Power corrections to the CP-violation parameter ε K

: We present the calculation of the short-distance power corrections to the CP-violation parameter ε K coming from dimension-8 operators in the ∆ S = 2 eﬀective Hamiltonian. A ﬁrst estimate of this contribution, obtained for large- N c and in the chiral limit, was provided in ref. [1]. Here we evaluate and include the O ( m 2 K /m 2 c ) and O (Λ 2QCD /m 2 c ) contributions that, a priori, could induce O (1) corrections to previous esti-mates, as m K is numerically of order Λ QCD . Our computation shows that there are several dimension-8 operators besides the one considered before. Their eﬀect on ε K , however, accidentally cancels out to a large extent, leaving the ﬁnal correction at the level of 1%.


Introduction
The parameter ε K , which describes indirect CP-violation in the K 0 −K 0 system, represents one of the most interesting observables in Flavor Physics. It plays an important role in the Unitarity Triangle Analysis both within the Standard Model and beyond, see [2][3][4], references therein and [5]. As the phenomenon of K 0 −K 0 mixing is a loop process further suppressed by the GIM mechanism, ε K turns out to be a powerful constraint on New Physics models, for which it is important to have experimental and theoretical uncertainties well under control.
On the one hand ε K is experimentally measured with a 0.4% accuracy, ε exp K = (2.228 ± 0.011) · 10 −3 [6], on the other hand, the theoretical accuracy of the SM prediction is approaching a few percent level, mainly thanks to the improvement of the lattice determination of the relevant bag parameter B K [7]. Given the improved accuracy, the ξ term and the deviation of the phase φ ε from 45 • appearing in the theoretical expression ε K = e i φε sin φ ε ImM 12 ∆m K + ξ , (1.1) are not negligible [8]. Furthermore, the dominant long-distance contribution to M 12 , due to the exchange of two pions, has been evaluated in [9]. The inclusion of these three contributions gives a 6% reduction of the predicted central value of ε K . Another improvement producing a 2% reduction of the central value is the inclusion of the available NNLO QCD

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corrections to the Wilson coefficients of the ∆S = 2 effective Hamiltonian [10,11]. We have not included these corrections, however, since the relevant matrix element, B K , computed on the lattice is matched to the M S coefficient at the NLO only. While the NNLO result for the Wilson coefficients is a crucial step to reduce the theoretical uncertainty of ε K , at present there is no gain in evaluating the Wilson coefficient with NNLO accuracy when the accuracy of the matrix element is only at NLO: the overall uncertainty on ε K remains at the NLO level in any case. In this respect, the perturbative calculation of the NNLO matching of B K from the non-perturbative RI-MOM/SMOM schemes used in lattice calculations to the M S scheme would be welcome. As discussed in ref. [12], the NNLO QCD correction for the charm contribution is actually not needed. Given the increasing precision of the theoretical ingredients entering ε K , it is becoming important to include all terms expected to contribute to the theoretical evaluation of ε K at the percent level. In this paper, we focus on the power corrections due to the finite value of the charm quark mass, denoted as δ mc , coming from dimension-8 operators in the effective Hamiltonian. The naive dimensional estimate of δ mc is of O(m 2 K /m 2 c ) ∼ 15%. Its size, however, is reduced to about 2-3%, because the dominant top quark contribution to ε K is unaffected by these corrections. A first estimate of δ mc , obtained for large-N c and in the SU(3) chiral limit, has been presented in [1]. In these limits, there is only one dimension-8 operator contributing to δ mc [13]. Numerically, ref. [1] finds δ mc ∼ 0.5 − 1% and it has been neglected in phenomenological analyses up to now. O(1) corrections to this estimate may, however, be expected, because m K is numerically of order Λ QCD . In addition, in the large-N c limit, an operator non vanishing in the chiral limit is also discarded. Indeed the full calculation, presented in this paper, shows that there are several new operators contributing to δ mc . We find, however, that their effect on δ mc accidentally cancels out to a large extent, leaving the final correction at the level of 1%.
The paper is organized as follows. In section 2 we discuss the ∆S = 2 effective Hamiltonian relevant for the calculation, including dimension-8 operators. In section 3 we describe the matching procedure followed to compute the Wilson coefficients of the operators entering δ mc . In section 4 we present the evaluation of the matrix elements of the relevant operators in the vacuum insertion approximation (VIA), on which we rely lacking more precise determinations of the matrix elements at the leading order in chiral perturbation theory. In section 5 we propose a new approach to compute non-local contributions on the lattice, while in section 6 we estimate δ mc and comment on its impact on the theoretical prediction of ε K . Some more technical details, such as the relations that have been used in the computation, based on equations of motion and Fierz transformations, are collected in three appendices.

