Hunting down massless dark photons in kaon physics

TL;DR

This paper explores the potential for massless dark photons to cause rare kaon decays via flavor-changing magnetic-dipole couplings, suggesting detectable signals in experiments like NA62.

Contribution

It introduces a novel analysis of massless dark photon effects in kaon physics, focusing on flavor-changing couplings and their experimental signatures.

Findings

Branching ratios up to 10^{-7} are possible.

Massless dark photons could produce detectable signals in rare kaon decay experiments.

The study provides a framework for future experimental searches.

Abstract

If dark photons are massless, they couple to standard-model particles only via higher dimensional operators, while the kinetic mixing with photons, which motivates most of the current experimental searches, is absent. We consider the effect of possible flavor-changing magnetic-dipole couplings of massless dark photons in kaon physics. In particular, we study the branching ratio for the process $K^+\rightarrow \pi^+\pi^0 \bar \gamma$ with a simplified-model approach, assuming the chiral quark model to evaluate the hadronic matrix element. Possible effects in the $K^0$-$\bar K^0$ mixing are taken into account. We find that branching ratios up to $O(10^{-7})$ are allowed---depending on the dark-sector masses and couplings. Such large branching ratios for $K^+\rightarrow \pi^+\pi^0 \bar \gamma$ could be of interest for experiments dedicated to rare $K^+$ decays like NA62 at CERN, where $\bar \gamma$ can be detected as a massless invisible system.

Figures

Tables

Table 1. Table 1: In the first two rows, relevant operators are numbered according to the notation in  Gabbiani:1996hi ; Ciuchini:1998ix . The matrix elements ⟨ K 0 | Q i | K ¯ 0 ⟩ quantum-operator-product superscript 𝐾 0 subscript 𝑄 𝑖 superscript ¯ 𝐾 0 \langle K^{0}|Q_{i}|\bar{K}^{0}\rangle (in the vacuum insertion approximation for the renormalized operators Q i subscript 𝑄 𝑖 Q_{i} at the low energy scale μ = 2 𝜇 2 \mu=2 GeV) are given in the third row multiplied by the respective bag factors B i ​ ( μ ) subscript 𝐵 𝑖 𝜇 B_{i}(\mu)   Ciuchini:1998ix evaluated at same scale, with X K ​ ( μ ) = ( m K / ( m d ​ ( μ ) + m s ​ ( μ ) ) ) 2 subscript 𝑋 𝐾 𝜇 superscript subscript 𝑚 𝐾 subscript 𝑚 𝑑 𝜇 subscript 𝑚 𝑠 𝜇 2 X_{K}(\mu)=\left(m_{K}/(m_{d}(\mu)+m_{s}(\mu))\right)^{2} . The fourth row gives the Wilson coefficients at the matching scale (the common factor at the matching being C 2 = ξ 2 / ( 16 ​ π 2 ​ Λ 2 ) superscript 𝐶 2 superscript 𝜉 2 16 superscript 𝜋 2 superscript Λ 2 C^{2}=\xi^{2}/(16\pi^{2}\Lambda^{2})   Biswas:2016jsh , where ξ = g L ​ g R / 2 𝜉 subscript 𝑔 𝐿 subscript 𝑔 𝑅 2 \xi=g_{L}g_{R}/2 ). Following Ciuchini:1998ix , we take m d ​ ( μ ) = 7 subscript 𝑚 𝑑 𝜇 7 m_{d}(\mu)=7 MeV, m s ​ ( μ ) = 125 subscript 𝑚 𝑠 𝜇 125 m_{s}(\mu)=125 MeV, m K = 497 subscript 𝑚 𝐾 497 m_{K}=497 MeV, f K = 160 subscript 𝑓 𝐾 160 f_{K}=160 MeV, and B 1 , 2 , 3 , 4 , 5 ​ ( μ ) = 0.60 , 0.66 , 1.05 , 1.03 , 0.73 subscript 𝐵 1 2 3 4 5 𝜇 0.60 0.66 1.05 1.03 0.73 B_{1,2,3,4,5}(\mu)=0.60,0.66,1.05,1.03,0.73 , respectively.

