Exploring charm decays with missing energy in leptoquark models

We investigate the possibility that scalar leptoquarks generate consequential effects on the flavor-changing neutral current decays of charmed hadrons into final states with missing energy ($\not\!\!E$) carried away by either standard model or sterile neutrinos. We focus on scenarios involving the $R_2$, $\tilde R_2$, and $\bar S_1$ leptoquarks and take into account various pertinent constraints, learning that meson-mixing ones and those inferred from collider searches can be of much significance. We find in particular that the branching fractions of charmed meson decays $D\to M\!\not\!\!E$, $M=\pi,\rho$, and $D_s\to K^{(*)}\!\not\!\!E$ and singly charmed baryon decays $\Lambda_c^+\to p\!\not\!\!E$ and $\Xi_c\to\Sigma\!\not\!\!E$ are presently allowed to attain the $10^{-7}$-$10^{-6}$ levels if induced by $R_2$ and that the impact of $\tilde R_2$ is comparatively much less. In contrast, the contributions of $\bar S_1$, which couples to right-handed up-type quarks and the sterile neutrinos, could lead to branching fractions as high as order $10^{-3}$. This suggests that these charmed hadron decays might be within reach of the Belle II and BESIII experiments or future super charm-tau factories and could serve as potentially promising probes of leptoquark interactions with sterile neutrinos.


I. INTRODUCTION
The flavor-changing neutral current (FCNC) decays of charmed hadrons into final states containing missing energy ( / E) have long been anticipated in the literature [1][2][3][4][5][6][7][8][9][10][11][12] to be among likely environments in which to discover hints of new physics (NP) beyond the standard model (SM). In the SM these processes arise at short distance from the quark transition c → uνν, which emits unobserved neutrinos (νν) and is very suppressed because it proceeds from loop diagrams and is subject to highly efficient Glashow-Iliopoulos-Maiani cancellation [1]. The effects of long-distance physics on these decays have also been estimated to be tiny [1]. Beyond the SM there could be extra ingredients causing modifications to the SM component and/or yield additional channels with one or more invisible nonstandard particles, which might translate into substantially amplified rates detectable by upcoming quests.
Experimentally, there still has not been a lot of activity to look for charmed hadron decays of this kind [13]. At the moment the sole result available is a limit on the branching fraction of charmed meson decay D 0 → / E, which has been set by the Belle Collaboration [14]. Due to the importance of these processes as valuable tools in the quest for NP, it is hoped that dedicated efforts will be increasingly made to pursue them. Since a clean environment and sizable luminosity are crucial for such endeavors, it is then timely that there are now heavy-flavor factories which are running and expectedly well-suited for them, namely Belle II [15] and BESIII [16]. In the future, further measurements with improved sensitivity reach would be feasible at the proposed super charm-tau factories [17,18] and Circular Electron Positron Collider (CEPC) operated as a Z-boson factory [19].
The foregoing motivates us in this paper to study these FCNC charm transitions with missing energy in the contexts of relatively simple NP scenarios. We entertain specifically the possibility that heavy leptoquarks (LQs) with spin 0 mediate the NP contributions to the FCNCs. Over the last several years LQs have attracted a great deal of attention because proposed models containing them are among those that could offer the preferred explanations for the so-called B-physics anomalies [7,20]. While more data are awaited in order to clarify whether or not these anomalies are attributable to NP, it is therefore of interest to examine if LQs can give rise to appreciable manifestations in the charm sector as well. There have been various analyses in the past, such as Refs. [6][7][8][21][22][23][24][25][26][27][28][29][30][31], addressing the effects of LQs on FCNC charm processes, but c → u / E was covered by only a few [6][7][8]. The outcomes of our work would be complementary to those of the latter.
Besides the LQs, we will incorporate light right-handed sterile neutrinos into the theory. They are singlets under the SM gauge groups and in the presence of the LQs can have renormalizable links to SM quarks. Our inclusion of the right-handed neutrinos is well motivated for a couple of reasons. First, their existence will be necessary if experiments in the future establish that neutrinos are Dirac in nature. Second, as will be demonstrated later on, compared to those with SM fermions alone, the LQ-mediated FCNCs that involve the sterile neutrinos and SM quarks might exert considerably enhanced influence on the charm decays, especially if the quarks are also right-handed. This latter LQ feature has not been discussed previously in the c → u / E context [6][7][8].
The organization of the remainder of the paper is as follows. In Sec. II we describe the interactions of the new particles, namely the scalar LQs and the light sterile neutrinos, with the SM fermions. Concentrating on three different scalar LQs, in Secs. III, IV, and V we explore three distinct scenarios, in each of which merely one of the LQs is responsible for c → u / E. In these sections, before presenting our numerical results, we first deal with the pertinent limitations on the Yukawa couplings of the LQs. We show particularly that meson-mixing constraints in the charm and down-type quark sectors are potentially relevant, but may be avoided in certain instances. We also take into account restrictions inferred from collider searches. Subsequently, we evaluate various FCNC decays of the lightest charmed pseudoscalar-mesons and singly charmed baryons manifesting the LQ effects on c → u / E. In Sec. VI we draw our conclusions. In the Appendix, we provide general expressions for the branching fractions of the hadron decay modes under consideration. For the meson channels we show that, assuming the invisible particles have negligible masses, the formulas for the LQ contributions can be determined with the aid of data on the corresponding semileptonic modes and without relying on the form factors in the mesonic matrix elements.

