On epsilon_K beyond lowest order in the Operator Product Expansion

We analyse the structure of long distance (LD) contributions to the CP-violating parameter epsilon_K, that generally affect both the absorptive (Gamma_12) and the dispersive (M_12) parts of the K0 -- K0bar mixing amplitude. We point out that, in a consistent framework, in addition to LD contributions to Im(Gamma_12), estimated recently by two of us, also LD contributions to Im(M_12) have to be taken into account. Estimating the latter contributions the impact of LD effects on epsilon_K is significantly reduced (from -6.0 % to -3.6 %). The overall effect of LD corrections and of the superweak phase being different from 45 degrees is summarised by the multiplicative factor kappa_epsilon = 0.94 +/- 0.02.


Introduction
Some of the most important tests of the Standard Model (SM) are offered by CP-violating observables, that in this model are supposed to originate from a single CP-odd phase in the CKM matrix [1].In particular, the crucial test is the hierarchy of CP-violating effects in B d , B s and K systems predicted by this model.Indeed the most prominent CP-violating observables in these three systems, S ψK S , S ψφ and K , predicted by the SM, differ by orders of magnitude Extensive analyses of the Unitarity Triangle have shown a spectacular consistency of the data for S ψK S and K , within the parametric and theoretical uncertainties in K , that until recently were rather sizable.The size of S ψφ measured by CDF [2] and DØ [3] appears to be by one order of magnitude larger than predicted by the SM, but the large experimental errors preclude any definitive conclusions.
Recently the consistency of the measured values for S ψK S and K within the SM has been challenged in [4,5] due to two facts: • The improved value of the relevant hadronic parameter BK from unquenched lattice QCD that enters the evaluation of K .This parameter is now not only known with an accuracy of 4% [6,7] but turns out to be significantly lower than previously found in lattice calculations, suppressing by 10% the previous estimates of K .
• A more careful look at K , that identified an additional suppression of | K |, summarised by a multiplicative factor κ = 0.92 ± 0.02 [5] to the previously adopted formula for K .
In view of these two suppressions, as demonstrated in [5], the size of CP violation measured in B d → ψK S might be insufficient to describe K within the SM.Clarifying this new tension is important as the S ψK S − K correlation in the SM is presently the most important direct relation between CP violation in the B d and K systems that can be tested experimentally.
The correction calculated in [5] originates from two factors: i) the difference of the superweak phase φ from 45 • , and ii) the long-distance contribution to K arising from the imaginary part of the absorptive amplitude of the K 0 − K0 mixing, Γ 12 .The latter effect has been estimated with the help of the ∆I = 1/2 dominance in K → ππ decays and the experimental value for / .
In the present paper we point out that at the same level of accuracy other effects should be considered, in particular the long distance contributions to the imaginary part of the dispersive amplitude M 12 .While this topic has been the subject of intensive discussions in the mid 1980's, it is important to have a fresh look at this issue in view of the decrease of the error in BK and of the theoretical advances during the last twenty five years.
Our paper is organized as follows.In Section 2 we present general formulae from which the different contributions to K can be clearly identified.In Section 3 we discuss K using the Operator Product Expansion (OPE).This allows us to identify the most important, still missing, long-distance contributions to ImM 12 .In Section 4 we estimate the size of these contributions in the framework of Chiral Perturbation Theory (CHPT), and briefly compare our findings with previous literature.We conclude in Section 5.

