Theory perspectives on rare Kaon decays and CPV

TL;DR

This paper reviews theoretical perspectives on rare Kaon decays, emphasizing their significance in probing new physics, CP violation, and their interplay with B-anomalies, especially focusing on decays like $K o u ar{ u}$ and $K^0 o \mu^+ \mu^-$.

Contribution

It provides a comprehensive review of rare Kaon decays in the LHC era, highlighting their role in understanding CP violation and potential new physics beyond the Standard Model.

Findings

Rare kaon decays are crucial for exploring new physics.

Interplay between kaon decays and B-anomalies offers insights into CP violation.

Recent measurements, such as $K^0 o \mu^+ \\mu^-$, impact theoretical models.

Abstract

The following proceedings contain a theory perspective on rare Kaon decays. I review rare kaon decays in the LHC era: we discuss interplay with B-anomalies and possible New Physics in direct CP violation in $K\to 2\pi$: very rare kaon decays like $K \to \pi \nu \bar{\nu}$ are very important to this purpose. We discuss also the decays $K^0 \to \mu ^+ \mu ^-$ due to the LHCB measurement

Figures

Tables

Table 1. Table 1: O( p 4 superscript 𝑝 4 p^{4} ) weak chiral countertems and their determination

	$K^{\pm }\to {\pi }^{\pm }{\gamma }^{\ast }$	$N_{14}^r-N_{15}^r$		$a_{+}=-0.578\pm 0.016$	NA48/2
	$K_S\to {\pi }^0{\gamma }^{\ast }$	$2N_{14}^r+N_{15}^r$		$a_S=\left({1.06}_{-0.21}^{+0.26}\pm 0.07\right)$	NA48/1
	$K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$	$N_{14}^r-N_{15}^r-N_{16}^r-N_{17}$		$X_E=\left(-24\pm 4\pm 4\right){\mathrm{GeV}}^{-4}$	NA48/2
	$K^{+}\to {\pi }^{+}\gamma \gamma$	$N_{14}^r-N_{15}^r-2N_{18}^r$		$\hat{c}=1.56\pm 0.23\pm 0.11$	NA48/2

Table 2. Table 2: Fitted values of coefficients in the vector form factor.

Channel	$a_{+}$	$b_{+}$	Reference
$ee$	$-0.587\pm 0.010$	$-0.655\pm 0.044$	E865 Appel:1999yq
$ee$	$-0.578\pm 0.016$	$-0.779\pm 0.066$	NA48/2 Batley:2009aa
$\mu \mu$	$-0.575\pm 0.039$	$-0.813\pm 0.145$	NA48/2 Batley:2011zz

Equations

$$\mathcal{H}$$

$$K_L=(2.9\pm 0.2)\times 10^{-11}{K}^{+}=(8.3\pm 0.9)\times 10^{-11}.$$

$$B(K_S\to \mu \overline{\mu }{)}_{LHCB}<9\times 10^{-9}\mathrm{at}90\%\mathrm{CL}$$

$$B(K_S\to \mu \overline{\mu }{)}_{SM}=(5.0\pm 1.5)\times 10^{-12}.$$

$$B(K_S\to \mu \overline{\mu }{)}_{SM}^{SD}$$

$$B(K_S\to \mu \overline{\mu }{)}_{NP}$$

$$\mathcal{B}(K_S^0\to {\mu }^{+}{\mu }^{-}{)}_{\mathrm{eff}}={\tau }_S{\left({\int }_{t_{\min }}^{t_{\max }}dt{e}^{-{\Gamma }_{S}t}\epsilon (t)\right)}^{-1}$$

$$\displaystyle\Biggl{[}\int^{t_{{\rm max}}}_{t_{{\rm min}}}dt\Biggl{\{}\Gamma(K^{0}_{S}\to\mu^{+}\mu^{-})e^{-\Gamma_{S}t}~{}+\frac{Df_{K}^{2}M_{K}^{3}\beta_{\mu}}{{8}\pi}\cdot$$

$$\displaystyle\textrm{Re}\left[i\left(A_{S}A_{L}-\beta_{\mu}^{2}{B_{S}^{\ast}}B_{L}\right)e^{-i\Delta M_{K}t}\right]e^{-\frac{\Gamma_{S}+\Gamma_{L}}{2}t}\Biggr{\}}\varepsilon(t)\Biggr{]}$$

