Closer look at the matching condition for radiative QCD θ parameter

: In this paper, we scrutinize a radiatively generated QCD θ parameter at the two-loop level based on both full analytical loop functions with the Fock-Schwinger gauge method and the effective field theory approach, using simplified models. We observe that the radiatively generated θ parameters at the low energy scale precisely match between them. It provides validity to perturbative loop calculations of the QCD θ parameter with the Fock-Schwinger gauge method. Furthermore, it is also shown that the ordinary Fu-jikawa method for the radiative θ parameter by using ¯ θ = − arg det M loop q does not cover all contributions in the simplified models. But, we also find that when there is a scale hierarchy in CP -violating sector, evaluation of the Fujikawa method is numerically sufficient. As an application, we calculate the radiative θ parameter at the two-loop level in a slightly extended Nelson-Barr model, where the spontaneous CP violation occurs to solve the strong CP problem. It is found a part of the radiative θ parameters cannot be described by the Fujikawa


Introduction
In the renormalizable QCD Lagrangian, the following terms are P -odd and T -odd, and then CP -odd terms under the CP T invariance, Gaµν . (1.1) Here, m q stands for the complex quark masses with m q ≡ |m q | exp(iθ q ), G a µν is the gluon field-strength tensor, Gaµν ≡ 1  2 ϵ µνρσ G a ρσ with ϵ 0123 = +1, and α s = g 2 s /(4π) is the SU (3) C coupling constant.It is known that one can turn off the first terms with the axial rotation of quarks in order to get the physical QCD θ parameter, θ (≡ θ G − q θ q ), defined as This QCD θ parameter would be expected to be O(1) naturally while it is constrained by the neutron electric dipole moments (EDM) measurements as | θ| ≲ 10 −10 [1,2].#1 This serious naturalness problem is called the strong CP problem.Some models have been proposed to solve the strong CP problem.One of them is the Peccei-Quinn (PQ) mechanism [6][7][8].The mechanism works well to solve the problem since the strong dynamics tunes the QCD θ parameter to be zero with the axion field, #1 Recently, EDMs of paramagnetic molecules such as ThO [3] or HfF + [4] are significantly improved, and the constraint on the QCD θ parameter from CP -odd semileptonic interactions by these experiments reaches | θ| < 10 −8 [5] if the source of CP -violation is dominated by the hadronic sector.
while an introduction of the PQ global symmetry suffers from the quality problem of the global symmetry in quantum gravity [9][10][11].The second option is to forbid the QCD θ term with the CP [12][13][14] or P symmetry [15][16][17][18].Those discrete symmetries are broken in the Standard Model (SM), so that the extensions of the SM with those symmetries are mandatory.In the left-right symmetric extension of the SM, the generalized P symmetry forbids the QCD θ term (and also the imaginary parts of quark masses) at the tree level.After the spontaneous breaking of the P symmetry to match the SM, the QCD θ parameter is radiatively generated.Also, it is known that there is no quality problem in the mechanism [19,20].On the other hand, in the Nelson-Barr models, the CP symmetry is imposed so that the QCD θ term is forbidden at the tree level.Then, the CP symmetry is spontaneously broken to generate the CP phase in the CKM matrix, and the model is tuned to keep the QCD θ parameter zero at the tree level, but it is generated radiatively.
In those models of the second option to solve the strong CP problem, the QCD θ parameter is radiatively generated and a finite neutron EDM is predicted even if the energy scale of the symmetry breaking is much higher than the weak scale [21].Thus, the precise evaluation of the radiative θ parameter is important to judge whether the models are viable under the current constraint from the neutron EDM measurements and also whether they are testable in future EDM experiments for the neutron and other developed systems.In past literature, e.g., Refs.[17,[20][21][22], the radiative corrections to the QCD θ parameter have been evaluated by using the radiative corrections to the imaginary parts of the colored particle masses [23], where M loop q is the renormalized (up-and down-type) colored fermion mass matrix.This relation is based on the Fujikawa method [24] or the Adler-Bell-Jackiw anomaly with Adler-Bardeen theorem [24][25][26][27][28].
