The Light Sgoldstino Phenomenology: Explanations for the Muon $(g-2)$ Deviation and KOTO Anomaly

In this work, we study the long-standing experimental anomaly in muon $(g-2)$ and also recent anomalous excess in $K_L\to \pi^0+\nu\bar\nu$ at the J-PARC KOTO experiment with sgoldstino. After supersymmetry breaking, the interactions between quarks and sgoldstino ($s$) make the decays $K\to \pi+s$ sizable through loop diagrams, which affects the measurements of decays $K\to \pi+\mathrm{invisible}$. Furthermore, the couplings between photons and sgoldstino contribute to $\Delta a_\mu$ as well as bino-slepton contribution. With satisfying all known experimental constraints such as from NA62, E949, Orsay, KTEV and CHARM experiments, these two anomalies can be explained simultaneously. The mass of CP-even sgoldstino is close to the neutral pion mass which does not violate the Grossman-Nir bound. The parameter space can be further tested in future NA62, DUNE experiments, as well as experiments in the LHC.


Introduction
After the Higgs particle was discovered at the Large Hadron Collider (LHC) [1,2], the Standard Model (SM) has been confirmed as a successful low energy description of Nature. Although most of the SM predictions are consistent with the experimental data, there still exists some problems in the SM. For example, gauge hierarchy problem, gauge coupling unification, dark matter, baryon asymmetry, and neutrino masses and mixings, etc. Supersymmetry provides a natural solution to the gauge hierarchy problem. In the Supersymmetric SMs (SSMs) with R-parity, gauge coupling unification can be achieved, the lightest supersymmetric particle (LSP) can be a dark matter candidate, and the electroweak gauge symmetry can be broken radiatively due to large top quark Yukawa coupling, etc. Therefore, supersymmetry is a promising scenario for new physics beyond the SM. However, we have strong constraints on the parameter space in the SSMs from the supersymmetry searches at the LHC. The interesting question is whether there exists some light particles in the SSMs which can be probed or can explain the anomalies at the current experiments. Therefore we can probe supersymmetry indirectly.
We shall present one example in this paper. Once supersymmetry is broken spontaneously, we have a Goldstone fermion G, i.e. goldstino. The superpartner of goldstino is called sgoldstino S = 1 √ 2 (s + ia) where s and a are a CP-even and CP-odd real scalars. In particular, in the low energy supersymmetry breaking, i.e. N = 1 supersymmetry is broken at low energy around TeV scale [3][4][5][6][7][8][9][10], sgoldstino is light so that it can be probed in low energy experiments. Therefore, we would like to explore the sgoldstino phenomenology.
Experimentally, there exists a 3.7 σ discrepancy for the anomalous magnetic moment of the muon a µ = (g µ − 2)/2 between the experimental results [11,12] and theoretical predictions [13][14][15][16] ∆a µ = a exp µ − a th µ = (2.74 ± 0.73) × 10 −9 , (1.1) which is a well-known long-standing deviation. Computing the hadronic light-by-light contribution with all errors under control by using lattice QCD, several groups are trying to improve the precision of the SM predictions [17][18][19][20]. The recent lattice calculation for the hadronic light-by-light scattering contribution has confirmed the ∆a µ discrepancy [21], and then a new physics explanation of the discrepancy is needed. Also, the ongoing experiment at Fermilab [22,23] and one planned at J-PARC [24] will try to reduce the uncertainty. In addition, the flavor changing processes like rare K meson decays, K L → π 0 νν and K + → π + νν, which are loop suppressed in the SM [25,26], are very sensitive to the new physics beyond the SM [27][28][29][30][31][32][33]. The SM predictions are [27] Br(K L → π 0 νν) SM = (3.00 ± 0.30) × 10 −11 , (1.2) These processes are studied at the KOTO experiment [34,35] at J-PARC [36] and NA62 experiment [37] at CERN. In particular, four candidate events have been observed in the signal region of K L → π 0 νν search at the KOTO experiment, while the SM prediction is only 0.10±0.02. One event can be suspected as a background coming from the SM upstream activity, and the other three can be considered as signals since they are not consistent with the currently known background. Note that the single event sensitivity is 6.9 × 10 −10 , three events are consistent with at 68(90)% confidence level (C.L.), including statistical uncertainties, whose central value is almost two orders of magnitude larger than the SM prediction. This new result includes the interpretation of photons and invisible final states as νν and is in agreement with their previous bounds [34] Br(K L → π 0 νν) KOTO18 < 3.0 × 10 −9 . (1.5) However the charged kaon decay searches have not observed any excess events. The recent update from NA62 puts a bound Br(K + → π + νν) NA62 < 2.44 × 10 −10 (1.6) at 95% C.L., which is consistent with the SM prediction of Eq. 1.3. Furthermore, the generic neutral and charged kaon decays satisfy the following Grossman-Nir (GN) bound [38] Br which depends on the isospin symmetry and kaon lifetimes. Because the explanations for the KOTO anomaly might be strongly constrained by the GN bound, the new physics explanation for the KOTO anomaly is required to not only generate three anomalous events, but also satisfy the GN bound. Recently, the KOTO anomaly has been studied extensively in the literatures .