Outline of the calculation
In this work we calculate the corrections of O(m 2 K /m 2 c ) and O(Λ 2 QCD /m 2 c ), relative to the dominant terms, to the ∆S = 2 effective Hamiltonian describing ε K . For simplicity, in the following we will denote the ensemble of these O(1/m 2 c ) corrections as δ mc .

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The imaginary part of the ∆S = 2 effective Hamiltonian at the leading order in the Operator Product Expansion (OPE) and at zero order in the strong interactions, has the well known expression where the subscript LT means dominant (leading) term in the 1/m c expansion; λ i = V id V * is and the short-distance Inami-Lim functions S 0 's are computed by matching full and effective amplitudes for external states of four quarks with zero momentum [14]. The full amplitude is represented by the box diagrams shown in figure 1. In eq. (2.1), the CKM factor λ u has been eliminated by using the unitarity relation λ u + λ c + λ t = 0 and, thus, the Inami-Lim functions are linear combinations of the functions (D ij ) corresponding to box diagrams with internal i and j quarks, namely An equivalent way to express the ∆S = 2 effective Hamiltonian comes from the fact that in the standard convention for the CKM matrix [6] λ u is real, so that Im(λ 2 u ) = Im(λ 2 c + λ 2 t + 2λ c λ t ) = 0. By using this relation in eq. (2.1), one can write At the leading order in the 1/m c expansion both the Im(λ 2 c ) and Im(λ 2 t ) contributions are relevant. The latter, in fact, which is CKM suppressed, is enhanced by the top quark mass, as S 0 (x t ) grows as x t = m 2 t /M 2 W in the large x t limit, while S 0 (x c ) and S 0 (x c , x t ) vanish as x c = m 2 c /M 2 W in the small x c limit. For the subleading contributions due to dimension-8 operators, instead, only the Im(λ 2 c ) term has some relevance. The Im(λ 2 c ) subleading contribution, indeed, represents a correction of O(1/m 2 c ) to the leading term, while in the Im(λ 2 t ) term the relative correction is of O(1/m 2 t ) and can safely be neglected.

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The calculation of the δ mc is based on the determination of the Wilson coefficients of dimension-8 operators from matching conditions. To this aim, box diagrams have to be calculated with non vanishing external quark momenta. The O(p 2 ) contribution to the full amplitudes is then computed taking arbitrary external momenta and performing derivatives with respect to the external momenta. Since we are evaluating small O(1/m 2 c ) corrections to ε K , we will use the effective Hamiltonian expanded at zeroth order in the strong interactions. At O(α s ), apart from the contribution of dimension-8 operators, other contributions cannot be written in terms of local operators, as discussed in the following.
We consider from the beginning only the case of p 2 < m 2 c . The matching condition schematically reads where the first term on the r.h.s. represents the long distance contribution to the amplitude A, coming from the double insertion of the effective ∆S = 1 Hamiltonian, whereas the last term is the contribution that can be expressed as linear combination of local operators multiplied by suitable Wilson coefficients C W i . By assuming the charm quark mass to be large, that is by neglecting terms of O(1/m 2 c ), the first term on the r.h.s. of eq. (2.4) is absent and the second, in the Standard Model, is simply given by the operator , and the intermediate states propagating in the s or t channels are light, perturbation theory is of no use and A(∆S = 1 ⊗ ∆S = 1) must be evaluated non perturbatively. This is to be contrasted with the case of the box diagrams where the non-perturbative part is encoded in the matrix elements of local operators. The contribution to the mixing amplitude of the double penguin diagrams of figures 2 corresponds to the long distance corrections to ε K estimated in ref. [9]. Note that this kind of diagrams are also extremely difficult to compute from first principles, in a lattice QCD simulation, even in the simpler case of the K L -K S mass difference where only the operators Q 1 and Q 2 of the ∆S = 1 Hamiltonian are considered. Although there is no theoretical issue in the lattice calculation, the difficulty is the standard one of achieving sufficient precision when evaluating disconnected diagrams. For more details see for example the discussion on the disconnected diagrams Type 3 and Type 4 of refs. [15,16], where the reader can also find the definition of the operators Q 1 and Q 2 . For a direct calculation of ε K see ref. [17]. In the present paper, for our final numerical estimate of the correction at O(1/m 2 c ) to ε K we will indeed use the estimate of JHEP02(2022)181 A(∆S = 1 ⊗ ∆S = 1) by ref. [9]. Some further discussion about this non-local contribution will be given in section 5. The diagrams in the right panels of figure 2 will be discussed in section 6.

