Quark level and hadronic contributions to the electric dipole moment of charged leptons in the standard model

We evaluate the electric dipole moment (EDM) of charged leptons in the standard model, where the complex phase of the Cabibbo-Kobayashi-Maskawa matrix is the only source of CP violation. We first prove that, at the quark-gluon level, it is suppressed by a factor of $m_b^2 m_c^2 m_s^2$ at all orders of perturbation due to the GIM mechanism. We then calculate the hadronic long distance contribution generated by vector mesons at one-loop level. The $|\Delta S|=1$ weak hadronic interaction is derived using the factorization, and the strong interaction is modeled by the hidden local symmetry framework. We find that the EDMs of charged leptons obtained from this hadronic mechanism are much larger than the perturbative four-loop level quark-gluon process, by several orders of magnitude.

In the SM, the Cabibbo-Kobayashi-Maskawa (CKM) matrix [149] has a CP violating complex phase, so it may generate the EDM. In the search for new physics beyond the SM, this contribution must be assessed as the leading background. It is known that, in most cases, it is unobservably much smaller than the experimental sensitivity [150][151][152][153][154][155][156][157][158][159][160][161][162][163][164][165]. However, the hadronic contribution to the EDM of charged leptons has never been evaluated in the past. This is just the aim of this paper to quantify it.
In this paper, we study the vector-meson contribution to the lepton EDM at one-loop level. We first prove that the contribution at the quark-gluon level is suppressed by a factor of m 2 b m 2 c m 2 s at all orders of perturbation due to the GIM mechanism. Next, we calculate the hadronic long distance contribution generated by vector mesons at one-loop level. The |∆S| = 1 weak hadronic interaction is derived using the factorization, while the strong interaction is given by the hidden local symmetry framework. Part of the results have been briefly reported in [166]. A complete report of our study is given in this article.
This paper is organized as follows. In the next Section, we review the quark-gluon level calculation of the CKM contribution to the EDM of charged leptons and prove that it is actually suppressed by factors of quark masses at all orders of perturbation. We then describe in Sec. III the setup of the evaluation of the hadronic contribution to the EDMs of charged leptons in the hidden local symmetry framework, with the weak interaction derived with the factorization. In Sec. IV, we show the result of our calculation and analyze the theoretical uncertainty. The final section gives the summary of this work.

II. QUARK LEVEL ESTIMATION OF THE EDM OF CHARGED LEPTONS AND THE GIM MECHANISM
Let us first review the previous works on the calculation of the short distance (quark-gluon level) effect to the EDM of charged leptons in the SM. Since we are supposing that the CP violation is generated by the physical complex phase of the CKM matrix, the Feynman diagrams contributing to the lepton EDM must have at least a quark loop, with sufficient flavor changes so as to fulfill the Jarlskog combination [167]. The Jarlskog invariant is given by the product of four CKM matrix elements (J = Im[V us V td V * ud V * ts ] = (3.18 ± 0.15) × 10 −5 [168]), so the quark loop must have four W boson-quark vertices. By noting that the W boson must also be connected to the electron, the two-loop level diagram which has only two vertices in the quark loop does not contribute to the EDM due to the cancellation of the complex phase. The first plausible contribution appears then at the three-loop order (two-loop level diagrams of the EDM of W boson as shown in Fig. 1, which is attached to the lepton line). However, extensive three-loop level analyses revealed us that it exactly cancels due to the antisymmetry of the Jarlskog invariant under the flavor exchange (also called the GIM mechanism, a consequence of the CKM unitarity) [156][157][158]. The cancellation works as follows. If we can find two quark propagators of the same type (up-type or down-type) in the diagram with identical momenta and sandwiched by W boson vertices, the sum of the direct product of these two parts over the d-type quark flavors reads The projection P L ≡ 1 2 (1 − γ 5 ) comes from the W boson-quark vertices. The mass insertions of S D cancel since odd number of chirality flips is not allowed when S D is sandwiched by W boson-quark vertices. It turns out that the pair of propagators with the same (u-or d-) type quarks can always be found in the two-loop level contribution to the EDM of W boson, and consequently in the three-loop level diagrams of the EDM of charged leptons. The most trivial ones are the symmetric diagrams with two quark propagators of the same type, but there are also diagrams which have nonsymmetric insertions of the external photon. The latter ones can actually be recast into the symmetric form of quark propagators by using the Ward-Takahashi identity [156,157]. Similar cancellation also occurs in the case of the quark EDM/chromo-EDM [150][151][152][153][154][155] or the Weinberg operator (gluon chromo-EDM) [159]. The first nonvanishing contribution which avoids the above symmetric cancellation appears at the four-loop level (see Fig. 2). Although the four-loop level contribution has never completely been calculated, it