(g − 2)μ versus K → π + Emiss induced by the (B − L)23 boson

To address the long-standing (g − 2)μ anomaly via a light boson, in ref. [1] we proposed to extend the standard model (SM) by the local (B − L)23, under which only the second and third generations of fermions are charged. It predicts an invisible Z′ with mass O100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}(100) $$\end{document} MeV, and moreover it has flavor-changing neutral current (FCNC) couplings to the up-type quarks at tree level. Such a Z′, via KL→ π0 + Z′(→vv¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v\overline{v} $$\end{document}) at loop level, may be a natural candidate to account for the recent KOTO anomaly. In this article, we investigate this possibility, to find that Z′ can readily do this job if it is no longer responsible for the (g − 2)μ anomaly. We further find that both anomalies can be explained with moderate tuning of the CP violation, but may contradict the B meson decays.


Introduction and experiment reviews
A dark world far below the weak scale is introduced in many different contexts of new physics beyond the Standard Model (SM). Whether violating the flavor structure of the SM or not, members of the light dark world may imprint in the rare decays of K and B mesons, etc. For instance, it is known many years ago that, a light dark photon which does not have tree-level flavor-changing neutral current (FCNC) couplings to quarks can lead to flavor violation decay K → πZ [2,3]. Hunting hints of such a world is the target of many experiments like BaBar, Belle and LHCb, etc.

(2.3)
A similar strong bound is available from E949. Hereafter, we refer to the loophole region of m Z with the neighborhood of m π 0 indicated above removed. In our model, merely Z in this region is allowed to account for the three KOTO events; maybe only two events can be explained in terms of the analysis in ref. [42].

The profile of Z -induced FCNC for KOTO
The s → d transition hinted by KOTO can happen either at tree level via FCNCs in the down-type quark sector or at loop level due to FCNCs originating from the up-type quark sector. Alternatively, new physics does not introduce extra FCNCs, and that transition is proceeding in the framework of CKM theory. The simplest candidate, a spin-0 scalar mixing with the SM Higgs doublet is such one. Nevertheless, its spin-1 similarity, the dark photon does not work [21]. In other words, for a light massive gauge boson Z , additional FCNCs beyond the SM is indispensable.
To that end, as a simple consideration, we presume that Z comes from a gauged Abelian flavorful symmetry U(1) X which has the following features: • The SM fermions carry non-universal charges of U(1) X , which then may result in nonsimultaneous diagonalization of quark mass matrix and quark-Z current couplings. Obviously, quarks should be charged under this gauge group.
• Besides, in order to make Z dominantly decay into a pair of invisible particles, neutrinos or dark matter-like states are also supposed to be charged under it. Considering the lightness of Z , we do not need a hierarchy of charges as long as the coupling to electron is suppressed.
• The gauge coupling is tiny, in particular for the case that the tree-level FCNCs are in the down-type quark sector. However, the massive gauge boson is at the sub-GeV level, and hence the spontaneously breaking scale of U(1) X is high. Therefore, in general there is no light flavon associated with U(1) X .
Model building can be explored along a variety of lines, and in this paper we take advantage of a model proposed by us before [1], which naturally fits the outlined profiles.

The local (B − L) 23 model and its patterns of FCNCs
Originally, this model aims at addressing the long-standing (g−2) µ anomaly via the light Z from the flavored local B − L extension to the SM; under this gauge group, only the second and third generations of fermions are charged. This gauge group is dubbed as (B − L) 23 in this paper. Such an arrangement leads to an electron and proton phobic Z , which helps avoid the relevant strong exclusions such as Borexino [43][44][45] and COHERENT [46][47][48][49], thus allowing the desired Z having a mass ∼ O(10) MeV and a moderately small gauge coupling g B−L ∼ O(10 −4 -10 −3 ). Because the Z has mass below 2m µ and moreover has suppressed coupling to electron through the kinetic mixing between Z and the photon, the dominant decay channel is into a pair of neutrinos, having decay width size is L = 3m, while the size of the NA64 detector is much larger, L = 150m. Since Z here is invisible, its lifetime is irrelevant to our following discussions.