Notation and general formulae
Indirect CP violation originates in the weak phase difference between the (off-diagonal elements of the) Hermitian matrices M and Γ which control the time evolution of a neutral meson system.For the K 0 − K0 system one has Defining the eigenvectors the following phase-convention-independent relation holds: This represents indeed the indirect CP-violating parameter measured from the semileptonic charge asymmetries [8] or the Bell-Steinberger relation [9].The experimental smallness of Re(¯ ) makes the expansion to first non-trivial order in the weak phases an excellent approximation.At this level of accuracy we can identify Re(¯ ) with the real part of the complex quantity K , defined in terms of the K → 2π amplitudes, The two parameters are indeed related by K = ¯ + iξ, where ξ is the weak phase of the K 0 → (2π) I=0 amplitude, namely Expanding to first non-trivial order in the weak phases we have Introducing also the so-called superweak phase, φ = arctan (2∆m K /∆Γ), the expression for Re(¯ ) becomes A further simplification arises by the observation that the |(2π) I=0 final state largely saturates the neutral kaon decay widths.Since the |(2π) I=0 dominance in the sum over final states implies Expressing ReM 12 in terms of ∆m K and using Eq. ( 10) we arrive at which is consistent with The equation above allows us to calculate K by taking φ and ∆m K from experiment and calculating ImM 12 and ξ in a given model, in particular the SM.In Ref. [5] only short distance contributions to ImM 12 , represented by the well known box diagrams, have been included, while ξ has been calculated by relating it to the ratio / and taking the latter from experiment.As we will discuss in the following, this approach is not fully consistent: in this way ImΓ 12 and ImM 12 are evaluated at a different order in the OPE.In particular, long distance contributions to ImM 12 , which are of the same order of ImΓ 12 (the latter giving rise to the ξ term in Eq. ( 12)), are missing.
3 Decomposition of Re( K ) using the OPE As shown in Eq. ( 8) the evaluation of K requires the knowledge of the weak phases of both M 12 and Γ 12 .In this respect, we should emphasize that ImM 12 and ImΓ 12 are both generated at O(G 2 F ). Since ReM 12 and ReΓ 12 are very similar in size (φ ≈ 43.5 • ), we should consistently evaluate ImM 12 and ImΓ 12 at the same order in the OPE.
The relevant effective Hamiltonians are H ∆S=2 (contributing to ImM 12 only) and H ∆S=1 (contributing to both ImM 12 and ImΓ 12 ).The leading term in the OPE is the short-distance contribution to ImM 12 , where is the dimension-six ∆S = 2 effective Hamiltonian.The operator Q (6) does not mix with other operators and the imaginary part of its Wilson coefficient is dominated by terms proportional to the top-quark Yukawa coupling. 1At this order in the OPE one is neglecting terms generated by two insertions of ∆S = 1 operators (see Fig. 1) which cannot be absorbed into the coefficient of Q (6) .For consistency, this implies one should set ImΓ 12 to zero, since ImΓ 12 is the absorptive part of the diagrams in Fig. 1.In other words, the leading order result is obtained with the following substitutions in Eq. ( 11): Going one step forward requires taking into account: ).The structure of the subleading terms in ImM 12 is very similar to the O(G 2 F ) long-distance contributions to K → πν ν, discussed in Ref. [11].The relevant effective Hamiltonian changes substantially if we choose a renormalization scale above or below the charm mass.Keeping the charm as explicit degree of freedom, dimension-eight operators are safely negligible and the key quantity to evaluate is where the superscript in H (u,c) ∆S=1 denotes that the we have two dynamical up-type quarks.The absorptive part of T 12 contributes to Γ 12 , while the dispersive part contributes to M 12 .In the latter case the leading term in the expansion in local operators should be subtracted, being already included in ImM (6) 12 .In principle, extracting the subleading contribution to ImM 12 directly from Eq. ( 16) is the best strategy: the result would be automatically scale independent.However, in practice this is far from being trivial also on the lattice, given the disconnected diagrams in Fig. 1.
Following a purely analytical approach, we can integrate out the charm and renormalize H ∆S=1 below the charm mass.This allows to identify ξ with the weak phase of the A 0 amplitude, that, as mentioned, has already been estimated in Ref. [5] (see also [12]).On the other hand, ImM 12 assumes the form where ImM non−local 12 and ImM (8) 12 are not separately scale independent.The structure of the dimensioneight operators obtained integrating out the charm, and an estimate of their impact on K , has been presented in Ref. [13].According to this estimate, ImM (8) 12 is less than 1% of the leading term.The smallness of ImM (8) 12 can be understood by the following dimensional argument.First, it should be noted that the CKM suppression of the dimension-eight operators is (V * cs V cd ) 2 , namely the same CKM factor of the genuine charm contribution in H (6) ∆S=2 .Second, even if we are not able to precisely evaluate the hadronic matrix elements of the dimension-eight operators, we expect According to this scaling, the contribution of ImM (8) 12 is an O(m 2 K /m 2 c ≈ 15%) correction of the charm contribution (charm-charm loops) to ImM (6) 12 , which itself is an O(15%) correction of the total dimension-six contribution.We are thus left with an overall O(2%) naive suppression of ImM (8) 12 with respect to ImM (6) 12 .According to the explicit evaluation in Ref. [13], the actual numerical impact is even smaller.
The only potentially large long-distance contribution to ImM 12 is the contribution of the non-local terms enhanced by the ∆I = 1/2 rule.For this purpose, we observe that if we had a single weak operator in H ∆S=1 , this would generate the same weak phase to both ImM LD  12 and ImΓ 12 .As we discuss in more detail in the next section, this is what happens to lowest order in CHPT, where the ∆I = 1/2 part H ∆S=1 has only one operator, with effective coupling G 8 .Decomposing ImM LD  12 as a leading term proportional to G 2  8 , and a subleading term with different effective coupling and identify the weak phase of G 8 with ξ.As a result, This allow us to re-write Eq. ( 11) as follows where δ ImM12 encodes the subleading terms in ImM LD 12 | non−G 2 8 (including also ImM ).Note that, in the limit where the contribution of G 8 saturates ∆m K , the contribution of ξ would be absent.This is exactly what we should expect, since in this limit M 12 and Γ 12 would have the same weak phase but for the short-distance contribution to ImM 12 .