is possible to estimate its size by symmetry consideration. It is indeed possible to prove that the GIM mechanism [169,170] always brings additional suppression of quark mass factors m 2 q , independently of the order of perturbation. Let us first consider the quark loop with several insertions of vertices of flavor unchanging (neutral) bosons, i.e. gluons, photons, or Higgs bosons (see Fig. 3). We focus on the direct product of the U and U quark lines with vertex insertions of Fig. 3, which may be expressed by the Taylor expansion in terms of the quark masses, as follows: where a (1) n and a (2) n are polynomials of the electric charge of up-type quarks, the strong coupling, the inverse of the Higgs vacuum expectation value (appearing from the Yukawa coupling of the Higgs boson after factoring out quark  The gluon, photon, and the neutral Higgs boson are denoted by the wiggly, wavy, and dashed lines, respectively, and the ellipses means that they each may be of arbitrary number. The sum of the quark flavors removes the contribution without flip of chirality due to the GIM mechanism. The emitted bosons have O(mW ) ≈ O(mt) momenta, and they may also form loops, or be connected to other fermion loops, which are not interfering with the flavor structure of the one considered in this figure.
masses), and all momenta carried by the bosons attached to U and U , respectively, which depend on the diagram considered. Here we took the direct product ⊗ to show that the above Taylor expansion also works for the case where Dirac matrices are involved. We can actually prove that the terms involving a n m 2n U which are also not difficult to treat. For the former case, we have We may repeat the same calculation to show the cancellation for the case of a n m 2n U as well. We thus proved that the leading order CP violation of the quark loop is accompanied by two factors of squared mass of two different up-type quarks to all orders of perturbation in QED, QCD, and Higgs corrections. We may also exactly repeat the above procedure for the down-type quark contribution which is independent of the up-type ones. The CP violating part of the quark loop is then at least having a suppression factor of m 2 t m 2 b m 2 c m 2 s , which of course persists even if some of the neutral or W bosons are contracted each other or with other quark loops. It also appeared in the result of the calculation of the Weinberg operator which is also generated by a quark loop [159].
The presence of the suppression due to quark mass factors, i.e. the cancellation of the zeroth order terms of the Taylor expansion of the quark lines with neutral boson insertions, may also more elegantly be shown using the unitarity of the CKM matrix. At the order of four W boson-quark vertices, the general flavor structure of the quark loop, with the sum over the flavor taken, is expressed by the following trace where V is the 3 × 3 CKM matrix, and Q (k) n l m 2n l D (k, l = 1, 2) are the down-type and up-type quark lines with arbitrary number of neutral boson insertions, respectively. We note that Q Here we used the unitarity of the CKM matrix V † V = 1, the fact that R Next, we have to see higher order corrections with W boson-quark vertices which may be treated in a similar manner. Here again the unitarity of the CKM matrix plays a crucial role. Let us consider the case with six W boson-quark vertices. The general flavor structure of this quark loop, with the flavor summed, looks like We now show that the correction at this order (V 6 ) is not larger than that of O(V 4 ) which has the quark mass factors m 2 t m 2 b m 2 c m 2 s . A potentially large contribution may arise from the zeroth order terms of the Taylor expansion a where we again used the unitarity of the CKM matrix. By noting that R  (4)]. This means that the the O(V 6 ) quark loop having one zeroth order term of the Taylor expansion is also having the quark mass factor m 2 t m 2 b m 2 c m 2 s . We also note that the contribution with the three up-type quarks being all top quarks, which may potentially be larger than the O(V 4 ) terms, has no effect to the EDM, since it will be proportional to three factors of the absolute values of squared CKM matrix elements |V tD | 2 , i.e. at least a factor of m 2 c or m 2 u is needed. This analysis may be extended to arbitrary higher orders recursively, since the zeroth order terms a ) and at all other higher orders of W boson-quark vertices (V ). We can also show with the above approach the cancelation of the quark loop at O(V 2 ) and at the two-loop level in a more elegant manner. At O(V 2 ), we have where D is diagonal and real, its trace with H is taking only the diagonal elements, which are also real. There is no room for the imaginary part, so CP is conserved at O(V 2 ), even accounting for all order corrections of neutral bosons. At the two-loop level (of the quark loop), we previously saw that we can always find a symmetric set of either up or down-type propagators with the same momentum argument [156][157][158]. We may then write the trace as where we used the hermiticity of H ≡ V S D V † . Due to the absolute value, there is no CP violation, and there is thus no contribution to the EDM of charged leptons at the three-loop level. We also see that, if the symmetry between the two S D is destroyed, the two H will no longer be complex conjugates, and the imaginary part will be generated.
