Implication of $K\to\pi\nu\bar{\nu}$ for generic neutrino interactions in effective field theories

We consider the effective operators for neutrino bilinears coupled to SM quarks in the framework of LEFT obeying SU$(3)_{\rm c}\times$U$(1)_{\rm em}$ gauge symmetry. In the basis of LEFT for neutrinos, the only dim-5 operator contributing to the neutrino magnetic moments is Canas et al. (2016)

$\displaystyle{\mathcal{O}}_{\nu\nu F}$

$\displaystyle=$

$\displaystyle(\overline{\nu^{C}}i\sigma_{\mu\nu}\nu)F^{\mu\nu}+h.c.\;,$

(6)

where $F_{\mu\nu}$ is the electromagnetic field strength tensor and $\nu=P_{L}\nu$ denote left-handed active SM neutrinos. Its SMEFT completion has been investigated by Cirigliano et al. in Ref. Cirigliano et al. (2017). In principle, the neutrino magnetic moment operator can yield the $K\to\pi\nu\nu$ process through a long-distance contribution with one vertex connecting to the $s\to d\gamma$ transition operator $\bar{s}\sigma_{\mu\nu}P_{L/R}dF^{\mu\nu}$. The corresponding coefficient is estimated to be $C_{sdF}\sim 10^{-9}~{}{\rm GeV}^{-1}$ in the SM Tandean (2000). There also exists a strong constraint on $|C_{\nu\nu F}|\leq 4\times 10^{-9}~{}{\rm GeV}^{-1}$ Canas et al. (2016). We thus conclude that the contribution from this operator to $K\to\pi\nu\nu$ transition is negligible. There are also the dim-6 operators Jenkins et al. (2018a) with lepton number conservation (LNC, $|\Delta L|=0$)

	$\displaystyle{\mathcal{O}}_{u\nu 1}^{V}$	$\displaystyle=(\overline{u_{L}}\gamma^{\mu}u_{L})(\overline{\nu}\gamma^{\mu}\nu)\;,$	$\displaystyle{\mathcal{O}}_{d\nu 1}^{V}$	$\displaystyle=(\overline{d_{L}}\gamma^{\mu}d_{L})(\overline{\nu}\gamma^{\mu}\nu)\;,$		(7)
	$\displaystyle{\mathcal{O}}_{u\nu 2}^{V}$	$\displaystyle=(\overline{u_{R}}\gamma^{\mu}u_{R})(\overline{\nu}\gamma^{\mu}\nu)\;,$	$\displaystyle{\mathcal{O}}_{d\nu 2}^{V}$	$\displaystyle=(\overline{d_{R}}\gamma^{\mu}d_{R})(\overline{\nu}\gamma^{\mu}\nu)\;,$		(8)

and those with lepton number violation (LNV, $|\Delta L|=2$)

	$\displaystyle{\mathcal{O}}_{u\nu 1}^{S}$	$\displaystyle=(\overline{u_{R}}u_{L})(\overline{\nu^{C}}\nu)+h.c.\;,$	$\displaystyle{\mathcal{O}}_{d\nu 1}^{S}$	$\displaystyle=(\overline{d_{R}}d_{L})(\overline{\nu^{C}}\nu)+h.c.\;,$		(9)
	$\displaystyle{\mathcal{O}}_{u\nu 2}^{S}$	$\displaystyle=(\overline{u_{L}}u_{R})(\overline{\nu^{C}}\nu)+h.c.\;,$	$\displaystyle{\mathcal{O}}_{d\nu 2}^{S}$	$\displaystyle=(\overline{d_{L}}d_{R})(\overline{\nu^{C}}\nu)+h.c.\;,$		(10)
	$\displaystyle{\mathcal{O}}_{u\nu}^{T}$	$\displaystyle=(\overline{u_{R}}\sigma^{\mu\nu}u_{L})(\overline{\nu^{C}}\sigma_{\mu\nu}\nu)+h.c.\;,$	$\displaystyle{\mathcal{O}}_{d\nu}^{T}$	$\displaystyle=(\overline{d_{R}}\sigma^{\mu\nu}d_{L})(\overline{\nu^{C}}\sigma_{\mu\nu}\nu)+h.c.\;.$		(11)