However, it has been shown in Refs.[29,30], in the context of the SM, that the radiative correction to the QCD θ parameter can be evaluated directly and perturbatively, not via the imaginary parts of quark masses [23], using an external field technique, while details of the technique are not written.In Ref. [31], it was shown that such a radiative θ parameter can be evaluated by the Feynman diagrammatic calculations using the Fock-Schwinger gauge method [32][33][34][35][36][37][38].The QCD θ term is equivalent to a total derivative term, and hence the radiative correction to the QCD θ parameter is not perturbatively generated if the translational invariance is maintained.The operator-Schwinger method [38] using the log-determinant of the Dirac operators works for the evaluation of the radiative corrections to the action with keeping the translational symmetry.However, the operator-Schwinger method makes it difficult to go beyond the one-loop corrections to the QCD θ parameter.On the other hand, the Fock-Schwinger gauge method is applicable to the higher-loop corrections [31,39].In fact, the Fock-Schwinger gauge violates the translational invariance, while it is shown that the invariance is recovered in the physical overvalues [40,41].Some of the authors applied the Fock-Schwinger gauge method for evaluation of the QCD θ parameter in the minimal left-right model [31].It is found that the two-loop contribution to the QCD θ parameter vanishes in the model.The order of magnitude of the three-loop contribution from the diagrams is evaluated and it is found that it can be as large as the current experimental bound of the neutron EDM measurement.
In this paper, we derive the structure of the radiative correction to the QCD θ parameter at two-loop level.We compare the full two-loop corrections with those evaluated by the Fujikawa method taking into account only the radiative correction to the colored fermion masses at one-loop level in Eq. (1.3).In Sec. 2, we review several calculations related to the QCD θ parameter at one-loop level by using CP -violating low-energy effective theory.In Sec. 3, we study minimal models at two-loop level, in which colored fermion(s) and a real scalar have CP -violating Yukawa coupling(s) inducing the radiative QCD θ parameter.We find that if the colored fermions and the real scalar have masses comparable to each others, the Fujikawa method in Eq. (1.3) does not cover full contribution and we need to calculate the two-loop contribution to the QCD θ parameter directly.
In Sec. 4, we discuss the two-loop corrections to the QCD θ parameter in an extension of the Nelson-Barr model, as a concrete example of UV model.In the model, we introduce the four-point couplings of the SM Higgs (H) and the spontaneous CP -violating scalar (Σ a , a = 1, • • • , N Σ ), and a real scalar (S) fields, responsible to the real vector-like fermion mass.Both interactions of |H| 2 Σ a Σ * b and S 2 Σ a Σ * b , induce the QCD θ parameter at twoloop level.The former is determined by the radiative correction to the SM quark masses.On the other hand, the latter depends on the heavy particle mass spectrum, and it may not be determined by the radiative correction to the fermion mass.Section 5 is devoted to conclusions.Details of the two-loop calculations are given in the Appendix.

One-loop calculations for matching condition of QCD θ parameter
In this section, we evaluate the QCD θ parameter by integrating out a colored fermion q, whose mass and interactions are CP violating, in low-energy effective field theories at one-loop level.The general effective Lagrangian of the colored fermion is given in QCD up to dimension-five operators as where If the CP -violating phase of the colored fermion mass, m q ≡ |m q | exp(iθ q ), is removed by the axial rotation (chiral rotation) q → exp(− i 2 θ q γ 5 )q, one obtains [24-28] with μq = cos(θ q )µ q + sin(θ q )d q , dq = − sin(θ q )µ q + cos(θ q )d q , (2.3) where μq and dq correspond to the (anomalous) chromo-magnetic dipole moment (CMDM) and chromo-electric dipole moment (CEDM).#2 Here, we ignore the dimension-six four-Fermi operators.#3 They are mixed with the dipole moment operators at one-loop level renormalization-group evolution if they are present [42].The renormalization-group evolution effect will be taken into account in the next section.We integrate out the colored fermion with the operator Schwinger method to evaluate the QCD θ parameter at one-loop level.The method is transparent compared with the Fock-Schwinger gauge method when considering the one-loop corrections [31,38].They must be coincident with each other when we calculate the gauge invariant quantities.
