Revisiting Kaon Physics in General Z Scenario

New physics contributions to the Z penguin are revisited in the light of the recently-reported discrepancy of the direct CP violation in K → ππ . Interference eﬀects between the standard model and new physics contributions to ∆ S = 2 observables are taken into account. Although the eﬀects are over-looked in the literature, they make experimental bounds signiﬁcantly severer. It is shown that the new physics contributions must be tuned to enhance B ( K L → π 0 ν ¯ ν ), if the discrepancy of the direct CP violation is explained with satisfying the experimental constraints. The branching ratio can be as large as 9 × 10 − 10 when the contributions are tuned at the 10 % level.


Introduction
A deviation of the standard model (SM) prediction from the experimental result is recently reported in the direct CP violation of the K → ππ decays, which is called .The latest lattice calculations of the hadron matrix elements significantly reduced the theoretical uncertainty [1][2][3][4] and yield [5,6]  (1. 2) The deviations correspond to the 2.8-2.9σlevel.
Several new physics (NP) models have been explored to explain the discrepancy [11][12][13][14][15][16][17][18][19][20][21].In the literature, electroweak penguin contributions to / have been studied.#1 In particular, the Z penguin contributions have been studied in detail [11,13,15,22].The decay, s → dq q (q = u, d), proceeds by intermediating the Z boson, and its flavor-changing (s-d) interaction is enhanced by NP.Then, the branching ratios of K → πν ν are likely to be deviated from the SM predictions once the / discrepancy is explained.This is because the Z boson couples to the neutrinos as well as the up and down quarks.They could be a signal to test the scenario.
Such a signal is constrained by the indirect CP violation of the K mesons.The flavor-changing Z couplings affect the indirect CP violation via the so-called double penguin diagrams; the Z boson intermediates the transition, both of whose couplings are provided by the flavor-changing Z couplings.Such a contribution is enhanced when there are both the left-and right-handed couplings because of the chiral enhancement of the hadron matrix elements.This is stressed by Ref. [15] in the context of the Z -exchange scenario.In the Z-boson case, since the left-handed coupling is installed by the SM, the right-handed coupling must be constrained even without NP contributions to the left-handed one.Such interference contributions between the NP and the SM are overlooked in Refs.[11,13,15,22] [23].Therefore, the parameter regions allowed by the indirect CP violation will change significantly.In this letter, we revisit the Z-boson scenario.#2 It will be shown that the NP contributions to the right-handed s-d coupling are tightly constrained due to the interference, and thus, the branching ratio of K L → π 0 ν ν is likely to be smaller than the SM predictions if the / discrepancy is explained.We will discuss that NP parameters are necessarily tuned to enhance the ratio.A degree of the parameter tuning will be investigated to estimate how large B(K L → π 0 ν ν) and B(K + → π + ν ν) can become.
#1 QCD penguin contributions, e.g., through Kaluza-Klein gluons, have also been considered [11].#2 In this letter, we focus on the s-d transitions.The ∆F = 2 transitions such as ∆m B generally involve the interference contributions.

Z-penguin observables
In this section, we briefly review the Z-penguin contributions to ∆S = 2 and ∆S = 1 processes in the general Z scenario.Above the electroweak symmetry breaking scale, NP particles generate Wilson coefficients of the (dimension-6) effective operators, They are gauge invariant under the SM gauge transformations.In this letter, we focus on the operators, O L and O R , to demonstrate the impact of the interference between the SM and NP contributions.#3 After the electroweak symmetry breaking, they provide the flavor-changing (s-d) Z interactions, where the first term in the bracket is the Z-boson interaction, while the others are those of the Nambu-Goldstone boson, and we omitted the irrelevant terms for the interference effects.Here, the Wilson coefficients of O L and O R are normalized by the flavor-changing Z interactions.In the following, we omit the subscript "NP" in ∆ NP L and ∆ NP R for simplicity.