where $u_{L}(u_{R})$ and $d_{L}(d_{R})$ denote the left- (right-) handed up-type and down-type quark fields in mass basis, respectively. Note that the tensor operator $\overline{\nu_{\alpha}^{C}}\sigma^{\mu\nu}\nu_{\beta}$ vanishes for identical neutrino flavors (with $\alpha=\beta$). The flavors of the two quarks and those of the two neutrinos in the above operators can be different although we do not specify their flavor indexes here. For the notation of the Wilson coefficients, we use the same subscripts as the operators, for instance $C_{d\nu 1}^{V,xy\alpha\beta}$ together with ${\mathcal{O}}_{d\nu 1}^{V,xy\alpha\beta}$, where $x,y$ denote the down-type quark flavors and $\alpha,\beta$ are the neutrino flavors. In the following we will study the $K\to\pi$ transition and thus only consider the operators with down-type quarks $s$ and $d$.

The dim-6 quark-neutrino operators can be matched onto the meson-lepton interactions through the $\chi$PT formalism by treating the lepton currents together with the accompanied Wilson coefficients as proper external sources. The QCD-like Lagrangian with external sources for the first three light quarks ($q=u,d,s$) can be described as

$\displaystyle\mathcal{L}_{\rm QCD}=\mathcal{L}_{\text{QCD}}^{m_{q}=0}+\overline{q_{L}}l_{\mu}\gamma^{\mu}q_{L}+\overline{q_{R}}r_{\mu}\gamma^{\mu}q_{R}+\left[\overline{q_{L}}(s-ip)q_{R}+\overline{q_{L}}(t_{l}^{\mu\nu}\sigma_{\mu\nu})q_{R}+h.c.\right],$

(12)

where the flavor space $3\times$3 matrices $\{l_{\mu}=l_{\mu}^{\dagger},~{}r_{\mu}=r_{\mu}^{\dagger},~{}s=s^{\dagger},~{}p=p^{\dagger},~{}t_{r}^{\mu\nu}=t_{l}^{\mu\nu\dagger}\}$ are the external sources related with the corresponding quark currents. One can extract the relevant external sources from the above dim-6 effective operators. On the other hand, based on Weinberg’s power-counting scheme, the most general chiral Lagrangian can be expanded according to the momentum $p$ and quark mass. The chiral Lagrangian with external sources at leading order reads Gasser and Leutwyler (1984, 1985)

$\displaystyle\mathcal{L}_{p^{2}}=\frac{F_{0}^{2}}{4}{{\rm Tr}}\left(D_{\mu}U(D^{\mu}U)^{\dagger}\right)+\frac{F_{0}^{2}}{4}{{\rm Tr}}\left(\chi U^{\dagger}+U\chi^{\dagger}\right),$

(13)

where $U$ is the standard matrix for the Nambu-Goldstone bosons

$\displaystyle U={\rm exp}\Big{(}{i\Phi\over F_{0}}\Big{)},\ \ \Phi=\left(\begin{array}[]{ccc}\pi^{0}+{\eta\over\sqrt{3}}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\ \sqrt{2}\pi^{-}&-\pi^{0}+{\eta\over\sqrt{3}}&\sqrt{2}K^{0}\\ \sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-{2\over\sqrt{3}}\eta\\ \end{array}\right)\;,$

(17)

with the constant $F_{0}$ being referred to the pion decay constant in the chiral limit. The covariant derivative of $U$ and $\chi$ are expressed in terms of the external sources

$\displaystyle D_{\mu}U$

$\displaystyle=\partial_{\mu}U-il_{\mu}U+iUr_{\mu},$

$\displaystyle\chi$

$\displaystyle=2B(s-ip),$

$\displaystyle\chi^{\dagger}$

$\displaystyle=2B(s+ip),$

(18)

where the constant $B$ is related to the quark condensate and $F_{0}$ by $B=-\langle\bar{q}q\rangle_{0}/(3F_{0}^{2})$. For the later numerical estimation, we take $F_{0}=87~{}\text{MeV}$ Colangelo and Durr (2004) and $B\approx 2.8~{}\text{GeV}$ Cirigliano et al. (2017); Patrignani et al. (2016). The Nambu-Goldstone bosons parameterized by $U$ and the (pseudo-)scalar sources $\chi$ transform as $U\to LUR^{\dagger}$ and $\chi\to L\chi R^{\dagger}$, where $L$ ($R$) is $SU(3)_{L}$ ($SU(3)_{R}$) transformation.

