Dark-sector physics in the search for the rare decays K + → π + ν ¯ ν and K L → π 0 ν ¯ ν
DDark-sector physics in the search for the rare decays K + → π + ν ¯ ν and K L → π ν ¯ ν M. Fabbrichesi † and E. Gabrielli ‡†∗ † INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy ‡ Physics Department, University of Trieste, Strada Costiera 11, 34151 Trieste and ∗ NICPB, R¨avala 10, Tallinn 10143, Estonia (Dated: June 23, 2020)We compute the contribution of the decays K L → π Q ¯ Q and K + → π + Q ¯ Q , where Q is a darkfermion of the dark sector, to the measured widths for the rare decays K + → π + ν ¯ ν and K L → π ν ¯ ν .The recent experimental limit for Γ( K + → π + ν ¯ ν ) from NA62 sets a new and very strict boundon the dark-sector parameters. A branching ratio for K L → π Q ¯ Q within the reach of the KOTO sensitivity is possible. The Grossman-Nir bound is weakened by the asymmetric effect of the differentkinematic cuts enforced by the NA62 and KOTO experiments. This last feature holds true for allmodels where the decay into invisible states takes place through a light or massless intermediatestate.
INTRODUCTION
The search for the rare decays K + → π + ν ¯ ν and K L → π ν ¯ ν is a most promising testing ground forphysics beyond the standard model (SM) because theirSM values are “short-distance” dominated and can bepredicted with great precision [1]. The contribution ofmany models beyond the SM to these decays has beenstudied (see, for example, the review articles in [2] and[3]).Among the models beyond the SM, those based on adark sector containing light dark fermions Q (by defini-tion singlet of the SM gauge groups and experimentallyindistinguishable from the SM neutrinos) are unique be-cause they can introduce a contribution to these decaysthat is a three-body decay (without the neutrinos in thefinal states) mediated by a massless vector boson. Thisfeature leads to the possibility of evading the Grossman-Nir (GN) bound [4] by means of a kinematical suppres-sion which is asymmetric in the two decays.The idea that the width of the decay K L → π ν ¯ ν canexceed the value dictated by the GN bound purely be-cause of kinematical reasons is best illustrated by thefollowing, rather extreme, case. There exists a small re-gion of the phase space where the decay K + → π + Q ¯ Q vanishes while the decay K L → π Q ¯ Q remains open.This region is selected by taking values of m Q inside theinterval m K + − m π + < m Q < m K L − m π . (1)For m Q within the interval in Eq. (1), the K + cannotdecay into a charged pion and the pair of dark fermionswhile the K L , owing to its larger mass, can.In a more general (and perhaps more realistic) case,the width Γ( K + → π + Q ¯ Q ) can be suppressed by theevents selection in the experimental setting dedicated toits measurement more than the width Γ( K L → π Q ¯ Q ) isby the events selection applied in the corresponding ex-periment. Eventually, this asymmetry—which originatesin the dependence of the signal events on the kinemat- ical variables and their relationship to the experimentalcuts—gives rise to a Γ( K L → π Q ¯ Q ) larger than whatrequired to satisfy the GN relationship.In this paper, we analyze the two rare decays K + → π + Q ¯ Q and K L → π Q ¯ Q in a simplified model of thedark sector—which is inspired by dark sector scenarios in[5, 6] and contains new flavor changing neutral currents(FCNC) structures and CP violation independently ofthe SM. Because of the asymmetric selection of the eventsoutlined above, it is possible to bypass the GN bound andobtain a branching ratio BR ( K L → π Q ¯ Q ) compatiblewith all existing bounds on FCNC physics and in thesensitivity range of the current experiments. Below ashort summary of the experimental situation.This year the upper bound in the result from BNLE949 [7] BR ( K + → π + ν ¯ ν ) = 1 . +1 . − . × − (2)has been (preliminarily) updated by CERN NA62 [8] toBR ( K + → π + ν ¯ ν ) < . × −
90% CL (3)which is now very close to the SM prediction which is [9]BR ( K + → π + ν ¯ ν ) = (7 . +0 . − . ± . × − , (4)where the first error summarizes the parametric, the sec-ond the remaining theoretical uncertainties.Meanwhile the limit from the 2015 run at J-PARCKOTO [10]BR ( K L → π ν ¯ ν ) < . × −
90% CL (5)is being updated by data from the 2016-18 run with asingle event sensibility (SES) of 6 . × − [11]. This SESspans a large range of values above the SM prediction,which is [9]BR ( K L → π ν ¯ ν ) = (2 . +0 . − . ± . × − , (6)where, as before, the first error summarizes the paramet-ric, the second the remaining theoretical uncertainties. a r X i v : . [ h e p - ph ] J un FIG. 1: Summary of the experimental limits (90% CL) on K L → π ν ¯ ν ( KOTO ) and K + → π + ν ¯ ν ( NA62 ). Also indi-cated are the GN bound and the SM predictions. The blueregion is excluded (assuming the validity of the GN bound).