In this paper, we shall explain the muon anomalous magnetic moment and KOTO anomaly via a light sgoldstino. This paper is structured as follows: In Sec. 2, we review the basic motivation of sgoldstino and introduce relevant interaction between sgoldstino and SM particles. In terms of them, we can analysis the sgoldstino phenomenology quantitively. Sec. 3 investigates how we can fit the muon (g − 2) and KOTO anomaly simultaneously. In addition, all the relevant constraints are considered seriously. Finally, we make a summary on the results in Sec. 4.

Motivation of Sgoldstino and its implication on phenomenology
Once supersymmetry is spontaneously broken, there must exists a massless goldstino G [64]. Goldstino is a Goldstone fermion and becomes the longitudinal component of gravitino with its mass being lifted by gravity correction i.e. m 3/2 ∼ F/M P . Thus, the existence of gravitino is an inevitable prediction of local supersymmetry from super-Higgs mechanism. The goldstino chiral superfield is written as Generally we assume that supersymmetry is broken by the dynamics of Goldstino superfield Φ and F encodes the SUSY breaking information. Through some weak couplings of Goldstino to those of the MSSM fields, some non-renormalizable operator gives rise to soft breaking terms. For example, any supersymmetric theory contains operator i.e. L eff =M a /F d 2 θΦW α a W a α , which is used to generate non-vanishing gaugino mass in various SUSY breaking and mediation mechanism.
The reason the operator is non-renormalizable comes from super-trace theorem. We should mention that, the operator not only generates gaugino soft masses but includes an inevitable coupling between sgoldstino and SM gauge bosons. The generic Lagrangian of our model is thus given by where M i are gaugino masses from F -term of Φ, andF µν = 1 2 µναβ F αβ . Also, A U,D,E ij are the couplings, but not necessary the same as the Yukawa couplings in the SM in general.
and H d are the left-handed quark doublets, right-handed uptype quark, and right-handed down-type quarks, left-handed lepton doublets, right-handed charged letons, up-type and down-type Higgs doublets, respectively.
The mass of sgoldstino arises from higher order Kahler potential, To obtain the mass splitting between s and a, we consider the high-dimensional Kähler potential as follows [6] And the scalar potential becomes [6] Taking κ 2 real (κ 2 = κ * 2 ), m 2 S = |F S | 2 M 2 , and using S = 1 √ 2 (s + ia), we can rewrite Eq. (2.5) as below [6] Thus, there are two simple cases for mass splitting: Case (1): Single resonance. When κ 1 κ 2 , we have m s m a . Case (2): Twin resonances. When κ 1 κ 2 , then m s m a . In this paper, we shall consider Case (1), and can treat a nearly massless in our setup. Therefore the light scalar to mimic KOTO anomaly is the real component of sgoldsitino, s. Besides, there is a tension between the neutral and charged sector which can be used to fix our sgoldstino mass. If we believe that these decays are dominated by the transitions with isospin ∆I = 1 2 , these two decays are related by the so-called Grossman-Nir (GN) bound [38]: The numerical factor comes from the differences in widths, isospin breaking effects and QED corrections. This bound puts very strong constraints on any explanations of the KOTO anomaly with new physics, since both two decays are induced by the same transition s quark → d quark + s. Recent studies based on the effective theory showed that a violation of GN bound by the new physics contribution is quite nontrivial [57,58]. Considering the experimental sensitivities of the charged and neutral Kaon experiments, the KOTO anomaly can be explained without violating GN bound if a new scalar with a mass of about pion is stable or with a lifetime lower than about a nanosecond, which has been pointed out in [65]. Very recently, the authors in [39] pointed out that a light scalar, with mass different from the pion mass and with a long lifetime, but not necessarily stable, can also explain the observed KOTO excess. In short, the mass of the sgoldstino should be close to the mass of pion i.e. m s ∈ [50,200]MeV. The additional interaction for sgoldstino and SM particles in Eq. 2.3 can be ignored at high scale SUSY breaking, since it is suppressed by SUSY breaking scale √ F . But it actually provides large deviation from SM when SUSY breaking scale is low enough such as √ F ∼ [10 3 , 10 5 ]GeV. So the question becomes whether or not we can fit the signal of KOTO anomaly, muon (g − 2) and evade the current bounds for light sgoldstino and sparticles. The relevant interactions between sgoldstino and Standard Model particles that are responsible for these anomalies can be obtained from Eq.2.3.
