In the case of four-fermion operators the linear potential term again causes a shift in the potential minimum given by (33), with χ W (0) replaced by χ F (0). The induced θ then directly contributes to EDM observables such as the neutron EDM where The experimental limit arising from the neutron EDM gives the constraint which can now be used to obtain constraints on various heavy axion scenarios 4 .

QCD
We first consider the effect of dimension six operators in QCD with N L light fermions (i.e. m f Λ QCD ). The induced θ (33) that arises from including the Weinberg operator is given by where ξ W = 384π 2 /5, b 0 is the β-function coefficient and the CP -violation scale Λ CP is identified with Λ W . Note that in (36) the product of all light quark masses cancel and the induced θ becomes small (or decouples) as Λ W → ∞. Imposing the constraint (35) for QCD (b QCD 0 = 9, N L = 3 and Λ QCD ≈ 300 MeV), gives the limit Λ W 10 6 GeV on the effective scale of the Weinberg operator.
For the case of the CP -odd four-fermion operator, the 't Hooft vertex now has two fewer factors of ρ m f compared to the topological susceptibility resulting from (21). This gives a bound similar to Λ W when there is no chirality suppression in the four-fermion operator, otherwise the Λ F bound is much weaker. A calculation for θ ind using the chiral anomaly can be found in [60], which agrees with our estimate of the bound on Λ F within an order of magnitude.
As such, current constraints on the neutron EDM correspond to new CP -violating physics at ∼ 10 6 GeV. Thus, future neutron EDM experiments can probe new CP -violating sources at scales ranging from ∼ 10 6 − 10 9 GeV, beyond which the SM contribution due to the CKM phase becomes comparable in size.

Product gauge group
A heavy axion can be generated by extending the QCD gauge group into a product gauge group SU (3) k = SU (3) 1 × SU (3) 2 × · · · × SU (3) k which is spontaneously broken at a scale Λ SI [39,40]. Small instantons at the scale Λ SI associated with the product gauge groups lead to this enhancement. The SM quarks are assumed to be charged under only SU (3) 1 . In addition, there are k axions, labeled by i, which couple to the k SU(3) G G terms with decay constants f a i , eliminating the k theta terms.
At the scale Λ SI the QCD gauge coupling α is matched to the SU (3) k gauge couplings, α i via the relation 1 This relation implies that each individual coupling α i must be larger than the QCD coupling at the scale Λ SI . Therefore, the larger couplings α i (Λ SI ) can make the small instanton effects dominate over the usual QCD large instantons. This effect is most dominant in the limit k 1, where the axion masses scale as m a 1 ∼ Π f y f Λ 2 SI /f a 1 (with y f the quark Yukawa couplings) and m a i ∼ Λ 2 SI /f a i for i = 2 . . . , k, showing that the lightest axion mass (m a 1 ) can remain much heavier that the QCD axion mass for Λ SI Λ QCD . For concreteness, let us consider the case with small k, where there is some perturbative control and the instanton (or anti-instanton) background still gives us qualitatively accurate results. Assuming the product gauge group is broken by scalars with a VEV, v φ , the effective cutoff for the instanton size then becomes 2πv φ , in contrast to the naive expectation, Λ SI [40]. The constraint (35) can then be used to obtain limits on the scales associated with the sources of CP violation from the Weinberg and four-fermion operators. Since the QCD instanton contribution to χ W,F (0) is suppressed by at least Λ 2 QCD /Λ 2 W,F , the small instanton contribution from the UV gauge group dominates and results in where ξ F = 24N f /5, ξ W is defined under (36) and we have assumed Λ SI ≈ v φ in the second expression in (38). The constraint (35) then implies Λ SI /Λ W 10 −8 and Λ SI /Λ F 10 −7 or Λ SI 10 10 (10 11 ) GeV for Λ W (Λ F ) = M P where M P = 2.4 × 10 18 GeV is the (reduced) Planck mass 5 , b 0,1 = 13/2 and b 0,k = 21/2. For i = 2, . . . , k − 1, the same expression (38) holds with b 0,i = 10, and v φ → √ 2v φ , which does not change the bounds significantly 6 . Note that if UV couplings are included in (22) then the effective scale Λ W can be larger than M P . Assuming f a > Λ SI , the limits on Λ W,F correspond to a maximum possible axion mass enhancement of ∼ 10 7 for k = 3 relative to the QCD axion [39,40]. As such, axion masses m a 100 MeV with f a 10 7 GeV [61,62] can be explored in future experimental searches.
However, when f a < Λ SI , we need to UV complete the dimension five axion-G G coupling and explain the PQ breaking. This can be done in a minimal KSVZ-type scenario [7,8], by introducing a single heavy Dirac fermion Ψ, with mass m Ψ , charged under the U (1) P Q symmetry, which changes the instanton measure by a factor of e 0.292 ρ m Ψ . Combining this with the 5 The difference in these two bounds results from the size of the different prefactors ξW,F , where ξW results from the large number of color contractions in (23), while ξF arises from the smaller flavor multiplicity of the four-fermion operator (27). 6 It is possible that the axion mass could instead be dominated by QCD large instantons. But in this case the CP violation arising from small instantons of the product gauge group gives the much weaker constraint that ΛSI/ΛW 10 −8 × ma,QCD/ma 1 . For instance, assuming ma,QCD/ma 1 = 10 3 implies that ΛSI 10 13 GeV for ΛW = MP . contribution arising from the running of the gauge coupling between m Ψ and Λ SI , the topological susceptibility (or any similar correlator) is modified to where the Yukawa coupling between Ψ and the PQ scalar is assumed to be order one, i.e. m Ψ ≈ f a . Since m 2 a ∝ χ(0) this suppresses the axion mass enhancement by an amount (f a /Λ SI ) 1/6 [63]. Thus for the experimentally interesting region of m a 100 MeV and f a 10 7 GeV, the axion mass enhancement is reduced by up to a factor of 10 when f a < Λ SI .
A similar result is also obtained for an enlarged color group [32,35,36] where Λ SI is identified with the scale where the enlarged symmetry group is broken and the appropriate b 0 is used. In all these cases, there is again a nondecoupling effect that depends on the ratio Λ SI /Λ W,F .