$Q_1,{\tilde{Q}}_1$	$Q_2,{\tilde{Q}}_2$	$Q_3,{\tilde{Q}}_3$	$Q_4$	$Q_5$
${\overline{d}}_L^{\alpha }{\gamma }_{\mu }{s}_L^{\alpha }{\overline{d}}_L^{\beta }{\gamma }_{\mu }{s}_L^{\beta },(L\leftrightarrow R)$	${\overline{d}}_R^{\alpha }{s}_L^{\alpha }{\overline{d}}_R^{\beta }{s}_L^{\beta },(L\leftrightarrow R)$	${\overline{d}}_R^{\alpha }{s}_L^{\beta }{\overline{d}}_R^{\beta }{s}_L^{\alpha },(L\leftrightarrow R)$	${\overline{d}}_R^{\alpha }{s}_L^{\alpha }{\overline{d}}_L^{\beta }{s}_R^{\beta }$	${\overline{d}}_R^{\alpha }{s}_L^{\beta }{\overline{d}}_L^{\beta }{s}_R^{\alpha }$
$1/3m_K{f}_K^2{B}_1\left(\mu \right)$	$-5/2X_K{m}_K{f}_K^2{B}_3\left(\mu \right)$	$1/24X_K{m}_K{f}_K^2{B}_3\left(\mu \right)$	$1/4X_K{m}_K{f}_K^2{B}_4\left(\mu \right)$	$1/12X_K{m}_K{f}_K^2{B}_5\left(\mu \right)$
$-1/24C^2$	$0$	$1/12C^2$	$1/6C^2$	$1/6C^2$

Equations

$$\mathcal{L}\supset g_L({\bar{Q}}_L{q}_R)S_R+g_R({\bar{Q}}_R{q}_L)S_L+H.c.,$$

$$\mathcal{L}\supset {\lambda }_S{S}_0\left(S_L{S}_R^{†}{\tilde{H}}^{†}+S_L^{†}{S}_{R}H\right),$$

$$\Delta M_K=2\text{mbox}Re[⟨K^0|{\mathcal{H}}_{eff}^{\Delta S=2}|{\bar{K}}^0⟩].$$

$${\mathcal{H}}_{eff}^{\Delta S=2}=\sum _i^5{C}_i{Q}_i+\sum _{i=1}^3{\tilde{C}}_i{\tilde{Q}}_i.$$

$$\Delta M_K=8.47\times 10^{-13}\frac{{\xi }^2}{{\Lambda }^2},$$

$${\mathcal{H}}_{eff}^{\Delta S=1}=\frac{e_D}{64{\pi }^2}\frac{\xi }{\Lambda }\hat{Q},$$

$$K^{+}(p)\to {\pi }^{+}(q_1){\pi }^0(q_2)\bar{\gamma }(k),$$

$$\hat{M}=\frac{M(z_1,z_2)}{m_K^3}{\epsilon }_{\mu \nu \rho \sigma }{q}_1^{\nu }{q}_2^{\rho }{k}^{\sigma }{\epsilon }^{\mu }(k),$$

$$\frac{\text{mbox}{d}^2\Gamma }{\text{mbox}d{z}_1\text{mbox}d{z}_2}$$

$$\frac{M(z_1,z_2)}{m_K^3}$$

$$\mathrm{BR}(K^{+}\to {\pi }^{+}{\pi }^0\bar{\gamma })\simeq 1.31{\alpha }_D{\eta }^2\frac{{\xi }^2}{{\Lambda }^2},$$

$$\mathrm{BR}(K^{+}\to {\pi }^{+}{\pi }^0\bar{\gamma })to0.0pt\sim \text{hss}<1.6\times 10^{-7}.$$

Full text

The clarification of the origin of dark matter (DM) might require the existence of a dark sector made up of particles uncharged under the standard model (SM) gauge group. The possibility of extra secluded $U(1)$ gauge groups—mediating interactions in the dark sector via dark photons— is the subject of many experimental searches (see Raggi:2015yfk for recent reviews). These searches are mostly based on the assumption that the secluded $U(1)$ gauge group is broken, and the corresponding massive dark photon ($\gamma^{\prime}$) interacts directly with the SM charged fields through renormalizable (dimension-four) operators induced by the kinetic mixing between dark and electromagnetic photons. Experimental results are then parametrized in terms of the dark-photon mass $m_{\gamma^{\prime}}$ and mixing parameter $\epsilon$, with dark photon signatures that can either correspond to its decay into SM particles or assume an invisible decay into extra dark fields. Because the induced operators have dimension four, most studies necessarily explore regions where the couplings are very small (millicharges).