As shown in Fig. 1, it is still possible that new physicsdominates this channel and the current sensitivity ofKOTO—falling as it does in the interval between theSM prediction and the exclusion limit in Eq. (5)—couldfind it. Scenarios giving rise to events in the KOTO SESrange are discussed in [12].Yet there is a catch: most of the range of the SES ofKOTO and Eq. (3), when taken together, violate the GNbound [4]BR ( K L → π ν ¯ ν ) ≤ . K + → π + ν ¯ ν ) , (7)which is only based on isospin symmetry and the differ-ence in the Kaon respective lifetimes. For this reason,the very stringent new limit in Eq. (3) on the chargedKaon decay seems to imply a comparably stronger limiton new physics in the neutral Kaon channel, as depictedin Fig. 1 by the blue exclusion region.As anticipated, this bound can be bypassed in the sim-plified dark-sector model by either the vanishing of theBR ( K + → π + Q ¯ Q ) when the mass of the dark fermionsis taken in the interval in Eq. (1) or because of the dif-ferent selections of the events in the kinematical regionsexplored by the two experiments. As discussed below,only the second possibility is fully consistent with theKOTO events.The paper is organized as follows. In the next sec-tion we present the details of a model for the dark sec-tor, including the most relevant constraints. In section3 we give the predictions for the total decay width andbranching ratio of K L → π Q ¯ Q , while in section 4 weanalyze the impact of the experimental cuts selections on the branching ratios. Finally, in section 5 we presentour conclusions. A MODEL OF THE DARK SECTOR
Among the many models for the dark sector (see, forexample, the review articles in [13]), we use one made toresemble QED—that is, a theory of charged fermions. Ithas the advantage of being simple. It contains fermions Q Ui and Q Di , where the index i runs over generationslike in the SM, and these dark fermions are charged onlyunder a gauge group U (1) D —a proxy for more generalinteractions—with different charges for the Q U and Q D type. The dark photon is massless and directly only cou-ples to the dark sector [14] (in contrast with the caseof massive dark photons). We denote throughout with α D = e D / π the U (1) D fine structure constant.There is no mixing between the ordinary and the darkphoton because such a term in the Lagrangian can berotated away [15, 16] (again, in contrast with the case ofthe massive dark photon). The dark fermions carry anelectric millicharge, the value of which is severely limitedby existing constrains (see, for example, the relative dis-cussion in [17]). This millicharge and the dark photoncoupling e D are independent parameters and we considerthe case in which the millicharge is negligible with respectto e D .The dark model scenario we are using here is a sim-plified version of the models in [5, 6], where only therelevant interactions for the physical processes we aregoing to discuss are retained. The original proposal [5]and its extended version to left-right SU (2) L × SU (2) R gauge group [6], has been mainly introduced to pro-vide a natural solution to the flavor hierarchy puzzle ofSM fermion masses. This model predicts the existence ofdark fermions and messenger fields (with universal mass),the latter having the same quantum numbers of squarksand slepton of supersymmetric models. The additionalrequirement of an unbroken U (1) D gauge theory in thedark sector, under which both dark fermions and mes-senger