K and ∆m K
In the ∆S = 2 observables, there are the indirect CP violation K and the mass difference ∆m K in the K 0 -K 0 mixing.Since K has been measured precisely, and the SM prediction is accurate, it provides a severe constraint.The SM and NP contributions are shown as where ϕ = (43.51± 0.05) • .The NP contribution is given by the double penguin diagrams with the Z boson exchange (Fig. 1 (a)), #3 A similar discussion as follows is expected to hold for the effective operator O L .
Figure 1.The NP contributions to ∆S = 2 process.The black bubble denotes the vertices in Eq. (2.4) originating from the dimension-6 effective operators: O L and O R .The white bubble with "SM" denotes the SM flavor-changing Z interaction.Subfigures (b)-(e) correspond to the interference contributions between the NP and SM.A contribution from G 0 -exchange diagram is negligible because it receives a suppression factor by the external momentum, so that we omit it here.
where the right-hand side is [15] In these expressions, renormalization group corrections and long-distance contributions are included [25].In addition, one must take account of the interference terms between the SM and NP contributions (Figs.Here, the SM contribution, ∆ SM L , is generated by radiative corrections.At the one-loop level, it is calculated as where c W = cos θ W , λ i ≡ V * is V id with the CKM matrix V ij , and µ NP corresponds to the NP scale.#4 In this letter, the CKMfitter result [24] is used for the CKM elements, unless otherwise mentioned.The loop function is #5   C(x, µ NP ) = C(x) + ∆C(x, µ NP ).
(2.10) #4 In order to introduce how significant the interference contributions are, we ignore the renormalization group corrections to the dimension-6 operators above the electroweak scale except for a first leading logarithmic contribution ln(µ NP /m W ) which comes from Fig. 1 (e).This approximation is valid when the NP scale is not so far from the electroweak scale.#5 The loop function C(x, µ NP ) is consistent with the result in Ref. [26].
In the Feynman-'t Hooft gauge, the first term in the right-hand side corresponds to the Z-boson exchange diagram [22] (Fig. 1 while the second term is obtained from the Nambu-Goldstone boson loops (Figs. (2.12) The explicit form of ∆C(x) depends on the effective operators above the electroweak symmetry breaking scale.Here, they are supposed to be O L and O R .It is noted that the interference terms are gauge-independent.The interference terms (2.8) have been overlooked in Refs.[11,13,15,22].They cannot be ignored, as we will see in the next section.
The latest estimation of the SM value is [27 On the other hand, the experimental result is [10] They are well consistent with each other, and NP K must satisfy at the 2 σ level.#6  The kaon mass difference ∆m K consists of the SM and NP contributions: If we parameterize the NP contribution as the right-hand side is estimated as [15] The SM estimation SM K is sensitive to the CKM elements.If one uses V cb that is determined by the exclusive B → D ( * ) ν decays [28], Similarly to the case of K , there are interference terms between the SM and NP contributions, Here, ∆ SM L is given by Eq. (2.9), and the result is gauge-independent.These terms have been overlooked in the literature.
The experimental result is [10] ∆m exp K = (3.484± 0.006) × 10 −15 GeV. (2.20) Since the SM prediction involves sizable contributions of long-distance effects, the uncertainty is large.#7 Hence, we simply require that the NP contribution does not exceed the experimental value (with the 2 σ uncertainty): This constraint will turn out to be much weaker than NP K .

/
The flavor-changing Z interaction also contributes to ∆S = 1 observables.The direct CP violation / is shown as The NP contribution is estimated as [15 where B (3/2) 8 = 0.76 ± 0.05 from the lattice calculation.Here, the terms which are not proportional to B (3/2) 8 are omitted; the approximation is valid at the 10 % accuracy.A factor in the parenthesis gives c 2 W /s 2 W 3.33.Thus, the NP contribution can be enhanced easily by ∆ R .
As mentioned in Sec. 1, the SM prediction deviates from the experimental result at the 2.8-2.9 σ level.In this letter, we require that the discrepancy of / is explained at the 1 σ level as 10.0 where Ref. [6] is used for the SM prediction.