We address instead the case of an unbroken dark $U(1)$ gauge symmetry, with a massless dark photon ($\bar{\gamma}$). The role of massless dark photons in galaxy formation and dynamics has been discussed in Foot:2004pa ; Ackerman:mha ; Fan:2013tia ; Foot:2014uba ; Heikinheimo:2015kra . A strictly massless dark photon is very appealing from the theoretical point of view. Indeed, for massless dark photons it is possible Holdom:1985ag to define two fields, the dark and the ordinary photon, in such a manner that the dark photon only sees the dark sector. In this basis, ordinary photons couple to both the SM and the dark sector— the latter with millicharged strength to prevent macroscopic effects. Massless dark photons therefore interact with SM fields only through higher dimensional operators—typically suppressed by the mass scales related to new massive fields charged under the unbroken dark $U(1)$ gauge symmetry Dobrescu:2004wz —while their coupling constants can take natural values thanks to the built-in suppression associated to the higher dimensional operators. This makes the $\bar{\gamma}$ direct production in SM particle scattering/decay small and unobservable, consequently evading most of the search strategies for dark photons currently ongoing in laboratories. A possible exception is provided by the Higgs boson decay into dark photons in the nondecoupling regime. This scenario has been considered in Biswas:2016jsh , where observable $\bar{\gamma}$ production rates mediated by the Higgs decay $H\to\gamma\bar{\gamma}$ have been found at the LHC in realistic frameworks Gabrielli:2013jka ; Gabrielli:2016vbb . Flavor-changing-neutral-current (FCNC) decays of heavy flavors into a massless dark photon, $f\to f^{\prime}\bar{\gamma}$, can offer other search channels with potentially observable rates Dobrescu:2004wz ; Gabrielli:2016cut .

Here we focus on FCNC effects induced by massless dark photons $\bar{\gamma}$ in kaon physics, and discuss the change of picture with respect to the massive case.

The kaon system can be studied with great accuracy, allowing us to probe indirectly energy scales as large as tens of TeV, hence crucially constraining possible SM extensions. The detection of massive dark photons in $K$ decays is presently under scrutiny Raggi:2015yfk ; Pospelov:2017pbt . One can consider radiative $K$ decays where the (off-shell) SM photon $\gamma$ is replaced by a $\gamma^{\prime}$, and look for resonances at $m_{\gamma^{\prime}}$ for either $e^{+}e^{-}$ ($\mu^{+}\mu^{-}$) final states, or (in case $\gamma^{\prime}$ decays into dark particles) for invisible final systems with a peak structure at $m_{\gamma^{\prime}}$ in the missing mass distribution. Particular emphasis has been given to the decays $K^{+}\to\pi^{+}\gamma^{\prime}$ and $K^{+}\to\mu^{+}\nu\,\gamma^{\prime}$ Pospelov:2008zw ; Barger:2011mt ; Carlson:2013mya ; Batley:2015lha ; Chiang:2016cyf . However, if the secluded $U(1)$ gauge group is unbroken, these two channels are not viable. Indeed, $K^{+}\to\pi^{+}\bar{\gamma}$ violates angular momentum conservation, while $K^{+}\to\mu^{+}\nu\,\bar{\gamma}$ would require unsuppressed $\bar{\gamma}$ couplings.

Because $K^{+}\!$ decays into a dark photon $\bar{\gamma}$ must necessarily proceed through short-distance effects, we argue that the most interesting channel to look for massless dark photons in kaon physics could be the decay $K^{+}\to\pi^{+}\pi^{0}\bar{\gamma}$. This decay can be mediated by the FCNC transition $s\to d\,\bar{\gamma}$, prompted by a magnetic-dipole-type coupling generated at one loop by the dark-sector degrees of freedom. The dark photon gives rise in this case to a massless missing-momentum system inside the final state. Recently, the sensitivity of the NA62 experiment at the CERN SPS NA62:2017rwk to two-body $K$ decays into a light vector decaying invisibly [$K^{+}\!\to\pi^{+}\!+\!(\gamma^{\prime}\!\to\!E_{miss})$] has been emphasized Pospelov:2017pbt . For the three-body $K^{+}\to\pi^{+}\pi^{0}\bar{\gamma}$ channel, whose kinematics is less characterized, the detection efficiency is expected to be less favorable. Nevertheless—since the $K^{+}\to\pi^{+}\pi^{0}\bar{\gamma}$ channel has a unique potential to unveil the existence of a massless dark photon—we think that the NA62 Collaboration should consider search strategies aiming at detecting this newly proposed process, whose branching ratio (BR) can reach $10^{-7}$ in a simplified model of the dark sector, as we estimate in the following.