Let us now estimate the EDM of charged leptons according to the above discussion. The correct dimensional analysis of the four-loop level contribution according to the above proof therefore yields which is transcribed to Here we did not consider the logarithmic enhancement which may enlarge the above values by one or two orders of magnitude. Nevertheless, these results are actually telling us that the short distance contribution is extremely small. From this analysis, we see that the enormous suppression of the EDM of charged leptons is not due to the fact that it appears at the four-loop level, but rather due to the cancelation by the GIM mechanism. We stress that this suppression mechanism does apply only when all momenta involved are of O(m W ∼ m t ). In the case where nonperturbative physics is relevant in the infrared region, the coefficients a n , b n of Eq. (2) may be enhanced by In the next section, we recast the soft momentum physics into phenomenological hadron physics where the weak interacting hard part is given by low energy constants, which are also calculated with phenomenological models.

A. The long distance effect
The leading order contribution of the CKM matrix to the lepton EDM is constructed with at least two W boson exchanges. To avoid severe GIM cancellation as we saw in the previous section, we have to split the short distance flavor changing process at least into two parts at the hadron level (the long distance effect), while keeping the Jarlskog combination of the CKM matrix elements. The largest long distance contribution should involve unflavored and |S| = 1 mesons rather than heavy flavored (c, b) ones. Another important condition is that the charged lepton EDM is generated by one-loop level diagrams involving vector mesons, because the interaction of pseudoscalar mesons with the lepton will change the chirality, suppressing the EDM by at least by a factor of m 2 l (l = e, µ, τ ). The charged lepton EDM is then generated by diagrams involving a K * meson. The one-loop level diagrams must not have a neutrino in the intermediate state of the long distance process, since the small neutrino mass will not provide sufficient chirality flip required in the generation of the EDM. Moreover, if the process contains two weak K * -charged lepton vertices, the chirality selection will not allow an EDM. The K * meson must therefore change to an unflavored meson which in turn becomes a photon which will be absorbed by the charged lepton. Under such restrictions, we may draw diagrams shown in Fig. 4.

B. Hidden local symmetry
Let us now give the interactions to calculate the diagrams of Fig. 4. It is convenient to describe the |∆S| = 0 vector meson interactions with the hidden local symmetry (HLS) [171][172][173][174][175][176][177][178][179][180][181]. The HLS is a framework introduced to extend the domain of applicability of chiral perturbation to include vector meson resonances, and it is successful in phenomenology [181]. The effective Lagrangian for three vector mesons is given by Long distance contribution to the EDM of charged lepton l (= e, µ, τ ) in the SM. The diagrams (a) and (a ) ((b) and (b )) are the contribution with the weak (strong) three vector meson interactions. There are also diagrams with theK * propagator, which are not displayed. The grey blob denotes the |∆S| = 1 semi-leptonic effective interaction, while the black one is the |∆S| = 1 (two-and three-point) vector meson interactions which combine each other to form the Jarlskog invariant.
where the vector meson matrix V µ is given by where g = m ρ /f π with the pion decay constant f π = 93 MeV. The effective lagrangian for vector meson and photon is given by [179] where g γ = 5.7 and C. K * -lepton interaction Let us now model the weak interaction at the hadron level. From Fig. 4, the |∆S| = 1 weak interaction appears in the K * -lepton interaction and in the interacting vertices between K * and other vector mesons. Since the neutrino cannot appear in Fig. 4, the interaction between K * and the charged lepton must be at least a one-loop level process at the quark level. Then the best solution is to attribute the CKM matrix elements V cs V * cd or V ts V * td to the K * -lepton interaction, and V ud V * us to the K * -vector meson interactions. The latter attribution will maximize the |∆S| = 1 vector meson interactions, since V ud V * us is given from the tree level |∆S| = 1 four-quark interaction. The parity violating effective interaction between K * and the charged lepton is given by where K * µ is the field operators of the K * meson. In the zero momentum exchange limit, the coupling constant is given by where we fixed the complex phases of V ud V * us to be real. The K * meson matrix element is given by where ε K * µ , m K * = 890 MeV and f K * = 204 MeV [182][183][184][185] are the polarization vector, the mass, and the decay constant of K * , respectively. The quark level amplitude I dsll can be obtained by calculating the one-loop level diagrams of The diagram (c) is the largest, but all of them are of the same order. The numerical value of the total I dsll is which is quite consistent in absolute value with that of the naive dimensional analysis I dsll ∼ which is due to the GIM cancellation. This shows that if we invert the up-type and down-type quarks, the resulting meson-charged lepton couplings will be suppressed by a factor of m 2 D /m 2 W (D = d, s, b).