K
The (ultra-)rare kaon decay channels, K + → π + ν ν and K L → π 0 ν ν, are correlated with / as well as K and ∆m K in the general Z scenario.#8 They are represented as [15,32] B(K Here, X eff is estimated as (2.29) The SM predictions become (2.31) The branching ratios of K → π + − ( = e, µ) are also affected in the general Z scenario.However, K + → π + + − and K S → π 0 + − are dominated by a long-distance contribution through K → πγ * → π + − [31].On the other hand, such a contribution to K L → π 0 + − is forbidden by the CP symmetry, but is dominated by an indirect CP-violating contribution, K L → K S → π 0 + − [31].Therefore, it is challenging to discuss shot-distance NP contributions in these channels.
On the other hand, the experimental results are [33,34] (2.33) Although the current constraints on the NP contributions are very weak, the experimental values will be improved significantly in the near future.The NA62 experiment at CERN, which already started the physics run at low beam intensity in 2015, has a potential to measure B(K + → π + ν ν) at the 10 % precision by 2018 [35].The KOTO experiment at J-PARC is designed to improve the sensitivity for B(K L → π 0 ν ν), which enables us to measure it at the 10 % level of the SM value [36,37].As one can see from Eqs. (2.23) and (2.27), the NP contributions to B(K → πν ν) are correlated with those to / in the general Z scenario.Thus, if the / discrepancy is a signal of the scenario, these experiments would detect NP effects.

K
The branching ratio of K L → µ + µ − is also sensitive to the NP contributions to the flavor-changing Z couplings.Theoretically, only the short-distance (SD) contributions can be calculated reliably.They are shown as [15,38,39] where the first term in the right-hand side is the SM contribution, and the minus sign between ∆ L and ∆ R is due to the axial-vector current.The SM value is obtained as (2.36) On the other hand, it is challenging to extract a short-distance part in the experimental data (2.37) Since the constraint is much weaker than the SM uncertainties, we ignore them for simplicity and impose a bound on the Z couplings, The real parts of the NP contributions are constrained by B(K L → µ + µ − ).