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To generate the correct CKM structure, additional FCNCs associated with Z are unavoidable in this model. Therefore, qualitatively this Z fits the invisible light particle explanation to the KOTO events. If it succeeds quantitatively, accounting for both (g −2) µ and the KOTO anomalies at the same time, then the model should deserve top priority. Unfortunately, without new CP sources, we will find that the resulting s → d transition rate is too large for the typical Z parameters for (g − 2) µ . However, our Z is still a good spin-1 candidate for KOTO, as long as we abandon its responsibility in (g − 2) µ . 3 Let us discuss more on the FCNCs in this model. The (B − L) 23 forbids the mixings between the first and other two generations of fermions. Introducing flavons to regenerate these mixings then leads to FCNCs. Its patterns depend on the origins of the mixings, from the up-and/or down-type quark sectors. As a matter of fact, for the case that there are FCNCs in the down-type quark sector, in ref. [1] we have already taken into the constraints from the KOTO report which has not claimed the excess yet [25]. So, one can simply utilize the results there to derive the viable parameter space for three events.
We focus on the case that the FCNCs are present only in the up-type quark sector, described by the following terms where the Hermitian coupling matrices are defined as where g B−L is the gauge coupling; U u and W u are diagonalizing matrices of up quark Yukawa coupling Y u for left-and right-handed fields, respectively: Note that in the above expressions the parts giving rise to FCNCs are determined by (U u ) 1i and (W u ) 1i , which is traced back to the fact the FCNCs originate from the first and other two families of fermions carrying different B − L charge.
Using the above feature, and working in the favored scenario which takes advantage of a singlet flavon plus up-quark-like vector-like fermions to realize CKM, one can show that the coupling matrix g u L can be completely determined by the CKM elements, up to g B−L . The CKM matrix is defined by the mixing matrices for the left-handed quarks, 3 However, hopefully, (g − 2)µ can be explained by vector-like leptons which are introduced to produce the correct Pontecorvo-Maki-Nakagawa-Sakata matrix in the neutrino sector, even when there are no new CP sources. We leave this to a further publication.

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Moreover, U d is block diagonalized and can be parameterized as where the blank block denotes the 2 × 2 unitary matrix used to diagonalize the second and third families of down-type quarks; it may contain some CP phases, but dull in the FCNC processes in studying. By substituting eq. (3.6) for eq. (3.5) one obtains As a result, we obtain the explicit numerical form of g u L in terms of the CKM elements: The diagonal elements of g u L are real (true also for g u R ), and the suppression of (g u L ) 11 is a result of the neutrality of the first generation fermions under the B − L group. In particular, the largest CP violation is from the (1, 3)-and (3, 1)-element, ∼ 10 −3 , with others suppressed by orders of magnitude.
On the contrary, the structure of W u cannot be determined as U u in eq. (3.7) since there is no relation like in eq. (3.5). In principle it is regarded as a generic three by three unitary matrix, and in the later discussions we will make a detailed study on (W u ) 1i , to investigate its impacts on the meson rare decays. Of interest, W u can introduce some new CP sources which will largely contribute to K L → π 0 Z decay and has the potential to reduce its width by cancellation.
The readers may wonder if there are other advantages of the gauge group chosen here, since merely arranging the second or the third generation of fermions charged under B − L basically leads to a Z assembling this one. A strong support may be from neutrino physics. Letting the second and third generations of fermions charged under B − L gives a better understanding on neutrino masses and mixings: two right handed neutrinos are necessary to cancel anomalies, which is the minimal number to produce the acceptable neutrino mass pattern in the seesaw mechanism; moreover, the gauge symmetry leads to the approximate µ − τ symmetry demonstrated in neutrino mixings.

K → πZ from up-type quark FCNC insertion
In our last study, we merely studied the FCNCs in the up-type quark sector given in eq. (3.2), e.g., the top quark rare decay t → cZ , but we neglected the induced FCNCs in the down-type quark sector via the W -loop. They are the targets of this paper, and we will first calculate K → πZ and then investigate its implications to the model, facing the KOTO anomaly and as well the null results from E949/NA62.