D. |∆S| = 1 vector meson transition and three-vector meson interaction
We now model the |∆S| = 1 vector meson transition and three-vector meson interaction using the factorization. For that, we have to determine the Wilson coefficients of the quark level |∆S| = 1 processes. We chose the |∆S| = 1 case because it is the only allowed flavor change at low energy scale. At the scale just below the W boson mass (m W = 80.4 GeV), we have the following |∆S| = 1 effective hamiltonian with the Fermi constant G F = 1.16637 × 10 −5 GeV −2 [168]. Here Q q 1 , Q c 1 , Q q 2 , Q c 2 , and Q j (j = 3 ∼ 6) are defined as [186,187] where α and β are the fundamental color indices, and the summation over N f goes up to the allowed flavors at the given scale. The Hamiltonian of Eq. (25) keeps the same form down to µ = m c , but the Wilson coefficients run in the change of the scale.
The running is calculated in the next-to-leading order logarithmic approximation (NLLA) [163,186,187]. Below µ = m c , the charm quark is integrated out. The resulting |∆S| = 1 effective hamiltonian becomes Here we quote the values of Refs. [163,164]: We see that the Wilson coefficient of Q 2 is the largest. This is because Q 2 is the sole tree level operator at µ = m W , and the others were radiatively generated. Here we point that the coefficient of Q 1 is also important since the contribution of Q 2 obtains a factor of 1/N c after the Fierz rearrangement of the color (see below). The operators Q i (i = 3, · · · 6) cannot be neglected either, because they generate the φ meson which is impossible with Q 1 and Q 2 . We also note that Q 5 and Q 6 , after Fierz transformation, couple to the chiral condensate which may enhance the overall effect (see below).
For the calculation of the crossing symmetric contribution, it is convenient to Fierz transform the |∆S| = 1 four-quark operators Q i (i = 1, · · · 6). The Fierz transform of Eqs. (26), (28), (30), (31), (32), and (33) are where t a is the generator of the color SU (3) c group. The summation over the fundamental color indices runs inside each Dirac bilinear, so the indices (α and β) have been omitted. As for Eqs. (36), (39), and (41), we also displayed in the first equalities the Fierz rearrangement of the fundamental color indices to form color singlet Dirac bilinears. We note that an additional minus sign contributes due to the anticommutation of fermion operators. This sign change is important since there may be interference with crossing symmetric graphs. We use the standard factorization to derive the |∆S| = 1 vector meson interaction from the |∆S| = 1 four-quark interaction of Eq. (34). We first construct the |∆S| = 1 meson transition in the factorization with vacuum saturation approximation [188,189]. It works as where q = u, d. We note that the vacuum saturation approximation gives the leading contribution in the large N c expansion in the mesonic sector. The |∆S| = 1 four-quark interaction has two distinct contributions, as shown in Fig.  6 (a) and (b). The first contribution (a) is the factorization into two meson tadpoles [see Eq. (42)]. It requires the decay constants of vector mesons, as where ε µ is the polarization of the vector meson, and m ρ = 770 MeV, f ρ = 216 MeV, m ω = 783 MeV, f ω = 197 MeV, m φ = 1020 MeV and f φ = 233 MeV [182-185, 190, 191]. The second contribution [ Fig. 6 (b)] is the factorization into scalar matrix elements [see Eq. (43)]. It appears from the Fierz transformation of Q 5 and Q 6 . The chiral condensates relevant in this regard are 0|ss|0 ≈ 0|dd|0 ≈ − [192]. They are obtained at the appropriate renormalization scale µ = 1 GeV with m u ≈ 2.7 MeV and m d ≈ 5.9 MeV [168], calculated in the two-loop level renormalization group evolution [193,194]. The scalar matrix element of the vector meson is derived by using the result of the calculation of the chiral extrapolation of the vector meson mass in lattice QCD [195][196][197][198][199]. As derived in Appendix A, we obtain B φK * ≡ φ|ds|K * 0 = 2.14 GeV.