Analysis
In this section, we examine the general Z scenario.Although the discrepancy of / could be explained by the scenario, the parameter regions would be constrained by K , ∆m K and K L → µ + µ − .In particular, the interference between the SM and NP contributions, Eq. (2.8), affects K significantly.In this section, we choose the NP scale, µ NP = 1 TeV, as a reference.As we will see, wide parameter regions are excluded.Thus, the discrepancy of / will be explained by tuning the model parameters.Let us introduce a quantity which parameterizes the tuning: If NP K is dominated by a single term, one obtains ξ 1 and there is no tuning among the model parameters.If the maximal value of ( K ) Z i is about ten times larger than NP K , ξ ∼ 10 is obtained; the model parameters are tuned such that there is a cancellation among ( K ) Z i at the 10% level.
The blue regions are excluded by the K , and the orange regions are by the B(K L → µ + µ − ).The constraint from K is much severer in RHS than LHS due #9 Our definition is almost the same as that in Ref. [45], where the authors discuss correlations between the tuning parameter and flavor observables.#10 This scenario is realized by chargino contributions to the Z penguin in the supersymmetric model [17,19,[42][43][44].#11 Randall-Sundrum models with custodial protection [45,46] can generate large ∆ R .However, there are additional effects, e.g., from KK-gluon diagrams for NP K .#12 In axial-symmetric scenarios, ∆ L = −∆ R , there are no NP contributions to K → πν ν. to the interference contributions, Eq. (2.8).There is no constraint from ∆m K in the parameter regions of the plots.The red and black dashed contours represent B(K L → π 0 ν ν)/B(K L → π 0 ν ν) SM and B(K + → π + ν ν)/B(K + → π + ν ν) SM , respectively.Here and hereafter, B(K L → π 0 ν ν) SM and B(K + → π + ν ν) SM denote the central values of the SM predictions, Eqs.(2.30) and (2.31).It is found that B(K L → π 0 ν ν) cannot be as large as the SM value as long as / is explained in LHS or RHS.On the other hand, if the / discrepancy is explained by LHS, the NP contribution to B(K Next, we consider ImZS.Such a situation is often considered to amplify ( / ) NP but suppress NP K .In the left panel of Fig. 3, the Z-penguin observables are shown as functions of Im ∆ L,R .The most severe constraint is from K due to the interference between the SM and NP.The other bounds are weak and absent in the plot.Since there are no real components of ∆ L,R , B(K Finally, LRS is shown in the right panel of Fig. 3. Similarly to the cases of RHS and ImZS, most of the parameter regions are excluded by K .The NP contributions to B(K L → µ + µ − ) vanish because the process is the axial-vector current.
In Fig. 4, contours of the tuning parameter ξ are shown for the simplified scenarios: LHS, RHS, ImZS, and LRS on the plane of the branching ratios of K → πν ν.We scanned the whole parameter space of ∆ L,R in each scenario and selected the parameters where / is explained at the 1σ level, and the experimental bounds from K , ∆m K , and B(K L → µ + µ − ) are satisfied (see the previous section for the experimental constraints).Then, ξ was estimated at each point.Several parameter sets predict the same B(K + → π + ν ν) and B(K L → π 0 ν ν).Among them, the smallest ξ is chosen in Fig. 4 for each set of B(K + → π + ν ν) and B(K L → π 0 ν ν).In most of the allowed parameter regions, ξ = O(1) is obtained.Thus, one does not require tight tunings in these scenarios.
In the figures, B(K L → π 0 ν ν) is smaller than the SM value by more than 30%.Hence, the scenarios could be tested by the KOTO experiment.On the other hand, B(K + → π + ν ν) depends on the scenarios.In LHS, we obtain 0 < B(K is comparable to or larger than the SM value, but cannot be twice as large.In ImZS, the branching ratios are perfectly correlated and displayed by a line in Fig. 4.Then, B(K + → π + ν ν) is not deviated from the SM one.The right panel of Fig. 4 is a result of the tuning parameter ξ in LRS.It is found that B(K L → π 0 ν ν) does not exceed about a half of the SM value.On the other hand, B(K + → π + ν ν) is comparable to or larger than the SM value, but cannot be twice as large, as is similar to RHS.Here, "SM" in the axis labels denotes the central values of B(K L → π 0 ν ν) SM and B(K + → π + ν ν) SM , Eqs. (2.30) and (2.31).In the colored regions, / is explained at 1σ, and the experimental bounds of K , ∆m K , and B(K L → µ + µ − ) are satisfied.The right region of the blue dashed line is allowed by the measurement of B(K + → π + ν ν) at 1σ.The Grossman-Nir bound [47] is shown by the blue solid line.The NP scale is set to be µ NP = 1 TeV.