where M γ is the combination of M 1 and M 2 after electroweak symmetry breaking, i.e. M γ = M 1 cos 2 θ w + M 2 sin 2 θ w . It determines the interaction strength between sgoldstino and photon. There are several facts that we should mention the relevant lagrangian in Eq.2.8: • For naturalness, we can assume the universal trilinear soft terms as Therefore the trilinear soft terms do not lead to any tree-level flavor-chaning process. And then sgoldstino shares the same interaction strength for both quarks and leptons, which plays a crucial role in our phenomenological study.
where we denote λ q and λ l as the effective coupling between sgoldstino and quarks as well as leptons, respectively. Generically, λ q is around 10 −2 to obtain correct neutral Kaon decay. Meanwhile, such a tiny λ l can not generate ∆a µ at the required order 10 −9 . Here, we will not consider the non-universal trilinear soft terms as a solution since it requires about 1% fine-tuning. So this becomes a challenge for sgoldsino to explain the two anomalies at the same time. We will figure out how to solve the problem in next section.
• Sgoldstino can also couples to W-boson and Z-boson through gaugino mass M 1 and M 2 . But these two couplings have no effect on Kaon decay. Of course, the interaction with W-boson provides additional channel for sgoldstino decay into photon. But it is too small compared with the tree-level decay induced by M γ .
• MeV sgoldstino can not decay into gluon pair and quarks since it does not have enough energy to hadronization. Thus even gluino is much heavier than bino, wino, it can not decay into gluon pair channel.
• Coupling between s and a can be induced by higher order Kahler potential. But these couplings are highly suppressed by UV cut-off M . It can be thought that s has no coupling with a effectively.
With Eq.2.8 at hand, we can easily compute the neutral Kaon K L decay widths where triangle function λ(x, y, z) determines the absolute value of Γ.
the form factor f + (0) is set to be 0.9709 by lattice QCD. The charged Kaon decay can be obtained by replacing Re[g(λ q )] to |g(λ q )| and corresponding mass parameters. In figure  1, we show the contours for BR(K L → π 0 s) and BR(K + → π + s) (solid and dashed lines respectively) in the (m s − √ F ) plane. We see that in the range of sgoldstino mass, the branching ratio is fairly insensitive to m s and thus determined mostly by √ F , where a larger √ F corresponds to a smaller branching ratio. We can also find that for the same branching ratio for both processes, BR(K + → π + s) requires a bigger value of √ F , which also means that for a given set of parameters, the values of BR(K + → π + s) is smaller than that of BR(K L → π 0 s). It naturally explains why we only observe the neutral Kaon decay in KOTO experiment.  Figure 1. Contours for branching ratios of processes K L → π 0 s and K + → π + s. The solid colored lines indicate the contours for BR(K L → π 0 s). Dotted lines of corresponding colors show where BR(K + → π + s) achieves the corresponding value: blue for BR = 10 −8 , red for BR = 10 −6 . We set For the decay of s, we have two different channels in the mass range that we interested in: decay into photon pair and decay into electron pair, and the decay widths are where τ e = 4m 2 e /m 2 s . In figure 2, we depict the branching ratios of the possible decay modes of s, i.e. s → e + e − and s → γγ. Since the diphoton width receives tree level contributions from M 1 , M 2 , one can see that this process will be the dominant decay mode with the scalar mass larger than tens MeV in figure 2.
Naively, we can estimate the contributions to muon g − 2 from sgoldstino, since it couples to SM leptons with effective coupling λ l . However due to the fact that the quark coupling is equivalent to leptons, its contribution is very small when we impose KOTO anomaly requirement, see figure 3. Here we define the contribution from λ l = A 0 v/ √ 2F to be ∆a Lepton µ . The ratio between ∆a Lepton µ and central value of required ∆a µ is smaller than 0.2 in all the parameter space. That is to say, only λ l contribution is not enough to generate required muon (g − 2).