fields are charged, has the benefit to maintain sta-ble the dark fermions (provided the messenger sector isheavier) thus promoting them to potential dark mattercandidates.The dark fermions couple to the SM fermions by meansof a Yukawa-like interactions. The Lagrangian containsterms coupling SM fermions of different generations withthe dark fermions. In general the interaction is not di-agonal in flavor and, for the SM s and d quarks relevantfor Kaon physics, is given by L ⊃ g R ρ sdR S † R ¯ Q dL s R + g L ρ sdL S † L ¯ Q sR d L + H.c. (8)In Eq. (9), the fields S L and S R are heavy messengerscalar particles, doublets and singlets of the SM SU L (2) s R d L QS ± ¯ γs R d L Q ¯ γS ± FIG. 2: Vertex diagrams for the generation of the dipole op-erators in the model of the dark sector. gauge group respectively as well as SU (3) color triplets(color indices are implicit in Eq. (9)). The symmetricmatrices ρ sdL,R are the result of the diagonalization ofthe mass eigenstates of both the SM and dark fermions;they provide the generation mixing (and the CP-violationphases) necessary to have the messengers play a role inflavor physics. The messenger fields are heavier than thedark fermions and charged under the U (1) D gauge inter-action, carrying the same charges as the dark fermions.In order to fix the notation, we report below also theLagrangian for the flavor diagonal interaction L ⊃ g R ρ ssR S † R ¯ Q sL s R + g L ρ ddL S † L ¯ Q dR d L + H.c. . (9)The minimal flavor violation hypothesis requires the di-agonal couplings ρ to be ρ ssL,R , ρ ddL,R (cid:39) L ⊃ λ S S (cid:16) S L S † R ˜ H † + S † L S R H (cid:17) , (10)where H is the SM Higgs boson, ˜ H = iσ H (cid:63) , and S ascalar singlet. The Lagrangian in Eq. (10) gives rise tothe mixing after the scalars S and H take a vacuum ex-pectation value (VEV), respectively, µ S and v —the elec-troweak VEV. After diagonalization, the messenger fields S ± couple both to left- and right-handed SM fermionswith strength g L / √ g R / √
2, respectively. We canassume that the size of this mixing—proportional to theproduct µ s v of the VEVs—is large and of the same orderof the masses of the scalars.This model (see [6] for more details) has been used todiscuss processes with the emission of dark photons inHiggs physics [19], flavor changing neutral currents [18],kaon [20, 21] and Z boson [22] decays. Coupling SM fermions to the dark photon
SM fermions couple to the dark photon only via non-renormalizable interactions [14] induced by loops of dark-sector particles. The corresponding effective Lagrangianrelevant for the rare decays of the Kaons is equal to L = e D
2Λ ¯ s σ µν ( D M + iγ D E ) d B µν + H.c. , (11) where B µν is the strength of the dark photon field, Λ theeffective scale of the dark sector, which is the same orderof magnitude as the scalar masses m S . The magnetic-and electric-dipole are given by D M = ρ sd ρ ∗ dd g L g R (4 π ) and D E = ρ sd ρ ∗ dd g L g R (4 π ) , (12)respectively. For simplicity we take g L = g R real and D E = 0. A CP-violating phase comes from the mixingparameters: ρ sd ρ ∗ dd − ρ ∗ sd ρ dd = 2 i sin δ CP . (13) Constraints on the parameters of the model
The size of the coupling α D is constrained by galaxydynamics and cosmology (see [23–25]) if dark matter isamong the fermions charged under U (1) D . This limitdepends on the mass of the dark matter. The coupling α D can be as large as 0.1 for a mass around 10 TeV,while values around α D = 0 .