General scenario
Let us consider the full parameter space in the general Z scenario.Both ∆ L and ∆ R are turned on.Then, B(K + → π + ν ν) and/or B(K L → π 0 ν ν) can be enhanced if the tuning for NP K is allowed.In Fig. 5, the branching ratios of K → πν ν and the tuning parameter are shown for the case of ( / ) NP = 15.5 • 10 −4 and NP K = 0.37 • 10 −3 .The flavor-changing Z couplings, and namely the NP contributions to K → πν ν, are limited by B(K L → µ + µ − ) and the tuning parameter.
In Fig. 6, contours of the tuning parameter ξ are shown on the plane of the branching ratios of K → πν ν.The whole parameter space of the general Z scenario is scanned.In the colored regions, / is explained at the 1σ level, and the experimental bounds of K , ∆m K , and B(K L → µ + µ − ) are satisfied (see the previous section for the experimental constraints).For each set of B(K + → π + ν ν) and B(K L → π 0 ν ν), the smallest ξ is chosen among the parameter sets which predict the same branching ratios.
Compared to the simplified cases in Fig. 4, B(K L → π 0 ν ν) can be enhanced.The tuning parameter is not necessarily very large if only one of B(K L → π 0 ν ν) and B(K L → µ + µ − ) is enhanced.However, ξ 30-40 is required to amplify both of them.If ξ 10 (5) is allowed, B(K L → π 0 ν ν) can be as large as 6 × 10 −10 (2 × 10 −10 ).In other words, O(10)% tunings are required to enhance B(K L → π 0 ν ν) by an order of magnitudes compared the SM prediction.The KOTO experiment can probe such large branching ratios in the near future.

Conclusion and discussion
The recent discrepancy of / may be a sign of the NP contribution to the flavorchanging Z coupling.In this letter, we revisited the scenario with paying attention to the interference effects between the SM and NP contributions to the ∆S = 2 observables.They affect K significantly once the right-handed coupling is turned on.Consequently, B(K L → π 0 ν ν) is smaller than the SM prediction in the simplified scenarios as long as / is explained.
In the general Z scenario, B(K L → π 0 ν ν) can be large if parameter tunings are allowed.It was found that the branching ratio can be enhanced by an order of magnitudes compared to the SM prediction if the NP contributions to K are tuned at the O(10)% level.It can be as large as 6 × 10 −10 (2 × 10 −10 ) for ξ 10 (5), which implies that the NP contributions to K are tuned at the 10 % (20 %) level.The KOTO experiment could probe such large branching ratios in the near future.
In the analysis, the NP scale was set to be 1 TeV.The NP contributions to K as well as the tuning parameter depend on it through the interference terms of the SM and NP (see Eq. (2.12)).For µ NP 1 TeV, ∆ SM L is enhanced as the NP scale increases.Hence, one naively expects that tighter tuning is required in K .However, renormalization group corrections could be larger in such a case.Such contributions will be studied in elsewhere (see also Ref. [48]).
Note added: While resubmitting the manuscript, Ref. [48] appeared on the arXiv.In comparison with our analysis, the differences are as follows: • we considered O L and O R operators with the first leading logarithmic renormalization group contribution ln(µ NP /m W ) which comes from Fig. 1 (e).Hence, the operator O L , the operator mixing among them through the renormalization group above the electroweak scale, nor running effects of the coupling constants are not considered in our analysis.
• the present analysis focuses on the parameter region where the / discrepancy can be explained at 1 σ level.
The loop function C(x, µ NP ) in Eq. (2.9), which comes from O L and O R , is in agreement with a result of Ref. [48], which comes from O Hq and O Hd , at the first leading logarithmic approximation.Notice that µ NP corresponds to µ Λ .

Figure 3 .
Figure 3.The Z-penguin observables are displayed in ImZS (left panel) and LRS (right).Notations of the lines and shaded regions are the same as in Fig. 2.

Figure 4 .
Figure 4. Contours of the tuning parameter ξ are shown in the simplified scenarios: LHS, RHS, and ImZS (left panel) and LRS (right).Here, "SM" in the axis labels denotes the central values of B(K L → π 0 ν ν) SM and B(K + → π + ν ν) SM , Eqs. (2.30) and (2.31).In the colored regions, / is explained at 1σ, and the experimental bounds of K , ∆m K , and B(K L → µ + µ − ) are satisfied.The right region of the blue dashed line is allowed by the measurement of B(K + → π + ν ν) at 1σ.The Grossman-Nir bound[47] is shown by the blue solid line.The NP scale is set to be µ NP = 1 TeV.