By inspecting the dim-6 operators related to the $s\rightarrow d\nu\widehat{\nu}$ transition (the symbol  ‘ $\widehat{}$ ’  here indicating the neutrino pair can be either LNC $\nu\bar{\nu}$ or LNV $\nu\nu$¹¹1Below we use $K\to\pi\nu\bar{\nu}$ to generally denote the experimental processes. $K\to\pi\nu\widehat{\nu}$ appears when both $\nu\bar{\nu}$ and $\nu\nu$ final states can occur in the analytical expressions of the theoretical calculation unless a LNC or LNV process is specified in our discussion.), we find that only the LNC operators ${\mathcal{O}}_{d\nu 1}^{V},~{}{\mathcal{O}}_{d\nu 2}^{V}$ and LNV operators ${\mathcal{O}}_{d\nu 1}^{S},~{}{\mathcal{O}}_{d\nu 2}^{S}$ can contribute to the leading order chiral Lagrangian. The tensor operator ${\mathcal{O}}_{d\nu}^{T}$ only contributes to the next-to-leading order chiral Lagrangian at $\mathcal{O}(p^{4})$. They lead to the following external sources

	$\displaystyle(l^{\mu})_{sd}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{V,sd\alpha\beta}(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta}),$		(19)
	$\displaystyle(l^{\mu})_{ds}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{V,ds\alpha\beta}(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta}),$		(20)
	$\displaystyle(r^{\mu})_{sd}$	$\displaystyle=$	$\displaystyle C_{d\nu 2}^{V,sd\alpha\beta}(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta}),$		(21)
	$\displaystyle(r^{\mu})_{ds}$	$\displaystyle=$	$\displaystyle C_{d\nu 2}^{V,ds\alpha\beta}(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta}),$		(22)
	$\displaystyle(s+ip)_{sd}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{S,sd\alpha\beta}(\overline{\nu^{C}_{\alpha}}\nu_{\beta})+C_{d\nu 2}^{S,ds\alpha\beta*}(\overline{\nu_{\alpha}}\nu_{\beta}^{C}),$		(23)
	$\displaystyle(s+ip)_{ds}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{S,ds\alpha\beta}(\overline{\nu^{C}_{\alpha}}\nu_{\beta})+C_{d\nu 2}^{S,sd\alpha\beta*}(\overline{\nu_{\alpha}}\nu_{\beta}^{C}),$		(24)
	$\displaystyle(s-ip)_{sd}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{S,ds\alpha\beta*}(\overline{\nu_{\alpha}}\nu_{\beta}^{C})+C_{d\nu 2}^{S,sd\alpha\beta}(\overline{\nu^{C}_{\alpha}}\nu_{\beta}),$		(25)
	$\displaystyle(s-ip)_{ds}$	$\displaystyle=$	$\displaystyle C_{d\nu 1}^{S,sd\alpha\beta*}(\overline{\nu_{\alpha}}\nu_{\beta}^{C})+C_{d\nu 2}^{S,ds\alpha\beta}(\overline{\nu^{C}_{\alpha}}\nu_{\beta}).$		(26)

After expanding $U$, i.e. $U=1+i{\Phi\over F_{0}}+{1\over 2F_{0}^{2}}(i\Phi)^{2}+\cdots$ and the insertion of the above external sources into the Lagrangian in Eq. (13), we obtain the effective Lagrangians for $K^{0}\to\pi^{0}\nu\widehat{\nu}$ and $K^{+}\to\pi^{+}\nu\widehat{\nu}$ at the leading order

	$\displaystyle\mathcal{L}_{K^{0}\to\pi^{0}\nu\widehat{\nu}}$	$\displaystyle=$	$\displaystyle{B\over 2\sqrt{2}}\Big{[}\left(C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}\right)(\overline{\nu^{C}_{\alpha}}\nu_{\beta})+\left(C_{d\nu 1}^{S,ds\alpha\beta*}+C_{d\nu 2}^{S,ds\alpha\beta*}\right)(\overline{\nu_{\alpha}}\nu_{\beta}^{C})\Big{]}K^{0}\pi^{0}$		(27)
			$\displaystyle-{i\over 2\sqrt{2}}\left(C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right)(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta})(K^{0}\partial_{\mu}\pi^{0}-\partial_{\mu}K^{0}\pi^{0})\;,$
	$\displaystyle\mathcal{L}_{K^{+}\to\pi^{+}\nu\widehat{\nu}}$	$\displaystyle=$	$\displaystyle-{B\over 2}\Big{[}\left(C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}\right)(\overline{\nu^{C}_{\alpha}}\nu_{\beta})+\left(C_{d\nu 1}^{S,ds\alpha\beta*}+C_{d\nu 2}^{S,ds\alpha\beta*}\right)(\overline{\nu_{\alpha}}\nu_{\beta}^{C})\Big{]}K^{+}\pi^{-}$		(28)
			$\displaystyle+{i\over 2}\left(C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right)(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta})(K^{+}\partial_{\mu}\pi^{-}-\partial_{\mu}K^{+}\pi^{-})\;.$