By using the above parameters, the lagrangian of the weak vector meson transition is given by where ρ ν , ω ν and φ ν are the field operators of the ρ 0 , ω, and φ mesons, respectively. The coupling constants are given by Let us also construct the weak three-meson interactions. Again by using the vacuum saturation approximation, we have with q = u, d. The weak three-vector meson interaction is then The coupling constants are given by where V = ρ, ω, φ. The coefficients c V K * are obtained as the relative strength of the meson transition Tr

E. One-loop level calculation of the EDM of charged leptons
In this subsection, we perform the one-loop level calculation of the lepton EDM which is given by the amplitudes shown in Fig. 4. The diagrams in Figs. 4 (a) and (a ) are the contribution with the weak interaction of three vector mesons, while the diagrams in Figs. 4 (b) and (b ) are that with the strong interaction.
The scattering amplitudes with the weak three-meson interaction in Figs. 4 (a) and (a ) are given by The coefficients c V , c V are c ρ = 1, c ω = 1/3, c φ = − √ 2/3. In the soft photon limit (q 2 ∼ 0, p · q ∼ 0), the denominators of the integrands in Eqs. (59)-(60) are rewritten as where The numerators of the integrands in Eqs. (59)-(60) are reduced tō where the terms which do not contribute to the EDM are suppressed. By performing the integrals with respect to a and a for Eqs. (59)-(60), the amplitudes for Eqs. (59)-(60) are reduced to respectively, where we use the Gordon identitȳ The integrals in Eqs. (68) and (69), are performed numerically, with the results summarized in Table I.  The amplitudes iM K * (a) and iM K * (a ) are for the contributions with the K * propagator. In addition, the amplitudes with theK * propagator, denoted by iMK * (a) and iMK * (a ) , also contribute to the EDM. If we restrict to the CP violation, we have iMK * (a) = iM K * (a) and iMK * (a ) = iM K * (a ) . Thus the total scattering amplitude with the weak three-vector meson interactions is given by where In a similar manner, the charged lepton EDM contributions with the strong three-vector meson interactions shown in Figs. 4 (b) and (b ) are also calculated. The scattering amplitudes of the diagrams (b) and (b ) are obtained as In the soft photon limit, the denominators of the integrands in Eqs. (78)-(79) are rewritten as Performing the integrals with respect to b and b , Eqs. (78)-(79) are reduced to The numerical results of the integrals are summarized in Table II.  (88) for the leptons l = e, µ, τ and the vector mesons V = ρ 0 , ω, φ, given in units of GeV −6 . Finally, the total scattering amplitude with the strong three-vector meson interactions is obtained as IV. RESULTS AND ANALYSIS

A. Numerical results
From the scattering amplitudes derived in the previous section, we obtain the hadronic long distance contributions to the EDMs of charged leptons. From the amplitudes iM (a) and iM (a ) of Eqs. (76) and (77), we obtain the EDMs generated by the weak three-vector meson interactions as Similarly, the amplitudes iM (b) and iM (b ) of Eqs. (76) and (77) give the contribution from the strong three-vector meson interaction: We finally obtain the EDMs of e, µ, and τ generated by the hadronic long distance contributions as These values are much larger than the estimation at the four-loop level (11), (12), and (13). The most important reason of this enhancement is due to the relevance of the typical hadronic momenta in the loop. We recall that the elementary (quark) level contribution only had a typical momentum of O(m W ) ∼ O(m t ), and this feature, together with the GIM mechanism, forced the EDM of charged leptons to have a suppression factor m 2 b m 2 c m 2 s due to the cancellation between terms with very close values. We note that the GIM mechanism is also working at the hadron level. However, the cancellation among contributions with different flavors becomes much milder thanks to the fact that the typical momentum is replaced by the mass of the heaviest hadrons of each diagram. This is probably a general property of the hadronic CP violation in the SM, as similar enhancement is also relevant for the case of the EDM of neutron [155,158,161,162] or nuclei [163]. In this sense, the fact that the elementary contribution to the EDM appears only at the four-loop level is not truly essential in this strong suppression, but rather the GIM mechanism (or the antisymmetry of the Jarlskog invariant) is the main cause [158].