001 (like those we shall use)require a mass around 100 GeV.Anyway, the light dark fermions Q into which the darkphoton decays in K L → π Q ¯ Q and K + → π + Q ¯ Q arenot dark matter because they have annihilated beforethe current epoch into dark photons ¯ γ with a thermalaveraged cross section given by (cid:104) σ Q ¯ Q → ¯ γ ¯ γ v (cid:105) = 2 πα D m χ , (14)where v is the relative velocity of the annihilated pair.For a strength α D in the range we shall use (namely,between 0.0003 and 0.004, see Fig. 4 below), all darkfermions with masses of order 100 MeV have a large crosssection and their relic densityΩ χ h ≈ . × − GeV − (cid:104) σ Q ¯ Q → ¯ γ ¯ γ v (cid:105) (15)is below 10 − and therefore negligible.The scale Λ is constrained by astrophysical and cosmo-logical data [14, 26]. These limits only refer to the fla-vor conserving interactions—mostly electrons in the caseof stellar cooling, muons and s -quark in primordial nu-cleosynthesis and light quarks in the 1987A supernovaexplosion—and we assume here that they are not rele-vant because do not apply in our flavor-changing processcase. The only relevant limit is the one from Kaon mix-ing that we include in our analysis by means of Eq. (18)below.There are no bounds on the masses m Q of the darkfermions because of their very weak interaction with theSM states. There may be a question about a light massfor the dark fermion because of the impact of the darksector on the cosmic microwave background. This pointneeds to be investigated further [27].Laboratory limits apply to the mass of the messengerscalar states m S of the model, which is of the same or-der as Λ. The messenger states have the same quantumnumbers and spin of the supersymmetric squarks. At theLHC they are copiously produced in pairs through QCDinteractions and decay at tree level into a quark and adark fermion. The final state arising from their decay isthus the same as the one obtained from the ˜ q → qχ pro-cess. Therefore limits on the messenger masses can be ob-tained by reinterpreting supersymmetric searches on firstand second generation squarks decaying into a light jetand a massless neutralino [28], assuming that the gluinois decoupled. In particular we have used the upper limitson the cross section for various squark masses of [28] thatthe ATLAS collaboration provided on HEPData . Theselimits have been used to compute the bounds as a func-tion of the messenger mass using next-to-leading orderQCD cross section for squark pair production from theLHC Higgs Cross Section Working Group .We take into account the contributions to the totalevent yield given only by right-handed (degenerate) mes-sengers associated to the first generation of SM quarks,with the others set to a higher mass and thus with a negli-gible cross section. This corresponds to have only 2 lightdegrees of freedom, which are analogous to ˜ u and ˜ d insupersymmetry. With this assumption we obtain a lowerbound on their masses of 940 GeV, limit that increasesup to 1.5 TeV by assuming that messengers of both chi-ralities associated to the first and second generation ofSM quarks are degenerate in mass.These limits on Λ of the order of 1 TeV are muchweaker than those obtained in the next section from theKaon mass difference. Constraint from the Kaon mass difference
A direct constraint on the parameters of the modelarises because the same term driving the meson decayalso enters the box diagram that gives rise to the massdifference of the neutral meson. This quantity is givenby∆ m K = (cid:20) g L ( ρ Lsd ) ρ Ldd ρ Lss + g R ( ρ Rsd ) ρ Rss ρ Rdd Λ (cid:21) f K m K π (16)where we have identified m S = Λ and used the leadingvacuum insertion approximation ( B K = 1) to estimatethe matrix element (cid:104) K | (¯ s L γ µ d L ) (¯ s L γ µ d L ) | ¯ K (cid:105) = 13 m K f K B K η QCD (17)and a similar one for right-handed fields, where s L , d L represent the corresponding quark fields with left-handed chirality. Since we are just after an order of magnitudeestimate, we neglect the running (and contributions frommixing) of the Wilson coefficient η QCD of the 4-fermionoperator. Given the long-distance uncertainties, to sat-isfy the experimental bound on the mass difference, weonly impose that the new contribution does not exceedthe measured value. In order to simplify the analysis,in the expression of Eq.(16) we have neglected the CPviolating contributions, and, as already mentioned, as-sumed all couplings to be real and g L = g R . Moreover,to directly constrain the magnetic dipole interactions, weapproximate the diagonal couplings ρ dd = ρ ss = 1 inEq.(16), which is also in agreement with the minimal fla-vor violation hypothesis.The comparison requires the introduction of the fulleffective Lagrangian [30] inclusive of the new operatorsinduced by the dark sector. By using the results in [31],we obtain [20, 21] |D M | Λ ≤ π ∆ m exp K f K m K = 2 . × − MeV − , (18)with f K =159.8 MeV and ∆ m exp K = 3 . × − MeV [32].
