$(g-2)_{\mu}$ Versus $K\rightarrow\pi+E_{miss}$ Induced by the $(B-L)_{23}$ Boson

The $s\to d$ 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 Jho:2020jsa . In other words, for a light massive gauge boson $Z^{\prime}$, additional FCNCs beyond the SM is indispensable.

To that end, as a simple consideration, we presume that $Z^{\prime}$ comes from a gauged Abelian flavorful symmetry $U(1)_{X}$ which has the following features:

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  The SM fermions carry non-universal charges of $U(1)_{X}$, which then may result in non-simultaneous diagonalization of quark mass matrix and quark-$Z^{\prime}$ current couplings. Obviously, quarks should be charged under this gauge group.

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  Besides, in order to make $Z^{\prime}$ 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 $Z^{\prime}$, we do not need a hierarchy of charges as long as the coupling to electron is suppressed.

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  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 Kang:2019vng , which naturally fits the outlined profiles.

Originally, this model aims at addressing the long-standing $(g-2)_{\mu}$ anomaly via the light $Z^{\prime}$ 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 $Z^{\prime}$, which helps avoid the relevant strong exclusions such as Borexino Borexino ; Borexino1 ; Borexino2 and COHERENT Akimov:2015nza ; Akimov:2017ade ; Akimov:2020pdx ; Akimov:2020czh , thus allowing the desired $Z^{\prime}$ having a mass $\sim{\cal O}(10)$ MeV and a moderately small gauge coupling $g_{B-L}\sim{\cal O}(10^{-4}\mathchar 45\relax 10^{-3})$. Because the $Z^{\prime}$ has mass below $2m_{\mu}$ and moreover has suppressed coupling to electron through the kinetic mixing between $Z^{\prime}$ and the photon, the dominant decay channel is into a pair of neutrinos, having decay width

Then, the lifetime becomes $c\tau_{Z^{\prime}}\simeq 1.5\times 10^{-5}\times\left(\frac{10^{-4}}{g_{B_{L}}}\right)^{2}\left(\frac{0.1\rm GeV}{m_{Z^{\prime}}}\right)$ m. The KOTO detector size is $L=3$m, while the size of the NA64 detector is much larger, $L=150$m. Since $Z^{\prime}$ here is invisible, its lifetime is irrelevant to our following discussions.

To generate the correct CKM structure, additional FCNCs associated with $Z^{\prime}$ are unavoidable in this model. Therefore, qualitatively this $Z^{\prime}$ fits the invisible light particle explanation to the KOTO events. If it succeeds quantitatively, accounting for both $(g-2)_{\mu}$ 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 $s\to d$ transition rate is too large for the typical $Z^{\prime}$ parameters for $(g-2)_{\mu}$. However, our $Z^{\prime}$ is still a good spin-1 candidate for KOTO, as long as we abandon its responsibility in $(g-2)_{\mu}$ ³³3However, hopefully, $(g-2)_{\mu}$ 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..

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. Kang:2019vng we have already taken into the constraints from the KOTO report which has not claimed the excess yet KOTO . 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: $Y_{u}^{\rm diag}=U_{u}^{\dagger}Y_{u}W_{u}$. 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,

Moreover, $U_{d}$ is block diagonalized and can be parameterized as

where the blank block denotes the $2\times 2$ 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. (9) for Eq. (8) 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 $g_{L}^{u}$ are real (true also for $g_{R}^{u}$), 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, $\sim 10^{-3}$, with others suppressed by orders of magnitude.

On the contrary, the structure of $W_{u}$ cannot be determined as $U_{u}$ in Eq. (10) since there is no relation like in Eq. (8). 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 $K_{L}\to\pi^{0}Z^{\prime}$ 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 $Z^{\prime}$ 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 $\mu-\tau$ symmetry demonstrated in neutrino mixings.

At quark level, this process is described by the effective vertex $\bar{d}(p_{i})\Gamma^{\mu}(p_{i},p_{j})s(p_{j})$, 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 $Z^{\prime}$. We will not give a complete calculation of $\Gamma^{\mu}(p_{i},p_{j})$ which involves a couple of Feynman diagrams. Instead, here we just concentrate on the dominant one which is shown in Fig. 1, the $Z^{\prime}$-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. (12), $C\rightarrow 0$, because such terms require chirality flip, namely $C\propto m_{s/d}/m_{W}^{2}$. Moreover, the $qq^{\mu}$ term automatically vanishes after using the motion of equations for the fermions. Therefore, we expect that the $Z^{\prime}$-penguin diagram leads to an effective coupling $g_{dsZ^{\prime}}^{\rm eff}(q^{2})\bar{d}(p_{i})\gamma^{\mu}P_{L}s(p_{j})$.