Signal of (g − 2) µ , KOTO anomaly and Constraints
In this section, we focus on how to use sgoldstino to explain KOTO anomaly and muon g − 2 simultaneously. As discussed in last section, the coupling λ l does not contribute the desirable muon g − 2 that is several orders smaller than required one. But we are working with SUSY scenario, where neutralino and slepton can contribute to muon g − 2 too. Here we include all the five important one-loop diragrams: charigino-sneutrino loop (C), wino-slepton loop (W), bino-slepton loop (B), bino-higgsino loop (BHR and BHL) [66,67], where the loop functions are defined to be From Eq.3.1, slepton mass must be around O(400)GeV in order to boost the contribution to muon g − 2. However such a light slepton is highly constrained by null result of SUSY search in LHC. We have to set M 2 to be 1TeV to escape LHC exclusion limit. As a result, slepton mass can be around 300GeV without contradicting with observation.  In figure 4, we show the contour in plane (M 1 , M 2 ) by fixing mμ L = 400GeV, mμ R = mν = 300GeV. The green line is the lower bound of wino mass, red curve corresponds to the maximal values of ∆a µ , purple curve is the minimal value. Even though, wino is set to be 1TeV, O(400)GeV is still threaten by LHC constraint. Thanks to Barr-Zee two-loop contribution, we find slepton mass can be lifted. Recall that, sgoldstino contains direct interaction with photon. The two-loop Barr-Zee diagram can be effectively regarded as one-loop [68], where λ γ = √ 2M γ /F . Combining both of contributions can yield suitable ∆a µ . Therefore, we can use ∆a µ to reduce the number of input parameters. For example √ F can be solved by imposing muon (g − 2) constraint. Now we are in position to show how sgoldstino mimic the signal in the KOTO experiment. The effective branching ratio is The efficiency factor eff can be read from [34]. L is the detector size of KOTO experiment and chosen to be 3 meters. The charactermatic energy in KOTO experiment is 1.5 GeV. τ s is the lifetime of sgoldstino. In our sgoldstino model, we have 4 input parameters in total: We should mention that M 2 is fixed to be 1 TeV from LHC exclusion limit. The ratio A 0 / √ F and M 1 is set to be 0.2 and 100 GeV respectively, in order to boost muon g − 2 contribution. As a result, we have only two parameters m s and √ F to testify the signal and constraints. In figure 5, we show our final results in space (m s − √ F ) including both signals and constraints. In the blue band, we can obtain the KOTO signal events at the 95% C.L., in which the solid blue curve stands for the central value of KOTO data. To explain the muon g − 2 anomaly, we combine the contributions from sgoldstino, neutralino and slepton one-loop diagrams, Barr-Zee diagrams, then we find a red region in the parameter space where ∆a µ can be achieved in 2σ range. We choose a benchmark point to satisfy both anomalies in figure shown by a black point, sgoldstino mass m s ∼ 87 MeV and √ F ∼ 3.2 TeV.
Searching for rare decays of Kaons in a variety of beam dump experiments sets strong constraints on the light scalar. We demonstrate our analysis by showing the shaded excluded regions in the figure and discuss them in the following.
• E949 for green shaded region.
Measuring the decay K + → π + νν by the E949 collaboration has put constraints on the mimic process K + → π + X with X being long-lived particle. E949 collaboration [69] explored the possibility of such a process and provided the upper limits as function of scalar mass. We can thus use it to constrain our model easily with the effective branching ratio in Eq.3.4, except the characteristic energy of E949 is 0.71 GeV and detector length is 4 m.
• NA62 for light-orange shaded region. A similar constraint comes from the NA62, which sets a 95%C.L. bound on the Green: excluded regions at 95% C.L. by E949 experiment, which constraints on Br(K + → π + X) with X a long-lived particle. Light-Orange: limits from NA62 on Br(K + → π + νν) at 95% C.L.. Magenta: limits on displaced decays of the scalar from the CHARM experiment, which measures displaced decay of neutral particles into e + e − , γγ, µ + µ − . Black: the shaded region is excluded by the Orsay experiment which puts limits on the decaying of a scalar into electron pairs. Gray: limits from KTeV(γγ) on the process K L → π 0 γγ. Cyan: region excluded by K µ2 from searching of light scalars in process K + → π + s. The big black point in the parameters space corresponds to the benchmark that can explain both anomalies simultaneously. branching fraction of process K + → π + νν, Br(K + → π + νν) NA62 < 2.44 × 10 −10 . (3.5) To apply the NA62 limits we also should use the effective branching ratio as used for KOTO in Eq.3.4. The NA62 detector size L = 150 m; the scalar's energy is taken to be approximately half of the charged kaon energy at this experiment, E s = 37 GeV; for the NA62 effective branching fraction we set eff = 1. NA62 did not constrain the m s in the mass range [100 MeV, 161 MeV] because of large background in K + → π + π 0 → π + νν. So we do not need to compute the bounds in this mass range. NA62 excludes the parameter space in the light-orange shaded regions, and a gap also shown as expected (see figure 5).