Calculation of K → πZ
At quark level, this process is described by the effective vertexd(p i )Γ µ (p i , p j )s(p j ), and by taking advantage of the Lorentz invariance and Ward-Takahashi identity one reaches the following structure (up to possible chiral projection operators) where the coefficients are functions of q 2 with q = p j − p i the momentum carried by Z . We will not give a complete calculation of Γ µ (p i , p j ) which involves a couple of Feynman diagrams. Instead, here we just concentrate on the dominant one which is shown in figure 1, the Z -penguin diagram. Its contribution then is read from where g 2 is the SU(2) L gauge coupling, and m i is the mass of i-th generation of up-type quark; V ij is the (i, j) element of the CKM matrix, containing the SM flavor violations in the charged current. We further approximate the masses of the down and strange quarks to be zero. It leads to the vanishing dipole terms in eq. (4.1), C → 0, because such terms require chirality flip, namely C ∝ m s/d /m 2 W . Moreover, the / qq µ term automatically vanishes after using the motion of equations for the fermions. Therefore, we expect that the Z -penguin diagram leads to an effective coupling g eff dsZ (q 2 )d(p i )γ µ P L s(p j ).

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Now let us calculate g eff dsZ (q 2 ) explicitly, using the public codes, FeynCalc [50][51][52] 4 and LoopTools [53]. 5 The result of loop function from FeynCalc is where C a (a = 0, 1, 2, 00, 12) are Passarino-Veltman (PV) integrals [54]. Since we have taken m d,s → 0, the arguments for the PV integrals are reduced to 3) Among them, only C 00 does not scale as 1/m 2 W thus dominant in the effective coupling. One can gain more insights into the scaling behavior of the PV integrals by developing approximations like in ref. [55]. The effective coupling is given by (4.4) which depends not only on (g u L ) ij but also on (g u R ) ij , and note that generically both of them are complex. It is convenient to rewrite where C ds L,R are the combinations of CKM elements and PV integrals specified in eq. (4.4). With the effective vertex, now we can calculate the branching ratios for K → πZ processes by using the following results [56,57]: where m K and Γ K are mass and decay width of kaon, respectively; λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx, and f Kπ + (q 2 ) is the K → π form factor [58]. Remarkably, the charged kaon decay proceeds without CP violation and , whereas the neutral kaon decay requires it and BR K L → π 0 Z is proportional to the squared imaginary part of the effective coupling. In the SM, CP violation is known to be small, and therefore in general BR K L → π 0 Z is supposed to be at least moderately suppressed.

Analysis on C ds L,R
To develop the numerical impression on (C ds L,R ), we set q 2 (= m 2 Z ) = m 2 π 0 as a reference value, and then one obtains where the masses of up-type quark and the CKM matrix are taken from PDG 2019 [59]. Several observations are in orders: • It is clear that for most elements the size of (C ds L ) ij is much larger than that of (C ds R ) ij , due to the fact that the former receives the C 00 contribution. (C ds R ) 33 is an exception, because it benefits from the m 2 t enhancement.
• |(C ds L ) 12 |, without involving flavor violation from the charged current, is the largest element as expected and would be the dominant contribution to s → dZ processes.
• It is notable that some elements of C ds R , in particular (C ds R ) 32,23 and (C ds R ) 33 , have comparable size with those of C ds L . Hence, these may contribute to K L → π 0 Z decay process, depending on the size of (g u R ) ij , namely, the structure of W u . The last feature motivates us to consider two scenarios: I) omit contributions of (C ds R ) ij (g u R ) ij with i, j summed; 6 II) include contributions of (C ds R ) ij (g u R ) ij . It is interesting that for the Scenarios I, our predictions on K → πZ can be explicitly determined by the SM parameters except for g B−L . In this sense, the Scenario I corresponds to the model in which there are no new CP violation sources. On the other hand, the CP violation in the Scenario II is not completely determined by the SM parameters, owing to the arbitrariness of g u R . This additional CP violation may admit an elaborate cancellation between Im[(C ds R ) ij (g u R ) ij ] and Im[(C ds L ) ij (g u L ) ij ], thus opening the possibility to explain both of (g − 2) µ and KOTO anomalies in our model.