Matching
The Wilson coefficients that we want to determine can be derived by imposing the following matching conditions on amplitudes with four quarks and four quarks plus a gluon external states respectively, for some choice of external particle momenta p i m c , where "full" and "effective" denote the amplitudes in the full (Standard Model) and effective theories respectively.
The full amplitudes at O(α 0 s ) are calculated from the box diagrams of figure 1, with W propagators expanded at the lowest order in the large W mass, i.e. approximated by 4-fermion contact interactions. By choosing for the matching the color indices of the external quark and anti-quark states in a proper way, one can actually select the contribution of only one of the two box diagrams. For the four quark amplitude, for instance, we choose equal color indices for the quark and anti-quark in the initial state and similarly for the final state, but different between the initial and final states, which corresponds to selecting the box diagram in figure 1 (left panel). In this case, the amplitude has the form where p 1 , p 2 and p 3 , p 4 are initial and final momenta respectively and a and b are different color indices. The calculation of the box diagram with the four quarks in the external states

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then leads to the expression where, by omitting for brevity the external spinors, eliminating the combination p 4 − p 2 in favour of q = p 1 − p 3 and using the equations of motion, with Throughout this paper we use the notation At O(α 1 s ), the amplitude for the four quarks and a gluon includes the contributions of a gluon emitted either from an internal line or from an external leg. The expression for this amplitude is quite long and tedious, containing about 50 different Dirac structures. For this reason it is not reported here but given in appendix C. Having the full amplitudes at hand, we then need the effective amplitudes in order to impose the matching conditions of eq. (3.1). The computation of the effective amplitudes is based on the effective Hamiltonian where the C W i = G 2 F /(16 π 2 ) Im(λ 2 c ) C i are the Wilson coefficients that we want to determine and the O i are the local operators in the expansion [19], namely implying a sum over repeated color indices, and Tr(t A t B ) = 1/2 δ AB . The effective amplitude for a state of four quarks has then the form

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where the matrix elements for the various operators in eq. (3.8) read For the state of four quarks plus a gluon, the expressions of the matrix elements are quite long, see appendix C, and here we write the result only for the operator O 4 The operator O 4 , at variance with the other seven operators, has vanishing matrix element on the four quark state. As a consequence, the determination of the Wilson coefficient C 4 requires the matching between the full and the effective amplitude on a state with a gluon. From the matching conditions in eq. (3.1) we find

Matrix elements in the VIA
In this section we evaluate the matrix elements of the relevant four fermion operators in the VIA which, lacking a more precise quantitative determination, is expected to provide a reasonable estimate of the matrix elements. In this approximation the four fermion operators are expressed in terms of factored bilinear operators, the matrix elements of which, in this paper, are evaluated by using the chiral effective theory, at the leading order. Given the level of accuracy at which we are working we will consider neither the QCD corrections to the Wilson coefficients of the dimension-8 operators nor the usual problems related to the presence of renormalons in matching the original theory to its OPE at the subleading order [20].
In the chiral effective theory, one starts by defining the meson fields and the explicit symmetry breaking term,