The above Lagrangians fit to the relation

$\displaystyle{\langle\pi^{0}|\mathcal{L}_{K^{0}\to\pi^{0}\nu\widehat{\nu}}|K^{0}\rangle\over\langle\pi^{-}|\mathcal{L}_{K^{+}\to\pi^{+}\nu\widehat{\nu}}|K^{+}\rangle}=-\frac{1}{\sqrt{2}}\;.$

(29)

This relation is the result of the transition operators that change isospin by $1/2$. By neglecting the small CP violation in $K^{0}-\bar{K}^{0}$ mixing, for the $K_{L}\rightarrow\pi^{0}$ transition, the relevant effective Lagrangian becomes

	$\displaystyle\mathcal{L}_{K_{L}\to\pi^{0}\nu\widehat{\nu}}$	$\displaystyle=$	$\displaystyle{B\over 4}\Big{[}\left(C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}+C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right)(\overline{\nu^{C}_{\alpha}}\nu_{\beta})$
			$\displaystyle+\left(C_{d\nu 1}^{S,ds\alpha\beta*}+C_{d\nu 2}^{S,ds\alpha\beta*}+C_{d\nu 1}^{S,sd\alpha\beta*}+C_{d\nu 2}^{S,sd\alpha\beta*}\right)(\overline{\nu_{\alpha}}\nu_{\beta}^{C})\Big{]}K_{L}\pi^{0}$
			$\displaystyle-{i\over 4}\left(C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}-C_{d\nu 1}^{V,ds\alpha\beta}-C_{d\nu 2}^{V,ds\alpha\beta}\right)(\overline{\nu_{\alpha}}\gamma^{\mu}\nu_{\beta})(K_{L}\partial_{\mu}\pi^{0}-\partial_{\mu}K_{L}\pi^{0})\;,$

where the flavor indices $\alpha,\beta$ are summed over all three neutrino generations. Note that the Wilson coefficients for the scalar operators are symmetric in the neutrino flavor indices. From the effective Lagrangian we derive the branching ratios for the decays $K_{L}\to\pi^{0}\nu\widehat{\nu}$ and $K^{+}\to\pi^{+}\nu\widehat{\nu}$

	$\displaystyle\mathcal{B}_{K_{L}\rightarrow\pi^{0}\nu\widehat{\nu}}$	$\displaystyle=$	$\displaystyle J_{1}^{K_{L}}\sum_{\alpha\leq\beta}\left(1-{1\over 2}\delta_{\alpha\beta}\right)\left|C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}+C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right|^{2}$		(31)
			$\displaystyle+J_{2}^{K_{L}}\sum_{\alpha,\beta}\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}-C_{d\nu 1}^{V,ds\alpha\beta}-C_{d\nu 2}^{V,ds\alpha\beta}\right|^{2},$
	$\displaystyle\mathcal{B}_{K^{+}\rightarrow\pi^{+}\nu\widehat{\nu}}$	$\displaystyle=$	$\displaystyle J_{1}^{K^{+}}\sum_{\alpha\leq\beta}\left(1-{1\over 2}\delta_{\alpha\beta}\right)\left(\left|C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}\right|^{2}+\left|C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right|^{2}\right)$		(32)
			$\displaystyle+J_{2}^{K^{+}}\sum_{\alpha,\beta}\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right|^{2}.$

The details of the calculation are collected in Appendix A. The $J$ functions parameterize the kinematics of the three-body decay and are defined as