Implications to the model
In this subsection we investigate the implications of induced K → π + Z to the local (B − L) 23 model in two scenarios, with Scenario I simply dropping the contribution from (C ds R ) ij (g u R ) ij for illustration, while Scenario II highlighting its additional CP violation. We will find that, in general the KOTO events can be easily explained in the Scenario I if we give up the original motivation, to account for the (g − 2) µ discrepancy. Otherwise, we should fall back on the other scenario. 6 Actually, the contributions from (C ds R )ij(g u R )ij cannot be omitted in any structure of Wu since some elements in (g u R )ij still exist in our model. However, as long as we discuss the prediction of KL → π 0 Z , we can ignore its contributions by setting appropriate structure of Wu.

Scenario I: close the door for (g − 2) µ but open the window for KOTO
We first discuss the scenario where the contributions from (C ds R ) ij (g u R ) ij is ignored. In this limit, the strong exclusion on the (m Z , g B−L ) parameter plane from K rare decays is clear, so it is questionable that if the remaining parameter space that is capable of accounting for (g − 2) µ survives.
For illustration, let us choose a point characterized by Z mass very close to m π 0 , e.g., m Z = m π 0 and g B−L = 10 −3 to explain the (g − 2) µ anomaly. The resulting branching ratio for which is much larger than the measured value, BR(K L → π 0 Z ) = O(10 −9 ). Therefore, this example point must have been excluded by KOTO. For this m Z , the constraint in eq. (2.3) applies and imposes an even stronger bound The bound from E949 experiment at 90% C.L. [31] is much weaker, g B−L < 4.2 × 10 −5 . Nevertheless, in the loophole region, e.g., m Z = 128 MeV, the Z can readily explain the KOTO anomaly for within 1σ (2σ) error. The summary plot for the parameter spaces for (g − 2) µ (red band) and KOTO result (magenta band) in the Scenario I is shown in figure 2. The darker and lighter bands show the favored region at 1σ and at 2σ, respectively. Other shaded regions are excluded by these experiments: Borexino (blue) [43][44][45], COHERENT (gray) [46][47][48][49], 7 CCFR (green) [60], E949 (orange) [30,31], KOTO before the events (cyan) [25] and NA62 (pink) [32]. The dotted lines show the contours for the life time of Z , calculated from eq. (3.1). It is seen that, for any value of m Z inside the loophole region, the required size of g B−L to account for the (g − 2) µ discrepancy is about two orders of magnitude larger than the upper bound by KOTO; outside the loophole, E949 yields the strongest bound and definitely rules out the possibility to explain (g −2) µ . 8 In this figure, we also show the future prospect of NA64 with dedicated muon beam, denoted as NA64µ (dashed yellow). This prospect is calculated with 10 12 incident muons [61], and its upper bound on g B−L in this mass region is about two orders of magnitude smaller than the required value for (g − 2) µ explanation. Interestingly, this prospect can search the parameter space for KOTO results in Scenario I. In addition, the future prospect of COHERENT is also shown by dashed-dotted gray line. Note that the COHERENT constraint and prospect for our model can be translated from the ones for L µ − L τ model 9 by considering the difference of the kinetic mixing between two models. 7 This constraint is obtained mainly from the muon neutrino source with interactions with up and down quarks in nucleon. In ref. [62], the authors discuss about the constraints also from the strange quark content in nucleon, which yields a relatively weak bound. 8 Recently, NA62 experiment provides upper bounds on BR(K + → π + Z ) for the mass ranges of m Z < 110 MeV and 154 MeV < m Z < 260 MeV [63]. We do not show its constraints in figure 2 since it is irrelevant to the following discussion. 9 Its constraint and prospect can be found, for example in refs. [64][65][66][67][68].
In the next scenario, we will demonstrate that BR(K L → π 0 Z ) can be significantly reduced and then both the (g − 2) µ and KOTO anomalies can be explained, at least in the loophole region of m Z .