We should also comment on the observable effect of the electron EDM in experiments. The EDM of the electron is usually measured through the paramagnetic atomic or molecular systems, since the relativistic effect enhances its effect [94-122, 124, 125]. However, these systems also receive contribution from other CP violating mechanisms such as the CP-odd electron-nucleon interaction or the nuclear Schiff moment. Previously, the EDM of charged lepton was believed to be extremely small and the CP-odd electron-nucleon interaction was thought to be the dominant effect, with a benchmark value equivalent to the electron EDM as d (eN ) e ∼ (10 −39 − 10 −37 )e cm for paramagnetic systems [16,158,163]. By considering the strong enhancement at the hadronic level, we just obtained a value of the electron EDM which lies in this range. It is then an interesting question to quantify which one, between the electron EDM and the CP-odd electron-nucleon interaction, gives the leading contribution at the atomic level.

B. Error bar analysis
We now assess the uncertainty of our calculation. The first important source of systematics is the nonperturbative effect of the renormalization of the |∆S| = 1 four-quark operators. This was quantified to be about 10%, by looking at the variation of the Wilson coefficient of Q 2 in the NLLA in the range of the scale µ = 0.6 GeV to µ = m c = 1.27 GeV [163,186,187]. Another major systematics comes from the factorization of the vector meson matrix elements. According to the large N c analysis, the vacuum saturation approximation should work up to O(N −1 c ) correction. To be conservative, we set the error bar associated to it to 40%.
Let us now see the uncertainty related to the use of the HLS. The important point is that the one-loop level diagrams we evaluated are not divergent, so that the uncertainty due to the counterterms is absent and the stability of the coupling constants is guaranteed. However, we have to comment on the contribution from the other heavier hadrons which were overlooked in this paper. Here we consider the axial vector meson K 1 (1270) which, in the viewpoint of the mass difference, should be the most important hadron among the neglected ones, and show that its contribution is likely to be subleading. First, the decay constant of K 1 is not particularly enhanced, f K1 ∼ 170 MeV [200]. Regarding the other hadron matrix elements, the values do not exist in the literature, but it is possible to show that they are not enhanced either. The axial vector matrix element ρ|dγ µ γ 5 s|K 1 corresponds to the quark spin, so there should be a suppression due to the destructive interference generated by successive gluon emissions/absorptions [201,202]. The pseudoscalar matrix element ρ|dγ 5 s|K 1 has also no reasons to be enhanced, since this receives contribution from the Nambu-Goldstone boson pole, which is suppressed by the K meson mass in the present case. For the time being, we suppose that the error bar is represented by the contribution of K 1 (1270), so that the uncertainty associated to the neglect of heavier hadrons is given by ∼ 50%. In all, we conclude that the theoretical uncertainty is 70%.

V. CONCLUSION
In conclusion, we evaluated for the first time the hadron level contribution to the EDM of charged leptons in the SM, where the CP violation is generated by the physical complex phase of the CKM matrix. As a result, we found that this long distance effect is much larger than the previously known one, which was estimated at the elementary level. We could also rigorously show that, in the perturbative elementary level calculation at all orders, the EDM of charged leptons is always suppressed by quark mass factors due to the GIM mechanism. The main reason of the enhancement at the hadronic level is because we could avoid additional factors of m 2 b,c,s /m 2 W,t by embedding the heavy W boson or top quark contribution into the |∆S| = 1 low energy constants while keeping loop momenta of O(Λ QCD ) ∼ 1 GeV. In Fig. 7, we plot the EDM of the electron in the SM compared with the progress of the experimental accuracy. The electron EDM obtained in this work is d e ∼ 10 −39 e cm, which is still well below the current sensitivity of molecular beam experiments. The EDM experiments are however improving very fast, and we have to be very sensitive to their progress and to proposals with new ideas, with some of them claiming to be able to go beyond the level of O(10 −35 )e cm [203].
Yukawa Institute Computer Facility. This work is supported in part by the Special Postdoctoral Researcher Program (SPDR) of RIKEN (Y.Y.). limit, we obtain The low-energy parameters b 1 , b 3 B 0 and M were determined by fitting the Lattice QCD data in Ref. [198], where three fitting results denoted by Fit 1, Fit 2 and Fit 3 were obtained as summarized in Table III. The scalar matrix elements obtained by using these parameters are also summarized in Table III. In this study, we employ the averaged values obtained as ρ 0 |sd|K * 0 = −1.14 GeV, ω|sd|K * 0 = 1.88 GeV and φ|ds|K * 0 = 2.14 GeV.