First, we consider the contribution of the colored fermion mass phase to the QCD θ parameter.It is shown in Ref. [31] that in the operator-Schwinger method, the one-loop correction to the effective action ∆S after integrating out the colored fermion field is given as where P µ ≡ iD µ .The derivative of ∆S over m q for P R is [31,38] where [P µ , P ν ] = −ig s G µν is used.Since the G G term includes the Levi-Civita tensor, the QCD θ term comes from the second order of the perturbation of 1 2 g s (σ • G) in the denominator in the right-handed side of Eq. (2.5), where Tr(T a T b ) = (1/2)δ ab is used.Here, P 2 is replaced with (i∂) 2 , and it is integrated in the momentum space.Similarly, d∆S/dm * q leads to the G G term.By integrating d∆S/dm q with m q and d∆S/dm * q with m * q , we get As expected, the integration of the colored fermion gives the contribution to QCD θ term, consistent with the Fujikawa method in Eq. (2.2).
Next, we show the contribution of the dipole moments, μq and dq , in Eq. (2.2).The CEDM is T -odd and P -odd, and it also contributes to the QCD θ parameter as shown below.When the phase of the colored fermion mass is removed by the chiral rotation, the physical CEDM dq is given by dq = cos(θ q )d q − sin(θ q )µ q in Eq. (2.3).The evaluation of the physical CEDM contribution to the QCD θ parameter is more straightforward in the operator-Schwinger method.The one-loop correction to the effective action after integrating out the colored fermion field is Gaµν . (2.8) In the second line, we keep the leading term of the CEDM in the expansion.In the fourth line, P 2 is replaced by (i∂) 2 , and it is integrated out in the momentum space again.Since the integral includes UV divergence, we introduce the dimensional regularization d = 4−2ϵ with the renormalization scale µ.If the MS scheme is adopted, 1/ε (≡ 1/ϵ − γ E + log 4π and γ E is the Euler's constant) is subtracted by the counter term.It is found that the QCD θ parameter depends on the renormalization scale as [43] µ dθ dµ = 4|m q | dq , (2.9) and the threshold correction at the colored fermion mass (µ = |m q |) due to the CEDM vanishes.
It is known that the CEDMs of the colored fermions generate the T -odd and P -odd higher-dimensional gluon operators, such as the Weinberg operator with dimension six, G 2 G, in addition to the QCD θ parameter when the colored fermions are integrated out [39,[44][45][46][47]. On the other hand, the colored fermion mass phases induce only the QCD θ parameter, but not the higher-dimensional gluon operators.The axial rotations of the colored fermions turn off the colored fermion mass phases and simultaneously generate the QCD θ term in Eq. (2.2).This can be derived with the Fujikawa method [24] or the explicit calculation of the above one-loop colored fermion diagram [31].Now, we consider whether T -odd and P -odd higher-dimensional gluon operators, such as the Weinberg operator, are generated or not, when the quark mass phases are non vanishing.
In the Fujikawa method, the higher-dimensional gluon operators are not generated from the Jacobian of the chiral transformation [24].However, an UV cut-off parameter has to be introduced in order to define the anomalous Jacobian, which includes only the QCD The one-loop babble diagram contributing to the radiative corrections to the Todd and P -odd higher-dimensional gluon operators.It is shown that they are not induced from the quark mass phases.
θ term in the limit of the infinite UV cut-off parameter.Therefore, it might be unclear whether the higher-dimensional terms are generated when the UV cut-off is finite in the Fujikawa method, and we would like to derive the absence of the higher-dimensional gluon operators via the explicit one-loop calculation without the introduction of the UV cut-off parameter.Here, by using the operator-Schwinger method we show a theorem that the T -odd and P -odd higher-dimensional gluon operators are not generated at one-loop level even if the quark mass phases are not vanishing, whose diagram is shown in Fig. 1.