	$\displaystyle J_{1}^{K_{L}}$	$\displaystyle=$	$\displaystyle{1\over\Gamma_{K_{L}}^{\text{Exp}}}{B^{2}\over 2^{9}\pi^{3}m_{K_{L}}^{3}}\int ds\,s\left((m_{K_{L}}^{2}+m_{\pi^{0}}^{2}-s)^{2}-4m_{K_{L}}^{2}m_{\pi^{0}}^{2}\right)^{1/2}=40.4G_{F}^{-2},$		(33)
	$\displaystyle J_{2}^{K_{L}}$	$\displaystyle=$	$\displaystyle{1\over\Gamma_{K_{L}}^{\text{Exp}}}{1\over 3\cdot 2^{11}\pi^{3}m_{K_{L}}^{3}}\int ds\left((m_{K_{L}}^{2}+m_{\pi^{0}}^{2}-s)^{2}-4m_{K_{L}}^{2}m_{\pi^{0}}^{2}\right)^{3/2}=0.247G_{F}^{-2},$		(34)
	$\displaystyle J_{1}^{K^{+}}$	$\displaystyle=$	$\displaystyle{1\over\Gamma_{K^{+}}^{\text{Exp}}}{B^{2}\over 2^{8}\pi^{3}m_{K^{+}}^{3}}\int ds\,s\left((m_{K^{+}}^{2}+m_{\pi^{+}}^{2}-s)^{2}-4m_{K^{+}}^{2}m_{\pi^{+}}^{2}\right)^{1/2}=17.9G_{F}^{-2},$		(35)
	$\displaystyle J_{2}^{K^{+}}$	$\displaystyle=$	$\displaystyle{1\over\Gamma_{K^{+}}^{\text{Exp}}}{1\over 3\cdot 2^{9}\pi^{3}m_{K^{+}}^{3}}\int ds\left((m_{K^{+}}^{2}+m_{\pi^{+}}^{2}-s)^{2}-4m_{K^{+}}^{2}m_{\pi^{+}}^{2}\right)^{3/2}=0.22G_{F}^{-2}\;,$		(36)

where $m_{K_{L}}(m_{K^{+}})$ and $\Gamma_{K_{L}}^{\text{Exp}}(\Gamma_{K^{+}}^{\text{Exp}})$ denote the physical mass and decay width of $K_{L}(K^{+})$, respectively. $m_{\pi^{0}}(m_{\pi^{+}})$ is the mass of $\pi^{0}(\pi^{+})$, $G_{F}$ is the Fermi constant and $s$ is the invariant squared mass of the final-state neutrino pair. From the hermiticity of the effective Lagrangian and the Cauchy-Schwarz inequality we derive the following relations for the Wilson coefficients in LEFT

	$\displaystyle\left|C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}+C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right|^{2}$	$\displaystyle\leq$	$\displaystyle\left||C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}|+|C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}|\right|^{2}$
		$\displaystyle\leq$	$\displaystyle 2\left(\left|C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}\right|^{2}+\left|C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right|^{2}\right)\;,$
	$\displaystyle\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}-C_{d\nu 1}^{V,ds\alpha\beta}-C_{d\nu 2}^{V,ds\alpha\beta}\right|^{2}$	$\displaystyle\leq$	$\displaystyle\left||C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}|+|C_{d\nu 1}^{V,ds\alpha\beta}+C_{d\nu 2}^{V,ds\alpha\beta}|\right|^{2}$
		$\displaystyle\leq$	$\displaystyle 2\left(\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right|^{2}+\left|C_{d\nu 1}^{V,sd\beta\alpha}+C_{d\nu 2}^{V,sd\beta\alpha}\right|^{2}\right)\;.$

Note that in the second inequality we used $C_{d\nu 1/2}^{V,ds\alpha\beta*}=C_{d\nu 1/2}^{V,sd\beta\alpha}$ from the fact that the vector operator itself is hermitian. If we sum over neutrino flavors, the second relation above turns out to be

$\displaystyle\sum_{\alpha,\beta}\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}-C_{d\nu 1}^{V,ds\alpha\beta}-C_{d\nu 2}^{V,ds\alpha\beta}\right|^{2}\leq 4\sum_{\alpha,\beta}\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right|^{2}\;.$

(39)

Based on the above inequalities, the branching ratios in Eq. (31) and Eq. (32) lead to

$\displaystyle{\mathcal{B}_{K_{L}\rightarrow\pi^{0}\nu\widehat{\nu}}\over\mathcal{B}_{K^{+}\rightarrow\pi^{+}\nu\widehat{\nu}}}\leq{4J_{2}^{K_{L}}+2J_{1}^{K_{L}}\epsilon\over J_{2}^{K^{+}}+J_{1}^{K^{+}}\epsilon}\lesssim 4.5\;,$