THE DECAY WIDTH
This process is experimentally seen as two photons(from the decay of the pion) plus the missing energyand momentum carried away by the neutrinos. In thepresence of the dark sector, the same signature would beprovided by K → π ¯ γ , where ¯ γ is a dark photon, but thisdecay is forbidden by the conservation of the angular mo-mentum when the dark photon is massless. This meansthat the decay we are interested in can only proceed ifthe dark photon is off shell and decays into a pair of darkfermions.This signature could proceed also via box diagramsat 1-loop, where in the internal states are running mes-sengers and dark-fermions fields. However, the box dia-grams are suppressed—doubly, by an extra mass factor O ( m K / Λ) and an additional factor O ( g L g R / (4 π ) /α D )with respect to the diagram with an off-shell dark-photon, thus they are subleading and we neglect themin our analysis.Assigning the momenta as K L ( p K ) → π ( p π ) Q ( q ) ¯ Q ( q ) , we find d Γ( K L → π Q ¯ Q ) dz dz = 2 α D π |D M | Λ m K | f KπT ( z , z ) | Ω C ( z , z ) sin δ CP (1 + r π ) (cid:104) r π + 4 z z + r π (2 z + 2 z − (cid:105) , (19)where r π = m π /m K , z = q .p π /m K , z = q .p π /m K and sin δ CP is defined in Eq. (13) and comes from the CP-violation in the dark sector. We have found Package-X [33] useful in checking Eq. (19).The Sommerfeld-Fermi factor [29] is given byΩ C ( z , z ) = ξ ( z , z ) e ξ ( z ,z ) − , (20)with ξ ( z , z ) = − πα D (cid:113) − m Q / ( q − m Q ) , (21)and q = m K − m π − m K ( z + z ), arises from the(dark) attractive Coulomb interaction of the (dark) finalstates. This factor can be numerically important andpartially compensates the kinematical suppression dueto the smallness of the available phase space when m Q is sufficiently large; it is characteristic of having a darksector with QED-like interactions.In Eq. (19) we have taken for the hadron matrix ele-ment (cid:104) π | ¯ s σ µν d | K (cid:105) = ( p µπ p νK − p νπ p µK ) √ f KπT ( q ) m π + m K , (22)where the tensor form factor is given by f KπT ( q ) = f KπT (0)1 − s KπT q , (23)with q as before and f KπT (0) = 0 . s KπT =1 . − on the lattice [34].The phase-space integration is between z minmax = ( m ) minmax − m π − m Q m K z minmax = ( m ) minmax − m π − m Q m K , where ( m ) minmax = ( m Q + m π ) ( m K − m Q ) and( m ) minmax = ( E + E ) − (cid:18)(cid:113) E − m π ± (cid:113) E − m Q (cid:19) for E = (cid:0) m − m Q + m π (cid:1) (cid:112) m , E = (cid:0) m K − m − m Q (cid:1) (cid:112) m with m = 2 m K z + m π + m Q .The result for Γ( K + → π + Q ¯ Q ) is the same as that inEq. (19) but for the absence of the CP-violating sin δ CP and for a factor 0.954 coming from the isospin rotationand the difference in the masses. The two widths togethersatisfy the GN relationship in Eq. (7) once the differentlifetimes of the K + and the K L are taken into accountin the BRs. BR ( K L → π Q ¯ Q ) without experimental cuts We take m K L = 497 . m π = 134 .