$$\sum _{\mathrm{spin}}\mathcal{A}(K_1\to {\mu }^{+}{\mu }^{-}{)}^{\ast }\mathcal{A}(K_2\to {\mu }^{+}{\mu }^{-})\sim \underset{SD}{\underbrace{\text{Im}[{\lambda }_t]y_{7A}^{\prime }}}$$

$$\left\{\underset{LD}{\underbrace{A_{L\gamma \gamma }^{\mu }}}-2\pi {\sin }^2{\theta }_W\left(\text{Re}[{\lambda }_t]y_{7A}^{\prime }+\text{Re}[{\lambda }_c]y_c\right)\right\}$$

$$\begin{array}[]{ll}{{\cal L}}_{\Delta S=1}={{\cal L}}_{\Delta S=1}^{2}+{{\cal L}}_{\Delta S=1}^{4}+\cdots\\ \\ \quad={{G_{8}}F^{4}}\underbrace{\langle\lambda_{6}D_{\mu}U^{\dagger}D^{\mu}U\rangle}_{{K\rightarrow 2\pi}/3\pi}+\underbrace{{G_{8}F^{2}}\sum_{i}N_{i}W_{i}}_{K^{+}\rightarrow\pi^{+}\gamma\gamma,{{K\rightarrow\pi l^{+}l^{-}}}}\end{array}$$

$$V_{+}(z)=a_{+}+b_{+}z+V_{+}^{\pi \pi }(z),z=q^2/m_K^2.$$

$$\bar{s}{\gamma }^{\mu }(1-{\gamma }_5)d\leftrightarrow i{F}_{\pi }^2(U{\partial }^{\mu }{U}^{†}{)}_{23},U=U(\pi ,K,\eta ),$$

$$a_{+}^{\mathrm{NP}}=\frac{2\pi \sqrt{2}}{\alpha }{V}_{ud}{V}_{us}^{\ast }{C}_{7V}^{\mathrm{NP}}.$$

$$C_{7V}^{\mu \mu }-C_{7V}^{ee}=\alpha \frac{a_{+}^{\mu \mu }-a_{+}^{ee}}{2\pi \sqrt{2}{V}_{ud}{V}_{us}^{\ast }}.$$

$$C_9^{B,\mu \mu }-C_9^{B,ee}=-\frac{a_{+}^{\mu \mu }-a_{+}^{ee}}{\sqrt{2}{V}_{td}{V}_{ts}^{\ast }}\approx -19\pm 79,$$

$$V_{+}^{\pi \pi }(z)={\int }_{4M_{\pi }^2}^{\infty }dx\frac{{\rho }_{+}^{\pi \pi }(x)}{x-z{M}_K^2-i0}.$$

$${\rho }_{+}^{\pi \pi }(s)$$

Full text

I Introduction and $K\to\pi\nu\bar{\nu}$

Rare kaon decays furnish challenging MFV probes and will severely constrain additional flavor physics motivated by NP Crivellin:2017gks . SM predicts the $V-A\otimes V-A$ effective hamiltonian (Fig. 1)

[TABLE]

$x_{q}=m_{q}^{2}/M_{W}^{2}$, $\theta_{W}$ the Weak angle and $X$’s are the Inami-Lin functions with Wilson coefficients known at two-loop electroweak corrections and the main uncertainties is due to the strong corrections in the charm loop contribution. The structure in (1) leads to a pure CP violating contribution to $K_{L}\rightarrow\pi^{0}\nu\overline{\nu},$ induced only from the top loop contribution and thus proportional to $\Im m(\lambda_{t})$ ($\lambda_{t}=V_{ts}^{*}V_{td}$) and free of hadronic uncertainties. This leads to the SM prediction

[TABLE]

where the parametric uncertainty due to the error on $|V_{cb}|$, $\rho$ and $\eta$ is shown.