In the operator-Schwinger method, we start from the derivative of ∆S over m q for P R in Eq. (2.5).Then, by using the covariant derivative expansion (CDE) [48,49], the trace over the functional space can be evaluated by + (terms including two Gs and two Ds in the numerator) where the first line in Eq. (2.11) induces G G term, while the remaining ones we obtain where r GGG is a O(1) real number (for example, r GGG = 1 comes from the second line of Eq. (2.11)).In the above equation, G G term comes from the second order of 1 2 g s (σ • G) in the denominator and its coefficient is proportional to 1/m q .On the other hand, G 2 G term comes from the expansions of the covariant derivatives and/or 1 2 g s (σ • G) and its coefficient is proportional to 1/(m q |m q | 2 ).The coefficients are determined with the dimensions of the operators up to the O(1) coefficients.Similarly, the derivative of ∆S over m * q for P L is Therefore, the overall factor 1/m q in Eq. (2.12) is replaced by −1/m * q , where the relative phase comes from the difference of the projection operators P L/R .
The one-loop corrections to the effective action can be determined by adding the integrals d∆S/dm q with m q and d∆S/dm * q with m * q , Therefore, we find that only the QCD θ term is generated with the coefficient proportional to the quark mass phase, and the Weinberg operator is not induced from Fig. 1.
We conclude that such a cancellation occurs for all the T -odd and P -odd higherdimensional gluon operators with dimensions (D) larger than four.In general, d∆S/dm q includes all the T -odd and P -odd dimension-D gluon operators from the expansions of the covariant derivatives and/or 1 2 g s (σ • G) in the denominator, and their coefficients are proportional to m * q /|m q | (D−2) .On the other hand, d∆S/dm * q has the gluon operators with the coefficients proportional to −m q /|m q | (D−2) .By integrating d∆S/dm q with m q and integrating d∆S/dm * q with m * q , it is found that the coefficients of the T -odd and P -odd higher-dimensional gluon operators in Fig. 1 are canceled, while only D = 4 term survives.Thus, we showed that the quark mass phases do not generate the T -odd and P -odd higherdimensional gluon operators at the one-loop level.

Two-loop contribution to QCD θ parameter in simplified models
In this section, using the Fock-Schwinger gauge method, we evaluate the two-loop corrections to the QCD θ parameter in simple models, in which fermion(s) and a real scalar have CP -violating Yukawa interaction(s).We find that when the fermions and the scalar have masses comparable to each other, the Fujikawa method does not cover full contribution in the evaluation of the QCD θ parameter even if taking into account the radiative correction to the imaginary parts of the fermion masses at one-loop level.
Im(m q ) Figure 2: The one-loop and two-loop babble diagrams which contribute to the radiative corrections to the QCD θ parameter.The one-loop diagram gives a simple result δθ| 1L = −Im(m q )/Re(m q ).For the two-loop diagrams, the first one generates I (2;2) loop function in Eq. (3.4), while the others 2 Ī(3;1) in total.

One flavor case
First, we evaluate the two-loop contributions to the QCD θ parameter in a simplified model in which a CP -violating Yukawa coupling y q with colored fermion q is contained, where the number of the flavor is one for simplicity.Let us consider the following Lagrangian, where ϕ is a neutral scalar.Note that the bare QCD θ parameter θ G is set to be zero, cf.Eq. (1.1).In the following, we assume Im(m q ) ≪ Re(m q ) at the tree level in order to investigate the structure of the two-loop corrections using the low-energy effective theories.
On the other hand, we keep the one-loop correction formula since the one-loop diagrams include the counter terms to cancel divergence of subdiagrams in the two-loop diagrams and the combined results of one-and two-loop corrections are renormalization-scale independent.
When one sets θ G = 0, the physical QCD θ parameter can be perturbatively calculated by the Fock-Schwinger gauge method [31,38].The radiative corrections δθ are derived from one-loop (1L) and two-loop diagrams (2L) given in Fig. 2, δθ = δθ| 1L + δθ| 2L , as [31] ) where Q 2 ≡ 4πµ 2 e −γ E and the dimensional regularization is used with the renormalization scale.Here, the mass squared for the colored-quark masses in the loop functions is understood as The two-loop functions I (2;2) and I (3;1) are given in Appendix A.
The loop-function I (3;1) corresponds to the last two two-loop diagrams in Fig. 2, whose subdiagram gives a one-loop UV divergence ∝ y 2 q m q /ε, and this divergence must be canceled by a counter term of the mass renormalization.We adopt the MS scheme for the mass renormalization.The second term in Ī(3;1) (x 1 ; x 2 ; x 3 ), subtracting a 1/ε pole, corresponds to the counter term contribution, and the factor (1 − ϵ log x 1 /Q 2 ) comes from the higherorder term of ϵ in the dimensional regularization (see Eq. (2.15) in Ref. [31]).#5 Then, Ī(3;1) (x 1 , x 2 , x 3 ) is UV and IR finite function [31].