(40)

where $\epsilon$ is defined by

$\displaystyle\epsilon={\sum_{\alpha\leq\beta}\left(1-{1\over 2}\delta_{\alpha\beta}\right)\left(\left|C_{d\nu 1}^{S,sd\alpha\beta}+C_{d\nu 2}^{S,sd\alpha\beta}\right|^{2}+\left|C_{d\nu 1}^{S,ds\alpha\beta}+C_{d\nu 2}^{S,ds\alpha\beta}\right|^{2}\right)\over\sum_{\alpha,\beta}\left|C_{d\nu 1}^{V,sd\alpha\beta}+C_{d\nu 2}^{V,sd\alpha\beta}\right|^{2}}\geq 0\;.$

(41)

The upper bound on the ratio of branching ratios in Eq. (40) holds independent of the value of $\epsilon\geq 0$, because $4J_{2}^{K_{L}}/J_{2}^{K^{+}}\approx 4.49$ and $2J_{1}^{K_{L}}/J_{1}^{K^{+}}\approx 4.51$ agree to two significant figures. This result is nothing but the Grossman-Nir bound Grossman and Nir (1997) expected from the isospin relation in Eq. (29) and the CP-conserving limit for neutral Kaon system. The numerical value slightly differs from the standard G-N bound value of 4.3, because we do not consider isospin breaking and electroweak correction effects beyond the mass difference in the phase space integration Marciano and Parsa (1996). We obtain the G-N bound from the matching of LEFT to $\chi$PT. Thus, as expected, the Grossman-Nir bound holds for dim-6 LEFT operators in leading order $\chi$PT.

For the tensor currents in Eqs. (11), we have to go beyond the leading order of chiral Lagrangian. The relevant $\mathcal{O}(p^{4})$ Lagrangian at next-to-leading order is Cata and Mateu (2007)

$\displaystyle\mathcal{L}_{p^{4}}^{T}\supset i\Lambda_{2}{{\rm Tr}}\left(t_{l}^{\mu\nu}(D_{\mu}U)^{\dagger}U(D_{\nu}U)^{\dagger}+t_{r}^{\mu\nu}D_{\mu}UU^{\dagger}D_{\nu}U\right),$

(42)

where $\Lambda_{2}$ denotes the low-energy constant. In terms of the dim-6 tensor operator ${\mathcal{O}}_{d\nu}^{T}$, the relevant tensor sources are

	$\displaystyle(t_{r}^{\mu\nu})_{ds}$	$\displaystyle=C_{d\nu}^{T,ds\alpha\beta}(\overline{\nu_{\alpha}^{C}}\sigma_{\mu\nu}\nu_{\beta}),$	$\displaystyle(t_{r}^{\mu\nu})_{sd}$	$\displaystyle=C_{d\nu}^{T,sd\alpha\beta}(\overline{\nu_{\alpha}^{C}}\sigma_{\mu\nu}\nu_{\beta}),$		(43)
	$\displaystyle(t_{l}^{\mu\nu})_{ds}$	$\displaystyle=C_{d\nu}^{T,sd\alpha\beta*}(\overline{\nu_{\alpha}}\sigma_{\mu\nu}\nu_{\beta}^{C}),$	$\displaystyle(t_{l}^{\mu\nu})_{sd}$	$\displaystyle=C_{d\nu}^{T,ds\alpha\beta*}(\overline{\nu_{\alpha}}\sigma_{\mu\nu}\nu_{\beta}^{C}).$		(44)

After inserting these external sources into the $\mathcal{O}(p^{4})$ Lagrangian (42), we obtain the following interactions for $K\to\pi\nu\nu$ transitions

	$\displaystyle\mathcal{L}_{p^{4}}^{T}$	$\displaystyle\supset$	$\displaystyle i{\Lambda_{2}\over\sqrt{2}F_{0}^{2}}C_{d\nu}^{T,sd\alpha\beta}\left(\sqrt{2}\partial_{[\mu}K^{+}\partial_{\nu]}\pi^{-}-\partial_{[\mu}K^{0}\partial_{\nu]}\pi^{0}\right)(\overline{\nu_{\alpha}^{C}}\sigma_{\mu\nu}\nu_{\beta})$		(45)
			$\displaystyle+i{\Lambda_{2}\over\sqrt{2}F_{0}^{2}}C_{d\nu}^{T,ds\alpha\beta*}\left(\sqrt{2}\partial_{[\mu}K^{+}\partial_{\nu]}\pi^{-}-\partial_{[\mu}K^{0}\partial_{\nu]}\pi^{0}\right)(\overline{\nu_{\alpha}}\sigma_{\mu\nu}\nu_{\beta}^{C})+h.c.,$

where $\partial_{[\mu}A\partial_{\nu]}B=\partial_{\mu}A\partial_{\nu}B-\partial_{\nu}A\partial_{\mu}B$.