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where f is the pseudoscalar decay constant defined such that f π ∼ 130 MeV, M = diag(m u , m d , m s ) and Π is the pseudoscalar meson operator, Under SU(3) L ⊗ SU(3) R chiral transformations, the fields U and χ transform as U → R U L † and χ → R χ L † . Next we provide the expressions of the relevant two-quark operators in terms of the chiral fields, at the leading order in the chiral expansion. We start by considering the dimension-3 operators: 3) In the chiral expression of the tensor operators the low energy constant α T is given by [19]. Note, however, that these operators do not contribute in the VIA, since their matrix elements between the vacuum and a kaon state vanish. We now list the dimension-4 operators [21] at the lowest order in the chiral expansion: where on the r.h.s. of eqs. (4.9) and (4.10) we only show explicitly the terms which contribute in the factorized case. Eq. (4.10) can be derived from (4.9) using the identity The chiral expression for the operatorsq i − → D µ Lq j andq i ← − D µ Rq j can be derived by using the relation: Finally, we also need the dimension- They are related by the following identity: (4.14) Using eq. (4.5) in the above expression, one obtains In terms of the chiral fields one can write [19] g sqi γ µLG where δ 2 K is a low-energy constant. A QCD sum rules calculation [19] provides the estimate δ 2 K = 0.12 ± 0.07 GeV 2 . Note that the matrix element of the operator in eq. (4.16), as all the operators of O(g s G A µ ), is of O(α 0 s ). We are now ready to compute the matrix elements of the 4-fermion operators in the VIA and at the lowest order in the chiral expansion. We compute always the matrix element between an initial K 0 and a finalK 0 state. Thus ∂ µ K 0 = −ip K µ K 0 and ∂ µK 0 = ip K µ K 0 . As an example, we report the detailed calculation of the matrix element of the operator O 3 :

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where, in evaluating the matrix element between kaon states we have replaced the low energy constant f with the kaon decay constant f K . By neglecting the mass of the down quark, m d , and thus also replacing B = M 2 K /m s , one finally obtains The VIA for the matrix elements of the other relevant 4-fermion operators can be derived in a similar way. In the limit m d → 0, we obtain Note that in the chiral power counting the leading contribution is given by the operators O 3 and O 4 whose matrix elements are of O(p 2 ) (for O 3 , however, the O(p 2 ) contribution is suppressed by 1/N c ). The matrix element of O 6 is of O(p 6 ) while all other matrix elements are of O(p 4 ). In ref. [1] the chiral expansion was performed at the leading order and the large-N c limit was assumed. As a consequence, only the O 4 contribution was included. We observe that the contribution of the other seven operators, which we include in the present work for the first time, can a priori introduce an O(1) correction to the O 4 contribution, as m K is numerically comparable to δ K (δ 2 K /M 2 K 0.5).

A possible approach to compute non-local contributions at O(α s p 2 /m 2 c )
Some remark may be useful at this point. Concerning the non-local contributions that appear at O(α s p 2 /m 2 c ), let us take, for example, the diagram on the left upper panel of figure 2 at the lowest perturbative order in the Standard Model, by considering only the CP violating contribution to K 0 −K 0 mixing. By using unitarity (λ u + λ c + λ t = 0), neglecting λ t × O(p 2 /m 2 c ) corrections and without the resummation of large QCD logarithms, we find that there is only one contribution to the amplitude which goes as where α s is the strong coupling constant and the factor log m 2 t /m 2 c corresponds to the Wilson coefficient of the operator   figure 2, which is a standard connected diagram much easier to evaluate in lattice QCD [15,16]. Following the standard approach, even if the momentum flowing in the charm-top GIM penguin is of O(p 2 ) ∼ O(Λ 2 QCD ), in eq. (5.1) the coefficient of Q can be computed in perturbation theory since both the quarks in the GIM loop are heavy (they are either the top or the charm). Eventually, the most convenient choice is to take µ = m c as renormalisation scale. In the calculation of the double penguins, in order take into account higher order QCD corrections, we can imagine to substitute to the Standard Model the double insertion of the low energy ∆S = 1 Hamiltonian and compute on the lattice, with this effective theory, the charm-top and up-charm GIM penguins. This extreme accuracy, although feasible, is probably not necessary since we are evaluating one of the corrections to the leading term and the uncertainties entailed in other terms, specifically in the calculation of the matrix elements of the local dimension-8 operators, are probably of the size, if not larger, than the neglected higher-order QCD effects. The calculation of the other double penguin diagram, shown on the left lower panel of figure 2 could proceed in the same way by replacing it with the connected diagram on the right lower panel of the same figure. 6 Result for the power corrections to ε K By combining the results for the Wilson coefficients of the effective Hamiltonian given in section 3 and those for the matrix elements of the four fermion operators evaluated in section 4 in the VIA, we obtain