The above formula for the radiative corrections to the QCD θ parameter at two-loop level becomes simpler when the colored fermion is much lighter (or heavier) than the scalar ϕ, since the effective theory description works.Let us check such hierarchical situations.
First, when the scalar particle mass is much heavier than the fermion mass m ϕ ≫ m q , the two-loop contributions in Eq. (3.4) can be simplified to From the viewpoint of the effective field theory, first of all, the scalar ϕ is integrated out at a scale of µ = m ϕ , then the following CP -violating effective interactions may be obtained up to the dimension-six operators [42], where f abc is the structure constant.#4 In the model, θG − θq (mq = |mq| exp(iθq)) and yqm * q are invariant under the chiral rotation of the colored fermion.In addition, the Lagrangian is invariant under ϕ → −ϕ and yq → −yq.These imply that an accurate numerator of the two-loop correction is Im((yqm * q ) 2 ), and we expand it with Im(mq) ≪ Re(mq).#5 The second term in Ī(3;1) (x1; x2; x3) corresponds to the divergence contribution Īϵ(3;1) defined in Ref. [31] when one adopts the MS scheme for the mass renormalization.As a result, Q 2 in F0(p 2 , x1, x2) in Ref. [31] is replaced with µ 2 in F0(p 2 , x1, x2) in Eq. (3.9).
The radiative correction to the imaginary part of the fermion mass at µ ≃ m ϕ is given as where (3.9) The above radiative correction to the imaginary part of the fermion mass corresponds to that in a limit of the zero external momentum, since the external momentum is negligible compared with m ϕ .In the case, the radiative correction is explicitly given as It is found by explicit calculation that the matching conditions for the higherdimensional operators in Eq. (3.7) ) The term ∆θ| th in Eq. (3.7) is a threshold correction to the QCD θ parameter, which will be identified by comparing with the full result and the contributions from the effective operators in the effective theory.
In the effective Lagrangian (3.7), the CEDM operators are mixed with the four-Fermi operators via the renormalization-group equation [42], and then the dq depends on µ when m q < µ < m ϕ .Thus, dq is approximately given as This CEDM operator induces the QCD θ parameter via the renormalization-group equation [43], as reviewed in Sec. 2.
Using these results in the effective theory, the radiatively generated QCD θ parameter at the low-energy scale is evaluated as This result is consistent with the QCD θ parameter directly calculated from the diagrams in Fig. 2 (see Eq. (3.6)), where µ should be taken for the heaviest particle mass of the loop diagrams, µ ≃ m ϕ .In addition, by matching the above results one obtains the threshold correction to the QCD θ parameter as Note that these effective theory matchings are valid only when m ϕ ≫ 2m q [50].