• CHARM for magneta shaded region. The CHARM experiment, which is a proton beam-dump experiment, measures the displaced decay of neutral particles into γγ, e + e − and µ + µ − final states. Since our signal resulted from sgoldstino being produced from neutral and charged Kaon decay, then the sgoldstino decays into the γγ, e + e − final states, CHARM experiment is thus relevant for our model. The events number in the CHARM detector is [70] N det N s (e The exponential factors in determining the number of scalars that reach and decay within the detector volume. L dump = 480 m is the CHARM beam dump baseline, while L fid = 35 m is the detector fiducial length. The scalar momentum is obtained assuming an average scalar energy of E s = 12.5 GeV [71]. N s = 2.9 × 10 17 σ s /σ π 0 represents the number produced in the kaon decay, where σ s is the production cross section [72], with σ pp the proton cross section, M pp is the total hadron multiplicity and χ s = 1/7 is the fraction of strange pair-production rate [73]. For the neural pion yield we have σ π 0 σ pp M pp /3. Due to the fact that CHARM experiment has observed zero event for such decays, we can set 90% confidential level bound by requiring N det < 2.3. The magenta shaded region in the parameter space has been excluded.
• K µ2 for cyan shaded region.
The K µ2 experiment [74] searched for a neutral boson in a two-body decay of K + → π + X with X being the neutral scalar, and a momentum mono-chromatic π + was expected due to K + is stopped in the above 2-body decay. The null result of experiment thus set a constrain for our model. We translate the limits on our model parameters, as a result the cyan shaded region has been excluded as shown in figure 5.
• KTeV(γγ) for gray shaded region. The KTeV experiment is used to measure the process for neutral Kaon decay K L → π 0 γγ. The derived branching ratio for this process is [75] BR(K L → π 0 γγ) = (1.29 ± 0.03 ± 0.05) × 10 −6 . (3.8) As a result, we can use this bound to constrain sgoldstino by conservatively setting the bound BR(K L → π 0 s)BR(s → γγ) < 10 −6 [39,47]. Since the branching ratio of process s → γγ closes to 1 for most ranges of m s , the constraint is rather stringent, as shown by the gray region. Furhtermore, KTeV(e + e − ) can also put a constraint to our model. But it highly depends on the branching ratio into electron pair for sgoldstino. From figure 2, the photon final states dominates over electron. So we can safely ignore this constraint.
• Orsay for black shaded region. Orsay is an electron beam dump experiment which is sensitive to sgoldstino decaying into electron. It is similar with KTeV(e + e − ). We employ the method used in [76], and place the limits on our parameter space at the 95%C.L. As discussed about the KTeV(e + e − ) constraints, the limits of Orsay is much less constraining, see the black shaded region in figure 5.
As known to us, supersymmetry is one of the most attractive new physics models. Once the supersymmetry is broken, there exists a goldstino G, and its superpartner is accordingly called sgoldstino s. Its mass can be light if the couplings induced Kahler potential are O(1). In this work, we had explored whether the sgoldstino can explain KOTO anomaly and muon g − 2 simultaneously, which are viewed as hints of new physics. The interactions between sgoldstino and quarks generate the flavor-changing neutral-current transition from strange quark to down quark via penguin diagrams. The resulting K L → π 0 s transition followed by the decay of s → γγ explains the KOTO signal. Although the coupling between lepton and sgoldstino is too small to contribute the desirable muon g − 2, the contributions from sgoldstino, neutralino and slepton one-loop diagrams, Barr-Zee diagrams can explain this discrepancy. We also studied all known experimental constraints such as from NA62, E949, KOTO, Orsay, KTEV and CHARM experiments, and found that the mass of CPeven sgoldstino is in range [80,200] MeV without violating the Grossman-Nir bound. In particular we obtain a benchmark point which is sgoldstino mass m s ∼ 87 MeV and SUSY breaking scale √ F ∼ 3.2 TeV. The parameter spaces can be further tested in future NA62, DUNE experiments, as well as experiments in the LHC for sleptons.