Mirror QCD
A heavy axion can also be obtained by assuming that there exists a Z 2 mirror copy of QCD that becomes strong at a scale Λ QCD (≡ Λ SI ) Λ QCD [22,23,30,31,34]. The axion is Z 2 neutral and couples to both QCD and mirror QCD, via the interaction where G µν is the mirror QCD field strength. The axion now receives contributions from the mirror QCD instantons (which are small in size relative to those from QCD) and gives rise to limits on higher dimensional operators with scales Λ W,F involving gluons and fermions in the mirror sector. The mirror QCD expression for the induced θ due to the Weinberg operator can be obtained by substituting Λ SI in (36). This leads to the bounds Λ SI /Λ W 10 −7 or Λ SI 10 11 GeV for Λ W = M P , assuming the mirror Higgs VEV, v Λ SI such that QCD is a pure Yang-Mills theory at Λ SI with b QCD 0 = 11. These bounds for the Weinberg operator do not change appreciably if this assumption is relaxed.
The induced θ from the four-fermion operator can be obtained by considering N L ≥ 2 light flavors in QCD . Applying the QCD result (21) for QCD then gives where we have taken v ≈ Λ SI in the last expression in (41). Assuming b 0 = 9 and N L = 3, implies Λ SI /Λ F 10 −5 , or Λ SI 10 13 GeV for Λ F = M P . Again, the difference in the Λ W,F bounds arises from the different color and flavor multiplicity factors.

5D Small Instantons
Another way for the QCD coupling to become large at a UV scale and increase the effect of small instantons is to consider a 5D model where QCD gluons propagate in a fifth dimension of size R. The axion can be identified with a UV boundary localized field that couples to QCD via a coupling proportional to 1/f a , with f a an independent parameter of the theory. This allows the decay constant to be either above or below the small instanton scale and allows for more general possibilities. Above the scale 1/R the QCD coupling increases in strength until the coupling becomes strong at the cutoff scale Λ 5 which is defined by the relation [41] where α = g 2 /(4π) and ≤ 1 is a perturbativity parameter 7 . The small instanton scale can be identified as Λ SI ≡ Λ 5 . The 4D effective action is approximately given by [41] S eff ≈ 2π where the power-law term R/ρ arises from summing over the 5D Kaluza-Klein gluons. Thus, small instantons of size 1/Λ SI ρ R can now reduce the effective action and contribute greatly to the path integral. Using an approximate expression for the integrals in (14) and (26) with the effective action (43), the induced θ from 5D small instantons is where ξ W and ξ F are defined under (36) and (38), respectively. The induced θ no longer necessarily decouples in the limit Λ SI , Λ W,F → ∞. Imposing the constraint (35) leads to the limit Λ SI /Λ W (Λ F ) 10 −7 (10 −6 ). For Λ W (Λ F ) = M P this implies an upper bound Λ SI 10 11 (10 12 ) GeV on the 5D strong coupling scale. The limit on Λ W,F from an exact numerical evaluation of θ ind is shown in Figure 3. We see that the limit on Λ W,F deviates from (44) for small 1/R (and hence small Λ 5 ). The limits on the ratio Λ SI /Λ W,F imply that for the case when Λ W,F ∼ Λ 5 (= Λ SI ), the dimension six terms would need to be generated from some new physics in the UV completion of the 5D model with an additional suppression in the otherwise order-one coefficients.
The corresponding range of axion mass enhancement is depicted in Figure 4. Note that both effects of small instantons -the enhancement of the axion mass and the shift in the axion potential minimum due to CP -violating operators -are dominant only for large 1/R, since eventually large (QCD) instantons dominate the susceptibility at small values of 1/R. Furthermore, when f a < Λ SI the axion mass enhancement is reduced by the factor (f a /Λ SI ) 1/6 as obtained from (39). This means that in the experimentally viable region of m a 100 MeV and f a 10 7 GeV [61,62], the axion mass enhancement is reduced by up to an order of magnitude, as can be seen in Figure 4, where we have taken f a = 10 6 GeV as a representative value.