A simplified model of the dark sector.—We estimate BR($K^{+}\to\pi^{+}\pi^{0}\bar{\gamma}$) in a simplified model that makes as few assumptions as possible, while providing the dipole-type transition we are interested in.

The minimal choice in terms of fields consists of a SM extension where there is a new (heavy) dark fermion $Q$, singlet under the SM gauge interactions, but charged under an unbroken $U(1)_{D}$ gauge group associated to the massless dark photon. SM fermions couple to the dark fermion by means of a Yukawa-like interaction in the Lagrangian ${\cal L}$

[TABLE]

where new (heavy) messenger scalar particles, $S_{L}$ and $S_{R}$, enter as well. In Eq. (1), the $q_{L}$ and $q_{R}$ fields are the SM fermions [$SU(3)$ triplets and, respectively, $SU(2)$ doublets and singlets]. Flavor indices are implicit, and we assume common (*i.e. *flavor blind) couplings $g_{L}$ and $g_{R}$. The left-handed messenger field $S_{L}$ is a $SU(2)$ doublet, the right-handed messenger field $S_{R}$ is a $SU(2)$ singlet, and both are $SU(3)$ color triplets. These messenger fields are charged under $U(1)_{D}$, carrying the same $U(1)_{D}$ charge of the dark fermion.

In order to generate chirality-changing processes we also need in the Lagrangian the mixing terms

[TABLE]

where $H$ is the SM Higgs boson, $\tilde{H}=i\sigma_{2}H^{\star}$, and $S_{0}$ a scalar singlet. The Lagrangian in Eq. (2) gives rise to the mixing after both the $S_{0}$ and $H$ scalars take a vacuum expectation value (VEV), respectively, $\mu_{S}$ and $v$—the electroweak VEV. After diagonalization, the messenger fields $S_{\pm}$ couple to both left- and right-handed SM fermions with strength $g_{L}/\sqrt{2}$ and $g_{R}/\sqrt{2}$, respectively. We can assume that the size of this mixing—proportional to the product of the VEVs ($\mu_{s}v$) —is large and of the same order of the masses of the heavy fermion and scalars.

The SM Lagrangian plus the terms in Eqs. (1)–(2) (supplemented by the corresponding kinetic terms) provide a simplified model for the dark sector and the effective interaction of the SM degrees of freedom with the massless dark photon $\bar{\gamma}$. SM fermions couple to $\bar{\gamma}$ only via nonrenormalizable interactions, induced by loops of the dark-sector states. Two scales are relevant: the dark fermion mass $M_{Q}$, which parametrizes the chiral symmetry breaking in the dark sector, and the lightest-messenger mass scale $m_{S}$. Since we are considering the contribution to the magnetic dipole operator (assuming vanishing quark masses), the dominant effective scale associated with it will either be chirally suppressed (being proportional to $M_{Q}/m_{S}^{2}$, for $m_{S}\gg M_{Q}$), or scale as $1/M_{Q}$ (for $m_{S}\ll M_{Q}$) due to decoupling. In order to have only one dimensionful parameter, in our analysis we assume a common mass for the dark fermion and the lightest scalar field, which we identify with the new-physics scale $\Lambda$. This choice corresponds to the maximum chiral enhancement.

This scenario is a simplified version of the model in Gabrielli:2013jka ; Gabrielli:2016vbb ; Gabrielli:2016cut (possibly providing a natural solution to the SM flavor-hierarchy problem), as well as a template for many models of the dark sector.

Bounds from $K^{0}$-$\bar{K}^{0}$ and astrophysics.—A most stringent limit to the mass scale and couplings of the above simplified model comes from its extra contributions to the $K^{0}$-$\bar{K}^{0}$ mixing in the kaon system (related to the mass difference $\Delta M_{K}$ of the neutral mass eigenstates $K_{L}$ and $K_{S}$, assuming $CPT$).