Typical BSM predict new flavor structure that might affect $K\to\pi\nu\bar{\nu}$ that now can be tested at NA62 and KOTO Mirra ; we describe two different BSM effects i) new flavor structures for $\epsilon^{\prime}$ avoiding $\Delta S=2$ constraints (Fig. 1) Kitahara:2016otd ; Crivellin:2017gks and ii) attempts to describe B-anomalies Hurth:2017sqw , typically induce large flavor effects at O${(1)}$  TeV DAmbrosio:2017wis . i) the recent lattice results for $K\to 2\pi$ leave open the possibility of BSM for $\epsilon^{\prime}$; to isospin breaking terms in ${\Im}(A_{2})$ have been studied Kitahara:2016otd in Fig.1. We expect effects at most 10% in $K^{+}\to\pi^{+}\nu\bar{\nu}$ while are more sizable for $K_{L}\to\pi^{0}\nu\bar{\nu}$. Theoretically addressing flavor in Randall Sundrum models is more challenging: we have studied the so called flavor anarchy scenario with 5D MFV and custodial symmetry; the only sources of flavor breaking are two 5D anarchic Yukawa matrices. These matrices also generate also the bulk masses, which are responsible for the resulting flavor hierarchy. The theory flows to a next to minimal flavor violation model where flavor violation is dominantly coming from the 3rd generation. We show that it is possible to find a range of parameters for bulk masses satisfying experimental flavor constraints, but also we explain the neutral B-anomalies, requiring NP flavor scale at O${(1)}$  TeV. Then we address $K\to\pi\nu\bar{\nu}-$decays: we show the TH predictions as a function of the bulk fermion masses in Fig.2 DAmbrosio:2017wis . A natural issue is to test O${(1)}$  TeV physics at LHC; we are trying to apply the technique of Ref. Chakraborty:2017mbz to this purpose.

II $K_{L,S}\rightarrow\mu^{+}\mu^{-}$

Recent $K_{S}\rightarrow\mu\overline{\mu}$ LHCB measurement is very interesting and unexpected

[TABLE]

It represents an important milestone since it has improved the previous limit, $<3.2\times 10^{-7}\ {\rm at}\ 90\ \%\ {\rm CL}$, lasted 40 years. It is based on a production of $10^{13}\ K_{S}$ per fb*-1* inside the LHCB acceptance and it is obtained using 1.0 fb*-1* of pp collisions at $\sqrt{s}=7\ {\rm TeV}$ collected in 2011.

Two photon exchange generates the dominant contribution for both $K_{L}$ and $K_{S}$ decays to two muons DAmbrosio:2017klp . The structure of weak and electromagnetic interactions entails a vanishing CP conserving short distance contribution to $K_{S}\rightarrow\mu^{+}\mu^{-}$. Indeed the SM short diagrams (similar to $K\to\pi\nu\bar{\nu}$ in Fig. 1) lead to the SM effective hamiltonian similar to eq. (1).

The LD contributions to $K_{S}\to\mu^{+}\mu^{-}$ Fig. (4) have been computed reliably in CHPT ($B=(5.0\pm 1.5)\times 10^{-12}$). The relevant short distance contributions are

[TABLE]

We have shown that in some appealing susy scenario in Fig. (3) Chobanova:2017rkj large allowed new physics contributions (NP) can be substantially larger that SM SD contributions.

The short distance hamiltonian will contribute also to $K_{L}\rightarrow\mu\overline{\mu}$, through a CP conserving amplitude, $\Re(A_{\rm short})$, that has to be disentangled from the large LD two-photon exchange contributions, $A_{\gamma\gamma}$: the absorptive LD contribution is much larger than SD, in the rate respectively 25 times larger than dispersive; total $B_{\rm expt}=(6.84\pm 0.11)\times 10^{-9}$. To extract SD info the situation would be better if we would know the sign of $A_{\gamma\gamma}$, theoretically and experimentally unknown. While $K_{L}-$decays outside the LHCB fiducial volume the interference $A(K_{L}\rightarrow\mu\overline{\mu}^{*}A(K_{S}\rightarrow\mu\overline{\mu}$ may affect the LHCB $K_{S}-$rates: we can study the time interference $K_{S,L}\to\mu\mu$ ; this can be done by flavor tagging ${K^{0}}{\bar{K}}^{0}$ , specifically by detecting the associated $\pi^{\pm}$ and (or) $K^{\mp}$, determining the impurity parameter $D=\frac{K^{0}-\bar{K}^{0}}{K^{0}+\bar{K}^{0}}$. Then interference term will affect the measured branching DAmbrosio:2017klp :

[TABLE]

Then we are i) increasing the sensitivity to short distance and ii) possibly determining the sign ${A}_{L\gamma\gamma}$

[TABLE]

Experimentally, one can also access an effective branching ratio of $K_{S}^{0}\to\mu^{+}\mu^{-}$ DAmbrosio:2017klp which includes an interference contribution with $K_{L}^{0}\to\mu^{+}\mu^{-}$ in the neutral kaon sample.