Enhanced EDMs
Compared to QCD, the small instanton contributions provide an enhancement to the EDMs due to CP -violating sources. In particular, using (38), (41) and (44) we see that the neutron EDM (34) is enhanced by a factor of Λ 2 SI /Λ 2 QCD compared to the θ induced from new CP -odd sources in QCD (see (36)). Therefore, measuring the neutron EDM can be interpreted as a probe of the small instanton scale, Λ SI . For example, if Λ W,F = M P , this corresponds to modified strong dynamics at scales of order Λ SI ∼ 10 8 − 10 11 GeV, where the lower limit represents a neutron EDM value equivalent to the Standard Model CKM contribution. Furthermore, if the CP -violating sources appear at scales lower than the Planck scale, then any new contribution due to small instantons will appear at even lower scales 10 4 GeV Λ SI 10 8 GeV, where the model-dependent lower limit corresponds to the scale of axion mass enhancement.
Finally note that when f a Λ SI , the UV completion of the dimension five axion-gluon coupling does not affect the predictions for the induced θ. Since the neutron EDM (34) depends only on the ratio of the mixed correlators with the topological susceptibility, the suppression factor in (39) cancels, leaving the results for the induced θ unchanged.

Conclusion
The QCD axion solution provides an elegant mechanism for solving the strong CP problem in such a way that an arbitrarily large amount of CP violation at UV scales Λ CP can be sufficiently decoupled as Λ CP → ∞. This is in contrast with solutions to the strong CP problem that invoke exact discrete symmetries. For these solutions there is a nondecoupling of the additional sources of CP violation, which means that arbitrarily large amounts of CP violation cannot be tolerated at UV scales in models with exact parity or CP symmetry.
Heavy axion models represent a qualitatively different class of solution to the strong CP problem in which new dynamics at some UV scale Λ SI magnifies the effect of small instantons (which are normally exponentially suppressed), giving rise to a new contribution and enhancement of the axion mass. This has led to renewed interest in axion searches outside the usual QCD axion mass window. However, in the presence of additional sources of CP -violation, the enhanced effect of small instantons could also lead to enhanced EDM observables such as the neutron EDM as well as possible nondecoupling effects.
We have estimated these effects by calculating the topological susceptibility and mixed correlators in the presence of two CP -violating dimension six operators: the Weinberg gluonic operator and a CP -odd four-fermion operator. The calculation is performed using an instanton (or anti-instanton) background where Standard Model fermion chiral zero modes in the 't Hooft vertex are closed with the Higgs boson. Identifying the scale of the additional sources of CP violation with Λ CP we find that the axion potential minimum shifts by an amount θ ind ∝ Λ 2 SI /Λ 2 CP in several heavy axion models, where Λ SI is the scale where small instanton effects dominate. This result reveals that unlike the minimal QCD axion models, the amount of decoupling is limited, although not as restrictive as models with exact discrete symmetries. Imposing the neutron EDM derived limit |θ| 10 −10 , we obtain the constraint Λ SI /Λ CP 10 −8 , which is stronger than the naive estimate of 10 −5 due to sizable prefactors that depend on the particular heavy axion model. In particular, for a benchmark value of Λ CP M P requires Λ SI 10 10 GeV (as can be seen in Figure 4 for the 5D small instanton model).
The modification of the decoupling behavior is a direct consequence of the new dynamical scale Λ SI . Our results therefore imply that EDM observables such as the neutron EDM can be enhanced in heavy axion models up to the current experimental limit d n 10 −26 e·cm. This compares with the SM CKM prediction (∼ 10 −32 − 10 −31 e·cm). Thus, besides axion searches, EDM observables provide another probe of UV scales in heavy axion models associated with new dynamics, assuming that this class of models plays any role in solving the strong CP problem.