In order to compute the dark-sector effects on $\Delta M_{K}$, we need to evaluate the dark-sector contribution to the effective Hamiltonian for the $\Delta S=2$ transitions, ${\cal H}^{\Delta S=2}_{eff}$

[TABLE]

The scalar-fermion interaction in Eq. (1) induces a new set of operators, which are reported in Table 1, then obtaining

[TABLE]

The Wilson coefficients at the matching scale are computed by considering the exchange of the lightest messenger state in the loop, which provides a good estimate of the dominant contribution in the large-mixing limit of the messenger mass sector.

We compute the corresponding Wilson coefficients $C_{i}(\mu)$ at the ${\cal O}(\alpha_{s})$ next-to-leading order, after running them from the matching scale down to the low energy scale $\mu\sim 2$ GeV, where the corresponding matrix elements are estimated on the lattice Ciuchini:1998ix . We assume as matching scale the characteristic mass $\Lambda$ of the lightest-messenger and dark-fermion states, assumed to be equal. Following this procedure, the dark-sector contribution to $\Delta M_{K}$ (in TeV) is

[TABLE]

where $\xi=g_{L}g_{R}/2$, and $\Lambda$ is in TeV units. We then assume that the above contribution of the new operators to Eq. (3) does not exceed 30% of the measured $\Delta M_{K}$ value PDG . Eq. (5) turns then into an upper bound for the allowed values for the $\xi^{2}/\Lambda^{2}$ ratio.

While the flavor-changing dipole operator induced in the simplified model (see Eq. (6) below) per se is only bounded by kaon physics, if we make the (very conservative) assumption that the model also gives flavor-diagonal dipole operators and these are the same size in the quark and lepton sectors, a bound can be derived from stellar cooling carried out by the emission of massless dark photons. Under these assumptions, the limit from $K^{0}$-$\bar{K}^{0}$ mixing in Eq. (5) falls between the current astrophysical bounds Hoffmann:1987et —with the most stringent one from white dwarves being 1 order of magnitude stronger and that from the Sun 1 order of magnitude weaker.

Amplitude and decay rate.—The $K^{+}\rightarrow\pi^{+}\pi^{0}\bar{\gamma}$ decay originates from the dimension-five magnetic dipole operator $\hat{Q}=\left(\bar{s}\,\sigma^{\mu\nu}\,d\right)\bar{F}_{\mu\nu}$, where $\bar{F}_{\mu\nu}$ is the $\bar{\gamma}$ field strength, $\sigma_{\mu\nu}=\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$, and color and spin contractions are understood. $\hat{Q}$ enters the effective Hamiltonian for $\Delta S=1$ transitions as

[TABLE]

where $\alpha_{D}=e_{D}^{2}/(4\pi)$ is the $\bar{\gamma}$ coupling strength. The Wilson coefficient multiplying the magnetic operator in Eq. (6) is obtained by integrating the vertex function in our simplified model (see Fig. 1). We have checked Eq. (6) by means of Package X Patel:2015tea .

The operator in Eq. (6) contributes only to the magnetic component of the process

[TABLE]

while its contribution to the process $K^{+}\rightarrow\pi^{+}\bar{\gamma}$ identically vanishes. The amplitude $\hat{M}\equiv\langle\bar{\gamma}\;\pi^{+}\pi^{0}|\,{\cal H}_{eff}^{\Delta S=1}\,|K^{+}\rangle$ in the momentum space can be written as

[TABLE]

where $\varepsilon^{\mu}(k)$ is the $\bar{\gamma}$ polarization vector. The corresponding differential decay rate is

[TABLE]

where $z_{i}=k\cdot q_{i}/m_{K}^{2}$ and $r_{i}=M_{\pi_{i}}/m_{K}$ Cirigliano:2011ny .

The matrix element in Eq. (8) can be estimated by means of the chiral quark model ($\chi$QM) chiQM . In this model quarks are coupled to hadrons by an effective interaction so that matrix elements can be evaluated by loop diagrams (see Fig. 2). In general there are several free parameters, but in the present case only $M$, the mass of the constituent quarks, and $f$, the pion decay constant, enter the computation. The model has been applied to kaon physics in Antonelli:1995nv , where a fit of the $CP$ preserving amplitudes of the nonleptonic decay of neutral kaons has yielded a value $M=200$ MeV Bertolini:2000dy with an error of less of 5%.