LHCB has a beautiful kaon physics program Chobanova:2017rkj ; Junior:2018odx .

III The weak chiral lagrangian

In Ref. Cappiello:2017ilv we have studied how to determine the weak O($p^{4}$) chiral countertems in

[TABLE]

In fact as shown in Table I there is a subset of the 37 CT’s, $N_{14}^{r},\ N_{15}^{r},\ N_{16}^{r}$ and $N_{17}$, that can be determined from experiments. Due to the accurate NA48/2 study of the decays $K^{\pm}\to\pi^{\pm}\pi^{0}\gamma$ and $K^{\pm}\to\pi^{\pm}\pi^{0}e^{+}e^{-}$ the subset of CT’s in the table I can be determined

IV Lepton flavor universality violation in kaons

The dominant contribution to $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ is due to single virtual-photon exchange. The amplitude involves a vector form factor $V_{+}(z)$ which up to $O(p^{6})$ in the chiral expansion, can be decomposed in the general form DAmbrosio:1998gur

[TABLE]

Here the LECs $a_{+}$ and $b_{+}$ parametrise the polynomial part, while the rescattering contribution $V_{+}^{\pi\pi}$ can be determined from fits to $K\to\pi\pi$ and $K\to\pi\pi\pi$ data. Chiral symmetry alone does not constrain the values of the LECs,so instead, we consider the differential decay rate $d\Gamma/dz\propto|V_{+}(z)|^{2}$ as a means to extract $a_{+}$ and $b_{+}$ from experiment. The resulting fit to the decay spectra from all available high-statistics experiments is given in Table 2.

Now for the crucial point: if lepton flavour universality applies, the coefficients $a_{+}$ and $b_{+}$ have to be equal for the $ee$ and $\mu\mu$ channels, which within errors is indeed the case. Since the SM interactions are lepton flavour universal, deviations from zero in differences like $a_{+}^{\mu\mu}-a_{+}^{ee}$ would then be a sign of NP, and the corresponding effect would be necessarily short-distance Crivellin:2016vjc .

To convert the allowed range on $a_{+}^{\rm NP}$ into a corresponding range in the Wilson coefficients $C_{7V}^{\ell\ell}$, we make use of the $O(p^{2})$ chiral realization of the $SU(3)_{L}$ current

[TABLE]

to obtain

[TABLE]

Contributions due to NP in $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ can then be probed by considering the difference between the two channels

[TABLE]

If the framework of MFV, this can be converted into a constraint on the NP contribution to $C_{9}^{B}$:

[TABLE]

where we have averaged over the two electron experiments listed in Table 2.

Evidently, the determination of $a_{+}^{\mu\mu}-a_{+}^{ee}$ needs to be improved by an $O(10)$ factor in order to probe the parameter space relevant for the $B$-anomalies, whose explanation involves Wilson coefficients $C_{9,10}^{B}=O(1)$ Descotes-Genon:2015uva . Improvements of this size may be possible at NA62, especially for the experimentally cleaner dimuon mode which currently has the larger uncertainty.

V Determining $a_{+}$ and $b_{+}$

Recently we have addressed determining $a_{+}$ and $b_{+}$ from low energy data and short distance constraintsDAmbrosio:2018ytt ; DAmbrosio:2019xph . This is done by first part describes the contribution from the two-pion intermediate state to $V_{+}(z)$ in eq. 6. It is constructed upon assuming, in analogy with the electromagnetic form factor of the pion $F_{V}^{\pi}(s)$ that it is given by an unsubtracted dispersion integral,

[TABLE]

The absorptive part consists of the two-pion spectral density $\rho_{+}^{\pi\pi}(s)$, and is obtained upon inserting a two-pion intermediate state in the representation of the form factor given in eq. 6,

[TABLE]

Data provide the scattering amplitude ${f}^{K^{\pm}\pi^{\mp}\to\pi^{+}\pi^{-}}_{1}$ (fixing also the correct position of the $\rho$-poles !!), while matching short distance and resonances furnish also the remaining contributions in eq. 6. Our predictions are in Fig. 5 DAmbrosio:2018ytt and they have been improved in DAmbrosio:2019xph .

Acknowledgements.

I would like to thank the organizers of FCPC 2019 in particular Bob Kowalewski. I also acknowledge collaboration with L. Cappiello, O. Cata, D. Greynat A. Iyer, T. Kitahara, M. Knecht, D.Martinez Santos and K. Yamamoto.