Estimate of long-distance effects in CHPT
A convenient framework for estimating the long-distance contribution to M 12 is provided by Chiral Perturbation Theory (CHPT).In this framework π, K and η fields are identified with the would-be Goldstone bosons arising from the SU (3) L × SU (3) R → SU (3) L+R symmetry breaking of the QCD action in the limit of vanishing light quark masses.Low-energy amplitudes involving these mesons, expanded in powers of their masses and momenta, are evaluated by means of an effective Lagrangian written in terms of the pseudo-Goldstone boson fields.
The lowest-order effective Lagrangian describing non-leptonic ∆S = 1 decays has only two operators, transforming as (8 L , 1 R ) and ( 27L , 1 R ) under the SU (3) L × SU (3) R chiral group.Moreover, only the (8 L , 1 R ) operator has a phenomenologically large coefficient, being responsible for the enhancement of ∆I = 1/2 amplitudes.As a result, the only term in the effective Lagrangian relevant to our calculation is L (2) where, as usual, we define which implies |G 8 | ≈ 9 × 10 −6 (GeV) −2 .As far as the weak phase of G 8 is concerned, at this level of accuracy we have Im(G 8 )/Re(G 8 ) = ξ.
In principle L (2) |∆S|=1 could contribute to M 12 already at O(p 2 ), via the tree-level diagram Fig. 2 (left).However, considering the O(p 2 ) relation among π 0 , η and kaon masses (i.e. the Gell-Mann-Okubo mass formula), this contribution vanishes [14].As a result, the first non-vanishing contribution to M 12 generated by L |∆S|=1 arises only at O(p 4 ).At O(p 4 ) we should evaluate loop amplitudes with two insertions of L |∆S|=1 and tree-level diagrams with the insertion of appropriate O(p 4 ) counterterms.Among all these O(p 4 ) contributions, the only model-independent, and presumably dominant, contribution to M 12 is the non-analytic one generated by the pion-loop amplitude in Fig. 2 (right), with r 2 π = m 2 π /m 2 K and where we have absorbed all finite (mass-independent) terms in the definition of the renormalization scale µ.This is the only contribution which has an absorptive part.As a consequence, its weak phase can be unambiguously related to the weak phase of the K 0 → (2π) I=0 amplitude to all orders in the chiral expansion.In addition, it is the only contribution that survives in the limit of SU (2) L × SU (2) R CHPT, which is known to represent a good approximation of the full O(p 4 ) amplitude in several K-decay observables where contributions from counterterms are fully under control (see e.g.Ref. [15]).
A CHPT calculation of M 12 complete to O(p 4 ) would require consideration of loops involving kaons and η's, as well as O(p 4 ) local counterterms.However, all these additional pieces are not associated with any physical cut.As such, they can effectively be treated as a local term whose overall weak phase cannot be related to the phase of the K 0 → (2π) I=0 amplitude. 2On account of the above considerations, 3 we refrain from a full O(p 4 ) CHPT calculation, and we focus on the pion-loop nonanalytic contribution only.Using the relation T 12 (µ), the result in Eq. (26) implies The absorptive part in Eq. ( 26) is nothing but the leading |(2π) I=0 contribution to Γ 12 , which gives rise to the relation (10).The dispersive part is the dominant contribution to M 12 in the leading-log approximation.The close link of these two terms is a further confirmation that we cannot neglect the long-distance contribution to ImM 12 if we want to keep track of all the O(ξ) terms in K .Using the result in Eq. ( 27) we can estimate the contribution to ImM 12 proportional to G 8 , which enters in the phenomenological formula for Re( K ) in Eq. (22).Setting µ = 800 MeV and varying it in the interval 0.6 ÷ 1 GeV leads to Following the notation of Ref. [5], we summarise the corrections to K due to LD effects and φ = 45 • , via the introduction of the phenomenological factor κ , defined by According to our result in Eq. ( 28), and taking into account the estimate of ξ obtained in [5], namely ξ = −(6.0± 1.5) × 10 −2 × √ 2| K |, the new numerical value of κ is This should be compared with 0.92 ± 0.02 in [5] and 0.92 ± 0.01 in [19], where only the long-distance contributions to ImΓ 12 (not those to ImM 12 ) have been included.