Scenario II: one stone for two birds at the price of moderate tuning
In the Scenario I, the largeness of the branching ratio of For concreteness, from the g u L matrix eq. (3.8) and the C ds we set q 2 = m 2 π 0 for reference unless otherwise specified. However, in the Scenario II by switching on the Im[(C ds R ) ij (g u R ) ij ] contribution, there is a possibility to cancel this size by about two orders of magnitude, hence to explain both anomalies. The corresponding fine-tuning of CP violation may be not very serious, since we find that the elements Im(C ds L/R ) ij (g u L/R ) ij (not summed) already accidentally cancel each other out to a degree ∼ 90%. In the following we make a detailed discuss on this cancellation.
As mentioned before, Im(C ds R ) 32,33 ∼ O(10 −5 ) are large enough to contribute to Im(g eff dsZ ). Moreover, Re(C ds R ) 23,32 are sufficiently large and they, along with the sizable Im(g u R ) 23,32 (namely the CP violation from the corresponding elements of W u ), may play an important role in Im(g eff dsZ ). In order to understand what is the proper pattern of W u good for reducing Im(g eff dsZ ), we generate its elements randomly. From eq. (3.3), the relevant elements are (W u ) 1i , which in principle are free parameters except for satisfying the unitary condition: (W u ) 1i = (r 11 e iθ 11 , r 12 e iθ 12 , r 13 e iθ 13 ), (4.13) where |r 1i | ≤ 1 satisfying the relation |r 11 | 2 + |r 12 | 2 + |r 13 | 2 = 1. (4.14) For example, an illustrative choice is Then, when m Z = 128 MeV and θ 13 0.59π, BR(K L → π 0 Z ) 2.1 × 10 −9 is realized with g B−L = 10 −3 which is needed to explain the (g − 2) µ anomaly.
As a general survey, we generate 10 5 samples for the (W u ) 1i elements and check the prediction of the favored g B−L value for (g − 2) µ and KOTO anomalies. We show each element which can explain both anomalies within 2σ in figure 3. The green square and yellow circle denote the points where both anomalies are explained within 1σ and 2σ, respectively. For this figure, we set m Z = 140 MeV, but we find that the similar results are obtained for a different value of m Z within the loophole regions.
We can observe some important features of these elements. First, all of the elements are bounded from above, |(W u ) 1i | < 0.7-0.8. Then, at least two elements of (W u ) 1i are needed to satisfy eq. (4.14), like the example in eq. (4.15). Second, |(W u ) 11,12 | can be small, while |(W u ) 13 | should be 0.5 ∼ 0.8. The reason is understood by nothing but that Im(C ds R ) 33 tends to be even larger than Im[(C ds L ) ij (g u L ) ij ], and consequently a sizable |(W u ) 13 | is necessary to lower down (g u R ) 33 ∝ (1 − |(W u ) 13 | 2 ), thus allowing the cancellation to happen. Therefore, we cannot explain both anomalies with W u ∼ V CKM , and some  figure 3. The color manner is the same as in figure 2. In order to specify each band, we change the boundaries for 1σ and 2σ to solid and dashed lines, respectively. different and specific structure for W u is needed. Since this specific structure is due to the structure of C ds R in eq. (4.9), which is obtained only from the SM parameters, the required structure of W u is specific to our setup.
In figure 4, we show the summary plot for the benchmark point in figure 3. The color manner is the same as in figure 2, but we change the boundaries of each favored band for 1σ and 2σ to solid and dashed lines, respectively. It is clear that both anomalies can be explained with g B−L = O(10 −3 ), and these parameter space will be searched by the COHERENT experiment. We emphasize that the future prospect of NA64µ can be also applied to this scenario, whose expected upper bound can be read as g B−L (1.8-1.9) × 10 −5 . Therefore, we can expect some signal of our model in NA64µ as well as COHERENT, and moreover, if such signal predicts g B−L ∼ O(10 −3 ), the explanation of both anomalies can be done, based on our Scenario II.