In the above calculation, we implicitly assume that the light fermion is heavier than the Λ QCD scale.On the other hand, the light quarks in the SM cannot be integrated out.The neutron EDM is evaluated in the low-energy effective theory, which includes higherdimensional operators of light quarks in addition to those of gluons.Note that when fermions with mass larger than Λ QCD , such as heavier quarks in the SM, are integrated out, the fermion CEDMs induce the Weinberg operator as a threshold correction and it also contributes to the neutron EDM [39,[44][45][46][47], though it does not contribute to the QCD θ term.The contribution to the neutron EDM from the induced Weinberg operator would be smaller than the contribution from δθ, suppressed by O(Λ 2 QCD /m 2 ϕ ).We show the numerical examination in Fig. (3a) in order to stress the importance of the full two-loop diagram calculation in the case of m ϕ ≳ m q .Here, µ = m ϕ , since the effective theory of the colored fermion works after integrating out ϕ, as mentioned above.The blue line represents the contribution to the QCD θ parameter from the imaginary part of the colored fermion mass in Eq. (3.8) with the Fujikawa method in Eq. (1.3), while the orange one comes from the direct two-loop diagram calculation in Eq. (3.4).The CEDM of the colored fermion contributes to the QCD θ parameter, though it is suppressed by m 2 q /m 2 ϕ .The threshold correction in Eq. (3.19) is also suppressed by m 2 q /m 2 ϕ .As a result, the correction to the QCD θ parameter is ) < l a t e x i t s h a 1 _ b a s e 6 4 = " t E R P a J l q U D y g I 3 p D x 0 7 q v H B e O a 8 H p R M F y 7 l G f z X n / S 9 F S 9 j c < / l a t e x i t > µ 2 = m 2 q : Fujikawa method The radiative corrections to the QCD θ parameter, δθ which is normalized by Im(y 2 q ), evaluated from the full two-loop diagrams (Eq.(3.4)) are presented by the orange lines.For comparison, the corrections evaluated from the one-loop correction to the colored fermion mass in a limit of zero external momentum (Eq.(3.8)) with the Fujikawa method are also shown by the blue lines.In the left panel (a), a parameter region of m ϕ ≳ m q is shown with µ = m ϕ .On the other hand, in the right panel (b) m ϕ ≲ m q with µ = m q .dominated by the contribution from the correction to the imaginary part of the colored fermion mass when m ϕ ≫ m q , as expected in Ref. [51].However, it is also shown that when m ϕ /m q ≲ 10, the estimation of the Fujikawa method does not cover full contribution.
On the other hand, when the scalar particle mass is quite light compared to the fermion mass m ϕ ≪ m q , the two-loop functions can be simplified to [50] where all two-loop diagrams are contributing.#7  From the viewpoint of the effective theory, this contribution corresponds to a threshold correction to the QCD θ parameter when the fermion is integrated out, and hence µ ≃ m q .The µ dependence is canceled between the first and the second terms.It is found that the second term coincides with the one-loop correction to the imaginary part #7 For m ϕ ≪ mq, each two-loop function becomes .21) When one sets µ = mq, the two-loop contribution is predominated by I (2;2) (m 2 q ; m 2 q ; m 2 ϕ ).
of the fermion mass (Im[∆m q (µ)]) in a limit of the zero external momentum, normalized by the fermion mass, In Fig. (3b), we take m ϕ ≲ m q .Here, µ = m q .Again, the blue line represents the contribution to the QCD θ parameter from the imaginary part of the colored fermion mass in Eq. (3.8) with the Fujikawa method in Eq. (1.3), while the orange one comes from the direct two-loop diagram calculation in Eq. (3.4).The blue and orange lines coincide when m ϕ ≪ m q , while all three diagrams at two-loop level contribute to the QCD θ parameter (see footnote#7), as mentioned above.
Thus, when m ϕ ≫ m q or m ϕ ≪ m q , the radiative correction to the QCD θ parameter at two-loop level can be sufficiently evaluated by the Fujikawa method taking into account the radiative correction to the fermion masses in a limit of the zero external momentum.The coincidence is, however, non-trivial for the case of m ϕ ≪ m q .It would be an open question whether the coincidence is valid even if one includes the higher-order corrections to the QCD θ parameter than the two-loop one.

Two hierarchical flavor case
Next, we consider a more complicated situation: there are two kinds of fermions q l (light fermion, m l ≪ m ϕ ) and q h (heavy one, m h ≈ m ϕ ), and the interactions are where i, j run l (light) and h (heavy), and we take a flavor diagonal mass basis.We suppose that the Yukawa interactions y lh and y hl are CP -violating complex couplings but y ll and y hh are real ones, for simplicity.