According to the $\chi$QM we obtain that the magnetic component generated by the dipole operator in Eq. (6) is given by

[TABLE]

where $D_{0}$ and $D_{00}$ are four-point Passarino-Veltman coefficient functions (see Ellis:2011cr for their explicit form) to be evaluated numerically Patel:2015tea .

Inserting the amplitude in Eq. (10) in the differential decay rate in Eq. (9) yields, after integration and by normalizing $\Gamma$ by the total $K^{+}$ width $\Gamma_{\rm tot}=5.317\times 10^{-14}$ MeV PDG ,

[TABLE]

where we assumed $M=200$, $f=92.4$, $m_{K}=494$, and $m_{\pi^{+}}=m_{\pi^{0}}=136$ MeV. The coefficient $\eta$ accounts for the renormalization of the Wilson coefficient of the dipole operator in going from the $\Lambda$ scale to approximately $m_{K}$. We assume it equal to 1, and discuss the impact of possible uncertainties below.

BR$(K^{+}\rightarrow\pi^{+}\pi^{0}\bar{\gamma})$ is proportional to $\xi^{2}/\Lambda^{2}$, just as $\Delta M_{K}$ in Eq. (5). By taking for $\xi^{2}/\Lambda^{2}$ the value that saturates the $\Delta M_{K}$ constraint, we find an upper bound for the BR which is, for the representative value $\alpha_{D}=0.1$,

[TABLE]

Fig. 3 shows the BR$(K^{+}\rightarrow\pi^{+}\pi^{0}\bar{\gamma})$ contour plot versus the scale $\Lambda$ and the coupling $\xi$, for $\alpha_{D}=0.1$. We see that a rather large range of parameters is allowed for which the BR is sizable. The upper bound—given by Eq. (12)—is represented in Fig. 3 by the boundary of the gray area.

There are three main sources of uncertainties in the result in Eq. (12):

• •

The matrix element estimate computed in the $\chi$QM depends on the parameter $M$. The result in Bertolini:2000dy seems to indicate a rather small uncertainty on this parameter but one must be aware of the dependence. We find an increase by a factor 2.5 in the BR when going from $M=200$ to 250 MeV;

• •

Even though there are $O(p^{4})$ chiral perturbation theory corrections to $K^{+}\rightarrow\pi^{+}\pi^{0}\bar{\gamma}$, these have been shown to be small Ecker:1991bf ;

• •

By taking the QCD leading-order multiplicative value $\eta=0.5$ (at $\mu=$ 2 GeV) Buras:1993xp , we find a BR smaller by a factor 1/4. However, it is known that nonmultiplicative corrections go the opposite direction, and we thus need the (not yet available) complete evolution before trusting this correction. Moreover, the QCD renormalization introduces a strong dependence on the low-energy scale $\mu$, because the matrix element computed within the $\chi$QM is scale independent.

On top of these uncertainties, we have the overall dependence on the $\alpha_{D}$ strength on which the BR depends linearly. There exist cosmological relic density bounds on the ratio $\alpha_{D}/\Lambda^{2}$ Ackerman:mha . Our choice of $\alpha_{D}=0.1$ is then consistent with $\Lambda$ of the order of 10 TeV.

Similar predictions can be obtained in the specific flavor model of Gabrielli:2013jka ; Gabrielli:2016vbb ; Gabrielli:2016cut . In particular, for $\alpha_{D}=0.1$, the approximate upper bound is given by BR $\simeq 1.2\times 10^{-8}$. The lower BR is explained by the dark-fermion masses being related in this case to the radiative generation of SM Yukawa couplings, resulting in a stronger chiral suppression of the effective scale associated with the dipole operator $\hat{Q}$, which turns out to be proportional to the bottom-quark Yukawa coupling Gabrielli:2016cut .

Conclusions.— NA62 at the CERN SPS will soon provide a sample of 1013 $K^{+}$, with hermetic photon coverage and good missing-mass resolution NA62:2017rwk . We propose to look for the rare decay $K^{+}\rightarrow\pi^{+}\pi^{0}\bar{\gamma}$ (where $\bar{\gamma}$ gives rise to a massless invisible system) as a sensitive probe for massless dark photons, for which the presently most explored dark-photon channels mediated by kinetic-mixing interactions in kaon decays are nonviable.