Comparison with previous literature
As anticipated in the introduction, the relative role of short-and long-distance contributions to K has been widely discussed in the literature in the mid 1980's [20][21][22][23][24][25][26][27][28].It is therefore useful to compare our findings to those in these earlier works.First of all, we agree on the main conclusion of all these papers, namely that LD K / exp K is small as long as / K is small.This is certainly correct, but it is not the point of our analysis: the issue we are addressing in this work is the size of the subleading (long-distance) contributions to K , that vanish in the limit of vanishing .
Second, we agree that single-particle intermediate states (π 0 , η, η ) do not generate a significant long-distance contribution to ImM 12 .The cancellation of π 0 and η contributions at the lowest order in the chiral expansion was noted first in [24].The role of the η was more debated [24][25][26][27].The issue was clarified in [28], where it was shown that the full nonet contribution (π 0 , η, η ) vanishes in the large N c limit.This is consistent with our findings, which are based on the updated and detailed analysis of the η exchange amplitude in Ref. [17].
Having clarified that single-particle intermediate states do not generate a significant contribution to ImM LD  12 , we are left with the two-pion intermediate state as the potentially leading contribution to ImM LD  12 .A naive estimate of this contribution at the partonic level seems to indicate that it is totally negligible; however, as we have shown, this is not the case because of the ∆I = 1/2 enhancement of K → 2π amplitudes.Our key observation is that, thanks to chiral symmetry and to the ∆I = 1/2 dominance, the weak phase of this contribution can be related to ξ, and the problem is shifted to the evaluation of the two-pion contribution to ∆m K , as summarised in Eq. (21).The numerical impact of this contribution is then estimated in two ways: i) a direct computation of the ππ loop in the leading-log approximation, Eq. ( 28), which provides a definite sign for this term; ii) the difference between the experimental value of ∆m K and the sum of its short-distance contribution and the other large long-distance contribution provided by the η exchange, which allows us to perform the useful cross-check: We finally note that our estimate of the O(ξ) corrections to K is based on the dominance of the ∆I = 1/2 amplitude in K → 2π decays.Given the experimental smallness of ∆I = 3/2 transitions, and the overall size of the effect we have evaluated (a few % correction to K ), this is certainly a very safe approximation.

Conclusions
In this paper we have presented a complete analysis of K beyond the lowest order in the OPE.In particular, we have analysed the structure of long distance (LD) contributions that affect both the absorptive (Γ 12 ) and dispersive (M 12 ) parts of the K 0 − K0 mixing amplitude.We have pointed out that, in a consistent framework, in addition to LD contributions to ImΓ 12 , estimated recently in [5], also LD contributions to ImM 12 have to be taken into account.Estimating the latter contributions in chiral perturbation theory, we found that they reduce by 40% the total impact of LD corrections on K .
The overall multiplicative factor κ in K , summarising the effect of LD corrections and of the superweak phase being different from 45 • , is increased to κ = 0.94 ± 0.02, to be compared with 0.92 ± 0.02 obtained without LD contributions to ImM 12 .

Figure 1 :
Figure 1: Contractions of the leading |∆S| = 1 four-quark effective operators contributing to M 12 at O(G 2 F ).

Figure 2 :
Figure 2: Tree-level and one-loop diagrams contributing to K0 -K 0 mixing in CHPT.