We also investigate dim-7 tensor operators in LEFT. There happens to be only one such operator related with the transition $K\to\pi\nu\bar{\nu}$ under consideration, that is

$\displaystyle{\mathcal{O}}_{d\nu,\text{dim-7}}^{T}$

$\displaystyle=$

$\displaystyle(\overline{d_{L}}\sigma_{\mu\nu}d_{R})(\overline{\nu}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu)+h.c.\;,$

(46)

which leads to the tensor sources to be

	$\displaystyle(t_{l}^{\mu\nu})_{ds}$	$\displaystyle=C^{T,ds\alpha\beta}_{d\nu,\text{dim-7}}(\overline{\nu_{\alpha}}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu_{\beta}),$	$\displaystyle(t_{l}^{\mu\nu})_{sd}$	$\displaystyle=C^{T,sd\alpha\beta}_{d\nu,\text{dim-7}}(\overline{\nu_{\alpha}}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu_{\beta}),$		(47)
	$\displaystyle(t_{r}^{\mu\nu})_{ds}$	$\displaystyle=C^{T,sd\beta\alpha*}_{d\nu,\text{dim-7}}(\overline{\nu_{\alpha}}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu_{\beta}),$	$\displaystyle(t_{r}^{\mu\nu})_{sd}$	$\displaystyle=C^{T,ds\beta\alpha*}_{d\nu,\text{dim-7}}(\overline{\nu_{\alpha}}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu_{\beta}).$		(48)

By analogy we expand the $\mathcal{O}(p^{4})$ Lagrangian (42) to obtain the interactions with mesons

$\displaystyle\mathcal{L}_{p^{4}}^{T}\supset{i\Lambda_{2}\over\sqrt{2}F_{0}^{2}}(C^{T,sd\alpha\beta}_{d\nu,\text{dim-7}}+C^{T,ds\beta\alpha*}_{d\nu,\text{dim-7}})\left(\sqrt{2}\partial_{[\mu}K^{+}\partial_{\nu]}\pi^{-}-\partial_{[\mu}K^{0}\partial_{\nu]}\pi^{0}\right)(\overline{\nu_{\alpha}}\gamma^{[\mu}i\overleftrightarrow{\partial}^{\nu]}\nu_{\beta})+h.c.\;.$

One can see that, for the next-to-leading order chiral Lagrangian with dim-6 and dim-7 tensor operators in LEFT, the isospin relation in Eq. (29) and thus the Grossman-Nir bound still hold. Note that the Eq. (II.3) vanishes for massless neutrinos. This implies that the non-zero contribution from dim-7 tensor operator appears at ${\mathcal{O}}(p^{6})$ level, and therefore is further suppressed by additional $p^{2}/\Lambda_{\chi}^{2}$ factor. In addition, there are also two dim-7 vector-like LNV operators related to $K\to\pi\nu\nu$, which we list for completeness

$\displaystyle{\mathcal{O}}_{d\nu 1,\text{dim-7}}^{V}$

$\displaystyle=(\overline{d_{L}}\gamma_{\mu}d_{L})(\overline{\nu^{C}}i\overleftrightarrow{\partial}^{\mu}\nu)+h.c.\;,$

$\displaystyle{\mathcal{O}}_{d\nu 2,\text{dim-7}}^{V}$

$\displaystyle=(\overline{d_{R}}\gamma_{\mu}d_{R})(\overline{\nu^{C}}i\overleftrightarrow{\partial}^{\mu}\nu)+h.c.\;.$

(50)

These dim-7 operators are suppressed by $p/m_{W}$ compared with dim-6 operators and, like the above tensor operators, lead to sub-leading contributions. Thus, we neglect them in the following calculation and restrict us to only consider the scalar and vector dim-6 operators in LEFT.