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which represents the main result of this paper. In eq. (6.1) we have neglected small terms of O(p 6 ), proportional to M 2 K m 2 s , that contribute for ∼ 1.5% of the total correction. The above result can be compared with the leading 1 contribution to the effective Hamiltonian that can be written in the form Therefore, with respect to the term proportional to Im λ 2 c in the LT effective Hamiltonian, Im(λ 2 c ) , eq. (6.1) represents a relative correction given by Up to higher order terms, we may then write where δ BGI = 0.02 was computed in ref. [9] and the relative correction with respect to the full LT Hamiltonian is given by that is the δ mc correction increases the theoretical prediction for ε K by 1%. As is evident from eq. (6.5), the O(m 2 K /m 2 c ) correction, included here for the first time, though a priori comparable to the O(δ 2 K /m 2 c ) term, turns out to be numerically smaller. At the origin of this numerical result there is the larger numerical value of the Wilson coefficient C 4 w.r.t. the coefficients of the other seven operators, and some cancellation occurring among the terms proportional to f 2 K m 4 K . The dominant uncertainty on our result for δ mc is represented by the VIA. We evaluated it assuming for the matrix elements of the dimension-8 operators either a uniform distribution centered on the VIA with a 50% half width or a Gaussian distribution with σ = 30%; in both cases we obtain a 30% error on δ mc , which keeps a negligible impact on the ε K total uncertainty.

A Useful relations coming from the equations of motion
In the calculation of the contribution of the gluon on an external leg, both in the full and in the effective theory, we can eliminate some structures containing the completely antisymmetric tensor, by using some relations originated from the equations of motion. These relations are obtained from the Chisholm identity by saturating one Lorentz index with a momentum which induces the Dirac equation of motion. These relations show that four of the structures involving the completely antisymmetric tensor can be reduced in terms of other structures. They can be written as

B Useful Fierz relations
The matrix elements of the operators entering the effective Hamiltonian H eff 1/m 2 c of eq. (3.7) are estimated by using the VIA, in which Fierz transformations play an important role. The Fierz transformations can be obtained starting from the following general formula

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where we used the definition σ αβ = γ α , γ β /2. From eq. (B.1) one derives, in particular, It is also useful to report the general formula for the Fierz of color indices. It reads: where C 1 and C 2 are generic color matrices. From eq. (B.5) one derives, in particular,

C Matching with the external state made up of four quarks and a gluon
In this appendix we collect the amplitudes that enter the matching with the external state made up of four quarks and a gluon. As discussed in the main text for the case of the four quark external state, one can choose equal color indices for the quark and anti-quark in the initial state and similarly for the final state, which corresponds to selecting the box diagram in figure 1(a). We do the same also when the gluon is added in the external state. With the additional gluon, moreover, one can impose the matching condition by considering only contributions with the color generator t A appearing in the initial state bilinear or in the final state bilinear. We perform the first choice, that is we consider the amplitudes having the forms By omitting for brevity the external spinors (and the gluon polarization vector), the matrix elements for the various operators in eq. (3.8), read

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We finally report the full amplitude for the four quarks and a gluon

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.