Kaon Physics Without New Physics in ε K

Despite the observation of signiﬁcant suppressions of b → sµ + µ − branching ratios no clear sign of New Physics (NP) has been identiﬁed in ∆ F = 2 observables ∆ M d,s , ε K and the mixing induced CP asymmetries S ψK S and S ψφ . Assuming negligible NP contributions to these observables allows to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of | V cb | and | V ub | . Furthermore this method avoids the impact of NP on the determination of these parameters present likely in global ﬁts. Simultaneously it provides SM predictions for numerous rare K and B branching ratios that are most accurate to date. Analyzing this scenario within Z (cid:48) models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to ε K , signiﬁcant NP contributions to K + → π + ν ¯ ν , K L → π 0 ν ¯ ν , K S → µ + µ − , K L → π 0 (cid:96) + (cid:96) − , ε (cid:48) /ε and ∆ M K can be present. In the simplest scenario, this is guaranteed, as far as ﬂavour changes are concerned, by a single non-vanishing imaginary left-handed Z (cid:48) coupling g Lsd . This scenario implies very stringent correlations between the Kaon observables considered by us. In particular, the identiﬁcation of NP in any of these observables implies automatically NP contributions to the remaining ones under the assumption of non-vanishing ﬂavour conserving Z


1 Introduction
Despite a number of anomalies observed in B decays, it has recently been demonstrated [1] that the quark mixing observables can be simultaneously described within the Standard Model (SM) without any need for new physics (NP) contributions.As these observables contain by now only small hadronic uncertainties and are already well measured this allowed to determine precisely the CKM matrix on the basis of these observables alone without the need to face the tensions in |V cb | and |V ub | determinations from inclusive end exclusive tree-level decays [2,3].Moreover, as pointed out in [4], this also avoids, under the assumption of negligible NP contributions to these observables, the impact of NP on the values of these parameters , which are most likely present in global fits.Simultaneously it provides SM predictions for numerous rare K and B branching ratios that are the most accurate to date.In this manner the size of the experimentally observed deviations from SM predictions (the pulls) can be better estimated.
As over the past decades the flavour community expected significant impact of NP on ε K , ∆M s and ∆M d , these findings, following dominantly from the 2+1+1 HPQCD lattice calculations of B s,d − Bs,d hadronic matrix elements [5], are not only surprising but also putting very strong constraints on NP models attempting to explain the B physics anomalies in question.
Concentrating on the K system, which gained a lot attention recently [6][7][8], one could at first sight start worrying that the absence of NP in a CP-violating observable like ε K would exclude all NP effects in rare decays governed by CP violation such as K L → π 0 ν ν, K S → µ + µ − , K L → π 0 + − and also in the ratio ε /ε.Fortunately, these worries are premature.Indeed, as pointed out already in 2009, in an important paper by Monika Blanke [9], the absence of NP in ε K does not preclude the absence of NP in these observables.This follows from the simple fact that where g sd is a complex coupling present in a given NP model.Setting Re(g sd ) = 0, that is making this coupling imaginary, eliminates NP contributions to ε K , while still allowing for sizable CP-violating effects in rare decays and ε /ε.This choice automatically eliminates the second solution considered in [9] (Im(g sd ) = 0), which is clearly less interesting.But there are additional virtues of this simple NP scenario.It can possibly explain the difference between the SM value for the K 0 − K0 mass difference ∆M K from RBC-UKQCD [10] and the data1 Indeed, as noted already in [12] and analyzed in the context of the SMEFT in [13], the suppression of ∆M K is only possible in the presence of new CP-violating couplings.This could appear surprising at first sight, since ∆M K is a CP-conserving quantity, but simply follows from the fact that the BSM shift (∆M K ) BSM is proportional to the real part of the square of a complex g sd coupling so that With pure imaginary coupling, this suppression mechanism is very efficient.The required negative contribution implies automatically NP contributions to ε /ε and also to rare decays K → πν ν, K S → µ + µ − and K L → π 0 + − , provided this NP involves non-vanishing flavour conserving q q couplings in the case of ε /ε and non-vanishing ν ν and µ + µ − couplings in the case of the rare K decays in question.In the case of Z models this works separately for lefthanded and right-handed couplings and even for the product of left-handed and right-handed couplings as long as all couplings are imaginary, as discussed for instance in [13,14].This happens beyond the tree level and in any model in which the flavour conserving couplings are real.Indeed, ε K being a ∆F = 2 observable must eventually be proportional to g2 sd .However, in order to see the implications of the absence of NP in ε K for Kaon physics, we concentrate in our paper on Z models.
But there is still one bonus in this scenario.With the vanishing Re[g sd ] there is no NP contribution to K L → µ + µ − which removes the constraint from this decay that can be important for K + → π + ν ν.
In [13] we have analyzed the observables listed in the abstract in NP scenarios with complex left-handed and right-handed Z couplings to quarks in the context of the SMEFT.In view of the results of [1] it is of interest to repeat our analysis, restricting the analysis to an imaginary g sd coupling as it eliminates NP contributions to ε K and also lowers the number of free parameters.Moreover, having already CKM parameters determined in the latter paper, the correlations between various observables are even more stringent than in [13] so that this NP scenario is rather predictive.
However, it is also important to investigate the impact of renormalization group effects on this simple scenario in the context of the SMEFT to find out under which conditions NP contributions to ε K are indeed negligible.
Our paper is organized as follows.In Section 2, concentrating on left-handed couplings, we summarize our strategy that in contrast to our analysis in [13] avoids the constraints from K L → µ + µ − and ε K .We refrain, with a few exceptions from listing the formulae for observables entering our analysis as they can be found in [12,13] and in more general papers on Z models in [15] and in [16] that deals with 331 models.In Section 3, we define the Z setup, and the observables analyzed by us and briefly discuss the impact of SMEFT RG running on ε K .In Section 4 we present a detailed numerical analysis of all observables listed above in the context of our simple scenario including QCD and top Yukawa renormalization group effects.We conclude in Section 5.

