Dark photon effect on the rare kaon decay $K_L \rightarrow \pi^0 \nu {\bar \nu}$

We present an analysis of the effect of a dark photon on the rare kaon decay $K_L \rightarrow \pi^0 \nu {\bar \nu}$. All relevant couplings of the dark photon to the Standard Model particles are derived explicitly in terms of the dark photon mass and the mixing parameter. We find that the dark photon yields no more than a few percent correction to the Standard Model branching ratio ${\rm Br}(K_L \rightarrow \pi^0 \nu {\bar \nu})$ in the region of interest.


Introduction
The dark photon is an appealing hypothesis for new physics beyond the Standard Model (SM) [1,2].It has emerged as a canonical portal connecting the dark matter and SM sectors [3,4].While there have been numerous experimental searches at  +  − and hadron colliders [5,6,7,8,9], there is no direct evidence for its existence so far.Instead, rather strong constraints have been derived on the mixing parameter , leading to an upper limit of  ≤ 10 −3 in both light [7,8] and heavy [9] mass regions, with just a few gaps.
Amongst rare decays, the processes  + →  +  ν and   →  0  ν are the so-called golden channels, as their branching ratios can be computed with high precision.The most accurate SM values for the branching fractions are [23] Br( + →  +  ν) = (9.11± 0.72) × 10 −11 , while the latest experimental measurements from NA62 [24] and KOTO [25] set upper limits at 68% and 90% confidence level (CL), respectively Br( + →  +  ν) = (10.6 +4.0 −3.4 | stat ± 0.9 syst ) × 10 −11 , Br(  →  0  ν) KOTO < 4.9 × 10 −9 . ( Possible anomalies between experiments and the SM predictions have been investigated in many new physics models [26,27,28,29,30]. A light dark photon may be expected to yield a significant contribution to branching ratios such as these because of enhancement from the propagator, compared with the  boson.Moreover, for an ultralight dark photon   , it is possible that the two-body decay  →   will occur, followed by visible or invisible decays of   [27,31].Recently, a light dark photon was introduced, either on-shell or off-shell, to improve the agreement of the branching ratios between the SM predictions and the experimental measurements for a bunch of rare  decays [32], with three parameters, the mass    , and two independent mixing parameters (,   ).However, the best fit results required an unrealistically large value of the mixing parameter  [27].
Here, we perform a quantitative analysis of the rare kaon decays within the dark photon framework, focusing on the   →  0  ν channel in order to avoid the complexity arising from the charm quark contribution in the  + channel.All the couplings of the dark photon to the SM particles are explicitly dependent on just two parameters, the dark photon mass,    , and the mixing parameter, .We also explore the sensitivity of its branching ratio to the dark photon parameters.
In Sec. 2, we briefly review the rare kaon decays in the Standard Model.We derive the couplings of the physical  and the dark photon in Sec.3.1 and 3.2, and the correction to the branching ratio in Sec.3.3.We present our numerical results in Sec. 4. Finally, we summarize our analysis in Sec. 5.

Standard Model predictions
The SM contributions to the decay modes  + →  +  ν and   →  0  ν include the " penguin" and the "box" diagrams with up, charm, and top quark exchanges.The invariant amplitudes can be written in the following form [33], The function     represents the charm quark contribution, which is only relevant to the  + →  +  ν channel and results from the renormalization group calculation in next-to-leading-order logarithmic approximation.
In this work, we will focus on the   →  0  ν decay, which only depends on the corresponding function in the top quark sector,  (  ) ≡  (  ,   = 0), by neglecting the lepton masses.Including next-to-leading order (NLO) Quantum Chromodynamics (QCD) corrections, it is given by [33] where with  () Z (  ) and  () (  ,   ) the " Z penguin" and the "box" contributions, respectively.It is convenient to compute the loop functions in t Hooft-Feynman gauge ( = 1), in which both the induced s Z vertex and the box diagram also receive contributions from the would-be Goldstone bosons ( ± ) [34].The leading order (LO) terms are given by [34,35] (see the Appendix Appendix A), The QCD correction in the top quark sector is [21]  (1) where The branching ratio for   →  0  ν involves only the top-quark contribution and can be written as [36,33] Br where   =  *    are the Cabibbo-Kobayashi-Maskawa (CKM) factors, and   parametrizes the hadronic matrix element [37,38], (10)

Dark photon formalism
The dark photon is usually introduced as an extra  (1) gauge boson [1,2,39], interacting with the SM particles through kinetic mixing with hypercharge [40] L ⊃ L int SM − where   is the Weinberg angle. and Z denote the unmixed version of the dark photon and the SM neutral weak boson, respectively.  and Z are the SM field strength tensors.
After diagonalizing the mixing term through field redefinitions, the masses of the physical  and   are given by [41,14] where The Z −  mixing angle  is given by The mass difference between the physical  and   is always finite for non-zero , Thus there is a region of the dark photon parameter space which is inaccessible [14], which is known as the "eigenmass repulsion" region on the Because of kinetic mixing, the SM weak couplings of the  boson given by (B.1) and (B.2) will be modified.The dark photon will also couple to the SM particles.All the physical couplings depend on only two parameters,    and .