Now let us calculate $g_{dsZ^{\prime}}^{\rm eff}(q^{2})$ explicitly, using the public codes, FeynCalc FeynCalc1 ; FeynCalc2 ; FeynCalc3  ⁴⁴4https://feyncalc.github.io/ and LoopTools LoopTools  ⁵⁵5http://www.feynarts.de/looptools/. The result of loop function from FeynCalc is

where $C_{a}$ $(a=0,1,2,00,12)$ are Passarino-Veltman (PV) integrals PV . Since we have taken $m_{d,s}\to 0$, the arguments for the PV integrals are reduced to

Among them, only $C_{00}$ does not scale as $1/m_{W}^{2}$ 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. Baek:2015fma . The effective coupling is given by

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 $C_{L,R}^{ds}$ are the combinations of CKM elements and PV integrals specified in Eq. (15).

With the effective vertex, now we can calculate the branching ratios for $K\to\pi Z^{\prime}$ processes by using the following results Fuyuto:2014cya ; Fuyuto:2015gmk :

where $m_{K}$ and $\Gamma_{K}$ are mass and decay width of kaon, respectively; $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$, and $f_{+}^{K\pi}(q^{2})$ is the $K\to\pi$ form factor Mescia:2007kn .

Remarkably, the charged kaon decay proceeds without CP violation and BR$\left(K^{+}\rightarrow\pi^{+}Z^{\prime}\right)\propto\left|g_{dsZ^{\prime}}^{\mathrm{eff}}\right|^{2}$, whereas the neutral kaon decay requires it and BR$\left(K_{L}\rightarrow\pi^{0}Z^{\prime}\right)$ 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$\left(K_{L}\rightarrow\pi^{0}Z^{\prime}\right)$ is supposed to be at least moderately suppressed.

To develop the numerical impression on $(C_{L,R}^{ds})$, we set $q^{2}(=m_{Z^{\prime}}^{2})=m_{\pi^{0}}^{2}$ as a reference value, and then one obtains

where the masses of up-type quark and the CKM matrix are taken from PDG 2019 PDG2019 . Several observations are in orders:

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  It is clear that for most elements the size of $(C_{L}^{ds})_{ij}$ is much larger than that of $(C_{R}^{ds})_{ij}$, due to the fact that the former receives the $C_{00}$ contribution. $(C_{R}^{ds})_{33}$ is an exception, because it benefits from the $m_{t}^{2}$ enhancement.

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  $|(C_{L}^{ds})_{12}|$, without involving flavor violation from the charged current, is the largest element as expected and would be the dominant contribution to $s\to dZ^{\prime}$ processes.

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  It is notable that some elements of $C_{R}^{ds}$, in particular $(C_{R}^{ds})_{32,23}$ and $(C_{R}^{ds})_{33}$, have comparable size with those of $C_{L}^{ds}$. Hence, these may contribute to $K_{L}\to\pi^{0}Z^{\prime}$ decay process, depending on the size of $(g_{R}^{u})_{ij}$, namely, the structure of $W_{u}$.

The last feature motivates us to consider two scenarios: I) omit contributions of $(C_{R}^{ds})_{ij}(g_{R}^{u})_{ij}$ with $i,j$ summed ⁶⁶6Actually, the contributions from $(C_{R}^{ds})_{ij}(g_{R}^{u})_{ij}$ cannot be omitted in any structure of $W_{u}$ since some elements in $(g_{R}^{u})_{ij}$ still exist in our model. However, as long as we discuss the prediction of $K_{L}\to\pi^{0}Z^{\prime}$, we can ignore its contributions by setting appropriate structure of $W_{u}$.; II) include contributions of $(C_{R}^{ds})_{ij}(g_{R}^{u})_{ij}$. It is interesting that for the Scenarios I, our predictions on $K\rightarrow\pi Z^{\prime}$ 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 $g_{R}^{u}$. This additional CP violation may admit an elaborate cancellation between Im$[(C_{R}^{ds})_{ij}(g_{R}^{u})_{ij}]$ and Im$[(C_{L}^{ds})_{ij}(g_{L}^{u})_{ij}]$, thus opening the possibility to explain both of $(g-2)_{\mu}$ and KOTO anomalies in our model.

In this subsection we investigate the implications of induced $K\rightarrow\pi+Z^{\prime}$ to the local $(B-L)_{23}$ model in two scenarios, with Scenario I simply dropping the contribution from $(C_{R}^{ds})_{ij}(g_{R}^{u})_{ij}$ 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 $(g-2)_{\mu}$ discrepancy. Otherwise, we should fall back on the other scenario.