Strategy
Our idea is best illustrated on the example of a new heavy Z gauge boson with a ∆S = 1 flavour-violating coupling g sd (Z ) that is left-handed and purely imaginary 2 .
As the tree-level contribution of this Z is proportional to the imaginary part of the square of this coupling, it does not contribute at tree-level to ε K as stated in (2).However, it contributes to ∆M K as seen in ( 4).It contributes also to several rare Kaon decays and also to ε /ε, for which the NP contribution is just proportional to Img sd (Z ).
and consequently ¯ = 0.164 (12), η = 0.341( 11) , where λ t = V * ts V td .The remaining parameters are given in Table 1.As the SM prediction for ε /ε is rather uncertain [30,31], we will, as in [13], fully concentrate on BSM contributions. 3Therefore in order to identify the pattern of BSM contributions to flavour observables implied by allowed BSM contributions to ε /ε in a transparent manner, we will proceed in our scenario by defining the parameter κ ε as follows [12] In the case of ε K we allow only for very small NP contributions that could be generated by RG effects despite setting the real part of g sd (Z ) to zero at the NP scale that we take to be equal to M Z .Explicitly which amounts to 1% of the experimental value.The SM predictions for K + → π + ν ν and K L → π 0 ν ν are given in (12).For the remaining decays one finds with the CKM parameters in (13) [4] B(K L → π 0 e + e − ) SM = 3.48 3 General master formulae for the BSM effects can be found for example in [32][33][34][35].
3 Setup 6 where for the K L → π 0 + − decays the numbers in parenthesis denote the destructive interference case.These results, that correspond to the CKM input in (13), differ marginally from the ones based on [36][37][38][39] used in our previous paper.Note that the full K S → µ + µ − branching ratio estimated in the SM including long-distance contributions is B( The experimental status of these decays is given by [40][41][42]: 3 Setup

The Z Model
The interaction Lagrangian of a Z = (1, 1) 0 field and the SM fermions reads: Here q i and i denote left-handed SU (2) L doublets and u i , d i and e i are right-handed singlets.This Z theory will then be matched at the scale M Z onto the SMEFT, generating effective operators.The details of the matching onto SMEFT can be found in Ref. [13].As far as NP parameters are concerned we have the following real parameters The remaining parameters are set to zero.The latter flavour conserving couplings are required for the ratio ε /ε, for which significant differences between various SM estimates can be found in the literature [30].The relation between these couplings makes the electroweak penguin contributions to ε /ε the dominant NP contributions.This is motivated by the analyses in [12,13] in which the superiority of electroweak penguins over QCD penguins in enhancing ε /ε has been demonstrated.In fact the latter were ruled out by back-rotation effects [43].

The ε K due to RG Running
In our scenario, at the scale of the Z mass, the ∆F = 2 operators contributing to ε K are assumed to be suppressed.However, this assumption may not always hold at the low energy scales and the contributing operators might still be generated due to SMEFT RG running.We should make sure that this is not the case.At the high scale, we have the following four-quark operators [O (1)  qq ] 2121 = (q 2 γ µ q 1 )(q 2 γ µ q 1 ) , [O qd ] 2111 = (q 2 γ µ q 1 )( d1 γ µ d 1 ) .
Figure 1: The dependence of κ and κ on Im(g 21 q ) are shown.The Re(q 21 q ) at scale Λ is fixed to be zero.The diagonal quark and lepton couplings are fixed as discussed in the text.Further, the solid and dashed lines correspond to Z with and without SMEFT RG running effects.The WET (QCD+QED) running is included in all cases.
In our scenario, at the scale Λ = M Z , the Wilson coefficient of the first operator is real and the second one is imaginary but it does not have the right flavour indices, so naively these operators should not affect ε K .Through operator-mixing [44,45], at the EW scale, in the leading-log approximation, we have ∆ C (1)  qq 2121 (M Z ) ≈ − ∆ C (1) qd 2111 (M Z ) + λ 11 t C (1) here, λ 11 t ≈ 8.3 × 10 −5 , λ 22 t ≈ 1.7 × 10 −3 .From the above two equations, it is clear that the RG running cannot induce the imaginary parts of C qd 2121 at M Z .Therefore, our initial assumptions are stable under the SMEFT RGEs.However, we still find a small effect in ε K due to the back-rotation effect [43], which is basically due to the running of the SM down-type Yukawa couplings.This is illustrated in Fig. 1.We observe that while the RG effects on ε K for this range of Im(g 21 q ) are very small, they are large in the case of ε /ε.Observable

Observables
In our numerical analysis we investigate the following quantities: R ∆M K = ∆M BSM ∆F = 2 observables only [1], the paucity of NP parameters in this scenario implies strong correlations between all observables involved.In the coming years, the most interesting will be an improved measurement of the K + → π + ν ν branching ratio, which if different from the SM prediction will imply NP effects in the remaining observables considered by us.Figs. 2 and 3. illustrate this in a spectacular manner.The size of possible enhancements will depend on the involved couplings, in particular, leptonic ones so that correlations with B physics observables will also be required to get the full insight into the possible anomalies.With improved theory estimates of ε /ε and ∆M K , the improved measurements of K L → π 0 ν ν and K S → µ + µ − , and those of B decays, this simple scenario will undergo very strong tests.

Figure 2 :
Figure 2: The correlations between the observable R + ν ν and various other Kaon observables are shown.The NP parameters are given in (29) and are the same as in Fig. 1.