From the direct loop calculations, the radiative corrections to the QCD θ parameter are ) Here, I (2;2) (x 1 ; x 2 ; x 3 ) = I (2;2) (x 2 ; x 1 ; x 3 ) is used.When m l ≪ m h , m ϕ , these loop functions can be simplified to Let us compare the radiative corrections to the QCD θ parameter in the effective theory approach.When the heavy fermion and the scalar boson are integrated out from the full theory, the following effective interactions are obtained, In addition to the Weinberg operator, the CP -violating four-Fermi operators are not generated since it is assumed that only y lh and y hl have complex phases.The one-loop correction Im(∆m l ), which is the correction in a limit of the zero external momentum, is Note that the one-loop corrected mass (m l + ∆m l ) is µ-independent because m l is the MS masses.In addition, the CEDM for q l at one-loop level is given as where (3.32) Using the above results, the radiative correction to the QCD θ parameter at two-loop level in Eq. (3.28) is reduced to The first two terms come from the integration of q l in the effective theory.The CEDM contribution is proportional to a large log, log(m l /m ϕ ).It is expected since the renormalization-group equation of θ involves a term proportional to the CEDM in Eq. (2.9).These results are easily improved by the renormalization-group equations.The threshold correction ∆θ| th represents the contributions that come from the integration of q h and m ϕ in the full theory, and we obtain it from the above matching as, Here, Im[∆m h (µ)] is the radiative correction to the imaginary part of heavy fermion mass in a limit of the zero external momentum, The threshold correction to the QCD θ parameter is suppressed by Re(m l )/Re(m h ), since we assume that y lh and y hl are CP -violating complex couplings while y ll and y hh are real ones.The CEDM contribution in the effective theory is also suppressed by Re(m l )Re(m h )/m 2 ϕ .Thus, the correction to the QCD θ parameter can be evaluated by the Fujikawa method taking into account the radiative correction to the imaginary part of the light quark mass in Eq. (3.30) in this case.

Nelson-Barr Model
In this section, we show an example in which the QCD θ parameter is induced at twoloop level while it vanishes at tree level.The CP symmetry is spontaneously broken in the Nelson-Barr models [12,13].The QCD θ term vanishes at the tree level by assuming quark mass matrices M q (q = u, d) with arg det (M q ) = 0 even after the spontaneous CP -violating sector.
The minimal models were derived by Bento, Branco, and Parada [52].They are classified into two types of models, depending on whether up-or down-type quarks are coupled with the CP -violating sectors.In this paper, we consider the d-type model, #8 given by where the down-type quarks are coupled with the CP -violating sector.Here, Q, u, and d are for SU (2) L doublet left-handed quarks, singlet right-handed up and down quarks, #8 The u-type model is given by The matter contents and their gauge charges in a Nelson-Barr type (spontaneous CP violating) model.
respectively.The subscripts i, j, k are used for the generation in the SM (i, j, k = 1, 2, 3).We introduce an SU (2) L singlet vector-like quark, whose left and right-handed components are represented by ψ and ψ c , respectively.The vacuum expectation value of the SM Higgs doublet field H is ⟨H⟩ = (0, v/ √ 2) T .The conjugate is H = ϵH * .The complex singlet scalar fields Σ a (a = 1, • • • , N Σ , N Σ > 1) are assumed to get the complex vacuum expectation values so that the CP symmetry is spontaneously broken.The gauge charges of the matter contents are summarised in Table 1.In this model, the down-type quark mass matrix is given by where ξ * i = g ai ⟨Σ a ⟩.We have to forbid terms such as Σ a ψψ c and H Qψ c since they lead to arg det(M d ) ̸ = 0.Those terms are forbidden by the following discrete Z N symmetry imposed as, ) It is pointed out by Ref. [53] that the four-point scalar interactions of H and Σ a lead to arg det(M d ) ̸ = 0 at one-loop level and the QCD θ parameter is radiatively generated.#9 It corresponds to the two-loop correction to the QCD θ parameter, shown in the left diagram of Fig. 4. #9 When the four-point scalar interactions are absent, the QCD θ parameter is still generated at three-loop level.It is evaluated using the dimensional analysis in Refs.