Predictions in the B physics
We have studied the induced FCNCs in the Kaon system, and in particular explored the possibility to explain two anomalies simultaneously in the Scenario II, by means of a large g B−L but a fine-tuned CP violation in the loophole region of m Z . However, the loophole and as well fine-tuning may be not true in the B meson system, and hence it is important JHEP04(2021)238 to study the accompanied rare decays of the B mesons, e.g., by B → K + Z (→ νν). 10 Then, the B-factory may provide a promising way to test it. Actually, the Belle data already imposes a constraint.
In analogy to s → dZ , the transitions b → qZ (q = d, s) are through the effective q-b-Z couplings, with the effective couplings given by with C qb L,R again the known matrices at some q 2 , and the concrete forms at q 2 = m 2 π 0 are cast in appendix. B, from which one can see that the most sizable elements are (C qb L ) 13,23 ∼ O(1). These effective couplings are fixed as long as g u R or (W u ) 1i is chosen to realize the CP violation cancellation in the Scenario II.
We calculate the B meson decays by the following formulas [70,71]: where m B and Γ B are mass and width of B meson, f BP + (q 2 ) is B → P form factor [72], and κ 2 P,V are 1 for P = π + , K 0,+ and V = ρ + , K * 0, * + or 2 for P = π 0 and V = ρ 0 . H V 0 and H V ± are the helicity amplitudes which are given as where A BV 1 (q 2 ), A BV 2 (q 2 ) and V BV (q 2 ) are the form factors for B → V transition [73], and x V Z ≡ m 2 B − m 2 V − m 2 Z / (2m V m Z ). Note that the above formulas can be used for both neutral and charged B meson decays, and moreover, unlike K L → π 0 Z , the former decays do not need CP violation.
The results with m Z = 128 MeV and 140 MeV in the Scenario II are summarized in table 1. In the calculation of these branching ratios, we use the benchmark values for (W u ) 1i in figure 3. In addition, as the reference value, g B−L is chosen to realize the central value of the KOTO result, BR(K L → π 0 Z ) = 2.1 × 10 −9 . Note that each g B−L value is satisfied the CCFR constraint.
Remarkably, the branching ratios related to b → s transition are about four orders of magnitude larger than those related to b → d transition. This feature is one of our 10 It is also of interest to study the detect prospect of radiative B decay B → γZ which is recently proposed in ref. [69].  figure 3, and g B−L which realizes BR(K L → π 0 Z ) = 2.1 × 10 −9 (the central value of the KOTO result) is used as the reference value.
interesting predictions in B meson decays. Unfortunately, the current bounds for each decay mode are O(10 −5 ), and therefore, the b → s transition is strongly constrained. In order to satisfy these constraints, g B−L needs to be about 30 times smaller than the current chosen value, g B−L = O(10 −3 ). In this case, the explanation of both (g − 2) µ and KOTO anomalies fails. However, the cancellation in the Scenario II does not completely pine down (W u ) 1i , which still leaves sufficient degrees of freedom to reduce |g eff sbZ | by about one order, saving the Scenario II. We leave this issue to a future work. Note that in the Scenario I, the constraints of rare B meson decays are satisfied since the required value of g B−L for the explanation of KOTO result is O(10 −5 ).
It is notable that the Belle II experiment aims to search the decay mode for B → K + E miss . The reported sensitivity on the branching ratio is about 10% with 50ab −1 [74].

Conclusions and discussions
In this paper, we focus on the model in ref. [1], in particular, the case where Z couples to the up-type quark flavor-dependently is considered. The model originally was designed to explain the (g − 2) µ anomaly via a muonic force carrier Z . Although tree-level FCNCs in the down-type quark sector are forbidden by gauge symmetry, loop-level FCNCs are caused by the W boson exchange but not taken into account in our previous study. The different point from the SM case is that flavor violating Z couplings exist, and therefore, the CKM suppression becomes mild. We calculated related loop diagrams and obtained the effective flavor-violating coupling for s → d transition, g eff dsZ . Then, by considering the g eff dsZ contribution, we discuss its implications to the model, especially the possibility to explain the KOTO result, and the strong constraint on the viable parameter space for (g − 2) µ . Because of this mild CKM suppression, the branching ratio for K → πZ can be easily enhanced. For the generic Z mass, we found that the KOTO result can be explained with g B−L = O(10 −5 ), however, K + → π + νν constraint gives g B−L < 5.4 × 10 −6 . Then we cannot explain the KOTO result, and moreover, such small gauge coupling fails to explain

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For q 2 = m 2 π 0 as reference value, C qb L,R are obtained as Similar to C ds L , there is no CKM suppression for (C db L ) 13 and (C sb L ) 23 , and therefore, these elements will be dominant contributions to related B meson decays.
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