98 MeV [32] andspan the possible values within the window in Eq. (1)178 < m Q <
181 MeV assuming maximal CP violation(sin δ CP = 1). We vary the dark-photon coupling con-stant: 0 . < α D < .
15. After enforcing the limit inEq. (18), we obtain that the integration of Eq. (19) overthe phase space leads to3 . × − < BR ( K L → π Q ¯ Q ) < . × − , (24)a range that covers the entire region from below the SMprediction to above the KOTO SES region.The result in Eq. (24) only depends in a significativemanner on • the choice of m Q and α D . By taking m Q closer tothe upper end of the window in Eq. (1) we closethe phase space and in the end, Sommerfeld-Fermienhancement notwithstanding, the width goes tozero. Notice that the window in Eq. (1) can be(slightly) enlarged by having the Γ( K + → π + Q ¯ Q )not closed but only suppressed by the kinematicsbelow the experimental limit in Eq. (3) (and stillabove the SM prediction). • sin δ CP . The whole decay width is proportional tothe size of CP violation. Its size can be modulatedby taking sin δ CP smaller than one.Notice that the new limit in Eq. (3) would imply astrong bound on the dark-sector parameters if the chan-nel were to be open and not kinematically restricted. Weuse this constraint in the next section. The transverse momentum of the pion
The particular kinematic window in Eq. (1) constrainsthe possible transverse momenta p π T of the π and wehave p π T < (cid:112) [ m K − (2 m Q − m π ) ][ m K − (2 m Q + m π ) ]2 m K which gives p π T <
36 MeV—for the most favorable caseof taking m Q = 178 MeV. This value can be increasedto around 60 MeV if we allow m Q to drop below thethreshold for the K + decay while still suppressing thewidth of this channel by the smallness of the phase space.The signal region of KOTO cuts off pions with mo-menta smaller than 130 MeV to reduce the backgroundfrom K L → π + π − π [10]. It is a prediction of the sce-nario with the choice in Eq. (1) that the pions have smalltransverse momentum and are, therefore, in a kinemati-cal region excluded by the KOTO experiment. ENTER THE EXPERIMENTAL CUTS
Let us now relax the strict constraints in Eq. (1) andat the same time take into account the actual cuts imple-mented by the experiments in selecting the signal events.The NA62 experiment enforces a selection on thesquare of the missing mass m = − m π + m K (1 − z − z ) (25)and the momentum of the pion. These cuts aim to re-duce the background from K + → π as well as to 2 π .Accordingly, in order to compare the dark-sector modelwith experiments, we only include events within the tworegions [8] 0 . < m < .
068 GeV (26)and 0 < m < .
01 GeV . (27)The momentum of the pion is taken to be between 15and 35 GeV.The KOTO experiment excludes events with a cut onthe transverse momentum p T = m K (cid:112) ( r π + z + z ) − r π . (28)The actual cut is in part a function of the distance of thepion decay vertex [10]; we approximate it to a rectangularregion as 130 MeV < p T <
250 MeV (29)and assume that the pion decays within the distance in-cluded in the experiment.
Events selection and GN bound
The actual number of events seen by both NA62 andKOTO is related to the BR by the acceptances of therelative decay and the efficiency in the detection of theevents. We look at the effect on the GN bound of enforc-ing the kinematical cuts used by the NA62 experimenton the number of events in the case of the decay intodark-sector fermions. This estimate provides only a par-tial inclusion of the actual differences between the darksector and the SM decay because the losses in the accep-tance of NA62 include—on top of the kinematical cuts in π + and ∆ m —also the effect of the detector geometryand particle identification and association in the fiducialvolume. Whereas a complete analysis would require theMonteCarlo simulation of the entire experimental setup,we only include the change in the acceptance in goingfrom the SM decay into neutrinos to the decay into thedark-sector fermions with respect to the kinematical cuts.This change is a conservative estimate of the actual ef-fect because we assume that the efficiency of triggers andtracking as well as the overall geometric acceptance areunchanged. The change in acceptance thus included suf-fices in showing that the GN bound is weakened by theexperimental cuts implemented by NA62 when appliedto the dark-sector decay.This is best understood by looking at the Dalitz plotsfor the decays. In Fig. 3 we show the Dalitz plot for thewidth Γ( K + → π + Q ¯ Q ) (that for BR ( K L → π Q ¯ Q ) isthe same) and compare it with those for the kinematicalvariables used in the experimental cuts: m and p T .Because of the massless intermediate state throughwhich the decay takes place, the width takes its largestvalues in the region of the Dalitz plot where z and z are more or less equal and in the middle of their