[54,55].Furthermore, we introduce a real scalar field S whose real vacuum expectation value leads to the singlet vector-like quark mass, m, as m = f ⟨S⟩ with a Yukawa coupling −L = f S ψψ c + h.c.. From a viewpoint of model-building, however, the origin of m should be related to the vacuum expectation values of Σ a since the observed CKM CP phase of O( 1) is realized only when m ∼ |ξ i | = |g ai ⟨Σ a ⟩|.It is found that the four-point scalar coupling of S and Σ a also generates the QCD θ parameter at two-loop level, denoted as a right diagram of Fig. 4.Then, we consider the following four-point scalar interactions, We evaluate the QCD θ parameter at two-loop level assuming γ ab and γab are non-vanishing.The down-type quark mass matrix is diagonalized as where Md is the diagonal mass eigenvalue matrix Md = diag( md A ) (A = 1, • • • , 4).The inverse of M d is given by the unitary matrices as We assume that ⟨H⟩ ≪ m, |ξ k |.In this case, the unitary matrices U d and O d are given as #10 #10 This diagonalization comes from a mathematical identity [55], Here, where m .The mass eigenvalues of the M d are given by md 4 = M and eigenvalues of Y ij d ⟨H⟩ for three SM down quarks, Now we evaluate the corrections to the QCD θ parameter at two-loop level, as shown in Fig. 4.They come from mixings between H and Σ a and also between S and Σ a .Assuming γ ab and γab much less than one, we keep only the leading terms in the perturbation.In the case of diagrams in Fig. 4, they give the corrections to the QCD θ parameter as (4.11)where m h , m S , and m Σa are masses of H, S, and Σ a , respectively.These results still include fake IR bad behavior due to light quarks such as 1/ md k .However, using Ī(3;1) (x 1 , x 2 , x 3 ) ≃ −1/(2x 1 )F 0 (0, x 2 , x 3 ) for x 1 ≪ x 3 and inverse matrix of M d in Eq. (4.6), the IR behavior can be removed.Then, when ⟨H⟩ ≪ m, |ξ k |, the above results are reduced as Since the CP symmetry is spontaneously broken, the corrections to the QCD θ parameter are UV finite.It is found that ∆θ Σ comes mainly from the correction to light quark masses since it is given by the function F 0 .On the other hand, ∆θ S is generated by the vector-like quark loop.When M is comparable to m S and/or m Σa , the correction is not dominated by the correction to the vector-like quark mass, and the full evaluation of the two-loop diagram is required to evaluate them.

Conclusion and Discussion
The QCD θ parameter is generated radiatively in spontaneous CP or P symmetry-breaking models, which solve the strong CP problem.We scrutinized the QCD θ parameter at the two-loop level analysis.In the simplified models with CP -violating Yukawa interactions, we observed that the two-loop calculation of the radiative QCD θ parameter using the Fock-Schwinger gauge method is consistent with the effective field theory approach at the low-energy scale.Furthermore, we clarified the application scope of the Fujikawa method.When there is a scale hierarchy in the particle masses in CP -violating sector, the Fujikawa method is sufficient for evaluating the QCD θ parameter.On the other hand, in the case of a small hierarchy, the Fujikawa method does not cover full contribution.The Nelson-Barr model is an example that the Fujikawa method cannot evaluate the radiatively generated QCD θ parameter correctly.If the vector-like quark and additional scalars have comparable masses, the Fock-Schwinger gauge method should be used to evaluate the radiative QCD θ parameter.
It is an important subject to evaluate the three-loop contributions to the QCD θ parameter in some well-motivated models for the strong CP problem, such as the leftright models, and to compare the predictions with the experimental bounds on the QCD θ parameter.In the evaluation, the effective theory approach would be useful if the CPviolating interaction can be integrated out.I(x 1 ; x 2 ; x 3 ) (≡ I (1;1) (x 1 ; x 2 ; x 3 )) as I (n;m) (x 1 ; x 2 ; x 3 ) = 1 (n − 1)!(m − 1)!

2 9 5 µ 2 = m 2 :
k j H h g 7 i A y W j E y O f B x 7 c i 8 h e Y f y A k 7 p x I y I 6 v i I a f v c I C M Q I 0 0 t U P c S 4 p q D x 4 y q z T d n a 7 V l R x 7 G c 2 o k 3 A w R E a / S J w W X n / 1 3 7 c + G W z W K t W 7 l T t r t 0 s r D + 3 / Y J 6 u 0 X W 6 C V f 3 a I U e 0 y r V o d i n 1 3 R I 7 x z l v H Q O n L f j 0 r m C 5 V y h v 5 r z 8 R e T 6 9 p X < / l a t e x i t >

Figure 4 :
Figure 4: The two-loop diagrams that contribute to the radiative QCD theta parameter in the Nelson-Barr model.The external gluon fields are omitted.The cross (×) represents the chirality flip.Subscripts A and B are for mass eigenstates of fermions.