Weak couplings
The couplings of the physical  to the quarks are given by [14,18,41] where   , = 2/3 and   , = −1/3.Its couplings to the neutrinos will be shifted by  ,  = (cos  −   sin ) Z,  . (16) The SM couplings of the Z to the gauge bosons and the Goldstone bosons will also be modified as

Dark couplings
The dark photon interacts with the quarks ( = , ) through both vector and axialvector couplings [14,18,41], Its interaction with the neutrinos also has  −  form, with We also derive its couplings to the gauge bosons and the Goldstone bosons,

Branching ratio
The most general form of the matrix element  (0) (  ,   ), after restoring the  -boson propagator and its coupling to neutrinos, can be written as where  is the momentum transfer through the Z propagator.When dark photon effects are included, the above expression is generalised to where the functions  (0)  (  ) and  (0)   (  ) have the same form as  (0) Z (  ) (see Appendix Appendix A), with the Z couplings being replaced by those for  and   , respectively.Note that the inclusion of the dark photon does not affect the box contribution.
The dark photon effect can be characterised by a correction factor to the SM branching ratio, which is independent of   ,   , and .
The strongest experimental constraint on  comes from the CMS Collaboration [9], leading to an upper limit of  ∼ 10 −3 , while the region of parameter space with   <    < 110 GeV is unconstrained as the  boson dominates  +  − production there.
The current limit in connection with electroweak precison observables (EWPOs) [12,13] leads to  < 0.06 for dark photon mass up to 200 GeV, which becomes much stronger when    gets close to   .The recent  −  DIS analysis [15,19] placed relatively weak constraint on , which will go above 0.1 when    >   .Therefore, the dark photon parameters in the region  ≤ 0.2 in the (,    ) plane is of most interest, as this region has not been fully excluded by the existing constraints [18].The corrections to the SM prediction of the branching ratio Br(  →  0  ν) are shown in figure 1 as a percentage.Surprisingly, the sensitivity of the correction factor   to the dark photon parameters is quite similar to that of  1 at low scale in parity violating electron scattering [18].  is at most several percent, corresponding to the case where the dark photon parameters approach the "eigenmass repulsion" region.It might have been expected that a light dark photon with mass at sub-GeV region could lead to a large correction.However, the dark couplings become negligibly small in that region because they scale as    /  , eliminating the enhancement associated with the propagator.It would also be interesting to apply the current framework to two-body decay  →   for ultralight dark photon.The gap on the  −  plane is not accessible due to "eigenmass repulsion" associated with the  mass.The 95% CL exclusion limits on  from DIS and EWPO determinations are taken from Refs.[15,19] and Ref. [13], respectively.We also show the most stringent constraint from the CMS Collaboration [9].

Conclusion
We have presented a systematic calculation of the dark photon contribution to the rare kaon decays, focusing on the channel   →  0  ν.We explicitly derived the coupling constants of the dark photon to the Standard Model fermions, the gauge bosons, and the would-be Goldstone bosons in the t Hooft-Feynman gauge.
In contrast with naive expectations, the branching ratio of   →  0  ν deviates from the Standard Model prediction by at most a few percent.This deviation would be at its largest were a dark photon to exist with parameters close to the "eigenmass repulsion" region.Unfortunately this effect is too small to be observed in the near future, given the experimental accuracy anticipated in the next few years [42].However, the advances in precision of such important tests of the Standard Model improve inexorably.
On the other hand, we expect that the dark photon would have a similar effect on the branching ratio of the charged channel  + →  +  ν, while a quantitative analysis requires inclusion of the charm quark sector which adds 30% to the total branching ratio.Experiments at the Brookhaven National Laboratory [43] has measured Br( + →  +  ν) = (17.3+11.5 −10.5 ) × 10 −11 .As mentioned earlier, a more precise result was reported by the NA62 experiment [24], Br( + →  +  ν) = (10.6 +4.0 −3.4 | stat ± 0.9 syst ) × 10 −11 at 68% CL.The proposed HIKE experiment is expected to reach a branching ratio measurement with O (5%) precision [44], making the charged channel more promising as a probe of the dark photon.
Finally, it is worth noting that the dark photon may serve as a promising portal connecting dark matter and ordinary particles.One can also extend the current analysis by introducing the dark photon couplings to dark matter particles.The potential anomalies of rare kaon decays may then be applied to constrain these extra invisible modes, in addition to the neutrino final states.contributions given by [34] explicitly in terms of the weak couplings as

Figure 1 .
Figure1.The dark photon correction to the SM value of the branching ratio Br(  →  0  ν).The gap on the  −  plane is not accessible due to "eigenmass repulsion" associated with the  mass.The 95% CL exclusion limits on  from DIS and EWPO determinations are taken from Refs.[15,19] and Ref.[13], respectively.We also show the most stringent constraint from the CMS Collaboration[9].

Figure A1 .
Figure A1.The penguin diagrams contributing to the induced s Z vertex.