range(lighter color in Fig. 3).Comparing the first plot in Fig. 3 with those on theright, one can see how the cuts (in red) for m removea region where the width is at its largest while those in p T do not. The GN bound is not respected because ofthis asymmetric effect in the selection of the events afterimposing the cuts in the kinematical variables.This feature holds true not only for the model of thedark sector we considered but for all the models wherethe decay into invisible states takes place through a lightor massless intermediate state. Notice that if the decaywere to proceed through a contact interaction—as it doesin the SM—the width would be largest in the oppositerange (darkest color in Fig. 3) and the experimental cutsmore symmetrical and less crucial. A somewhat similar argument was discussed in the case of thetwo-body decay K + → π + X in [35]. ���� �� ��� ��� ��� ��� ������������������ � � � � ������ ����� Γ � ���� �� ���� �� FIG. 3: Dalitz plots for the width Γ( K + → π + Q ¯ Q ) (for m Q = 10 MeV and α D = 0 . m andthe transverse momentum p T . Comparing the first plot on the left with those on the right, one can see how the experimentalcuts for m (NA62) remove a region (hatched between the two red contour lines and between the single red line and theupper border) where the width is at its largest while those in p T (KOTO) (the hatched region below the red line) are lesscrucial. See text for more details on the kinematic cuts. The decays in the presence of the experimental cuts
FIG. 4: Range of values for BR ( K L → π Q ¯ Q ) as functionof m Q . The dark gauge coupling α D has been taken so as tosatisfy the NA62 bound in Eq. (3) for each value of m Q . TheBR ( K L → π Q ¯ Q ) < . × − (KOTO limit) because of thelimit in Eq. (5). Also indicated is the GN bound correspond-ing to Eq. (3). The hatched areas are excluded. The insetdepicts the values of α D as a function of m Q as obtained bythe procedure outlined in the text. In order to satisfy the bound in Eq. (3) for BR ( K + → π + Q ¯ Q ) for values of m Q outside the range in Eq. (1),in which this BR is zero, we must take smaller values of α D with respect to the range considered in the previoussection. The procedure to implement these constraints isthe following.For m K L = 497 . m K + = 493 . m π = 134 . m π + = 139 .
57 [32], while again assuming asbefore maximal CP violation (sin δ CP = 1) and enforcingthe limit in Eq. (18) from Kaon mixing, we can obtainan upper bound on α D by requiring that the number ofevents generated by the BR ( K + → π + Q ¯ Q ) satisfies the NA62 experimental bound in Eq. (3). This limit is com-puted after enforcing the experimental cuts in Eq. (26)and Eq. (27). The value of α D thus found can then beinserted, together with the corresponding value for m Q ,to obtain an upper bound on BR ( K L → π Q ¯ Q ).Fig. 4 shows the result of this procedure. TheBR ( K L → π Q ¯ Q ) is a function of m Q and the value of α D obtained by implementing the constraint in Eq. (3) onthe BR ( K + → π + Q ¯ Q ). The α D coupling varies withinthe range 0 . < α D < .
003 as indicated in the in-let of Fig. 4; the maximum allowed value of α D growsquadratically as the mass m Q comes closer to the kine-matical threshold. The red curve is the upper bound ofthe BR. The area below (indicated by the lighter red re-gion) covers the entire KOTO SES region (as depicted inFig. 1) as m Q varies between zero and 120 MeV. Largervalues of m Q give a BR too large and already excludedby KOTO. CONCLUSIONS
The recently announced new limit on the Kaon decay K + → π + ν ¯ ν [8] implies that very little room is left in thischannel for new physics. If the GN bound is applied, thedecay K L → π ν ¯ ν is constrained to be lower than most ofthe current KOTO sensibility [11]. The potential tensionbetween events to be found by the
KOTO collaborationand the GN bound can be resolved in a model of the darksector with light dark fermions Q behaving as neutrinosin the detector, via the decay channel K L → π Q ¯ Q . ABR( K L → π Q ¯ Q ) above the SM prediction and in theregion currently probed by KOTO can be attained if wetake into account the asymmetric effect of the selection ofevents by the different cuts on the kinematical variablesenforced by the NA62 and KOTO experiments. We thanks Gaia Lanfranchi for discussions on the NA62experiments. MF is affiliated to the Physics Department ofthe University of Trieste and the Scuola Internazionale Su-periore di Studi Avanzati—the support of which is gratefullyacknowledged. MF and EG are affiliated to the Institute forFundamental Physics of the Universe, Trieste, Italy.[1] V. Cirigliano, G. Ecker, H. Neufeld, A. Pich and J. Por-toles, Rev. Mod. Phys. , 399 (2012) [arXiv:1107.6001[hep-ph]].[2] A. J. Buras, F. Schwab and S. Uhlig, Rev. Mod. 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