II. LEPTOQUARK COUPLINGS
Among LQs that can have renormalizable interactions with SM fermions without violating the conservations of baryon and lepton numbers and the SM gauge symmetries, there are four which possess spin 0 and can at tree level bring about the quark transition c → u / E where the missing energy is carried away by either SM or sterile neutrinos [7]. 1 In the nomenclature of Ref. [7], these scalar LQs, with their assignments under the SM gauge groups SU(3) color × SU(2) L × U(1) Y , are denoted by R 2 (3, 2, 7/6),R 2 (3, 2, 1/6),S 1 3 , 1, −2/3 , and S 3 3 , 3, 1/3 . Here we pay attention to the first three because we have found that the couplings of S 3 are comparatively more restricted than those of the other three LQs. In terms of their components, where the superscripts refer to their electric charges.
As for the right-handed neutrinos, we assume that there are three of them (N 1 , N 2 , and N 3 ) and that they are of Dirac nature. Moreover, we suppose that they have masses sufficiently small to be neglected in the charmed-hadron processes of concern and are long-lived enough that they do not decay inside detectors.
We express the Lagrangian for the renormalizable interactions of R 2 ,R 2 , andS 1 with SM fermions plus N 1,2,3 as L lq = Y rl 2,jy u j R t 2 iτ 2 P L l y +ỹ lr 2,jy q jR2 P R N y +ȳ rr 1,jy u c j P R N yS1 + H.c. , where Y rl 2,jy ,ỹ lr 2,jy , andȳ rr 1,jy are generally complex elements of the LQ Yukawa matrices, summation over family indices j, y = 1, 2, 3 is implicit, q j (l y ) and u j symbolize a left-handed quark (lepton) doublet and right-handed up-type quark singlet, τ 2 is the second Pauli matrix, P L,R = (1 ∓ γ 5 )/2, and the superscript c indicates charge conjugation. In Eq. (2), we introduce only the minimal ingredients which serve our purposes pertaining to the c → u / E transitions to be studied. Now we investigate three different possibilities each involving one of the LQs, taken to be heavy, with the couplings specified above.

III. R 2
Expanding the R 2 portion of Eq. (2), we have where U 1,2,3 = u, c, t and ℓ 1,2,3 = e, µ, τ represent mass eigenstates. Given that the ordinary neutrinos in the decays of interest have vanishing masses and are not detected experimentally, we can work with the states ν ℓy associated with ℓ y .
From L R 2 , one can derive effective |∆C| = 1 quark-lepton operators which at low energies are expressible as where G F is the Fermi constant, summation over x, y = 1, 2, 3 is implicit, and GeV and m R 2 being the mass of R 2 . They induce FCNC charmed-hadron decays with missing energy as well as those with charged leptons in the final state. Before treating the former processes, we look at some potentially important constraints on the LQ parameters in Eq. (5).
It is long known that scalar-LQ interactions could influence the mixing of charmed mesons D 0 andD 0 via |∆C| = 2 four-quark operators arising from box diagrams [21][22][23][24][25][26]. In the presence of L R 2 in Eq. (3), the loops contain the SM charged and neutral leptons, besides R 2 . This results in the effective Hamiltonian [7] It affects the mixing observable ∆m D = | D 0 |H |∆C|=2 |D 0 |r/m D 0 , wherer = 0.74 accounts for the renormalization-group running of the coefficient in Eq. (6) from the scale m R 2 = 2 TeV down to 3 GeV [24]. Our choice for m R 2 is consistent with the negative outcome of a recent direct search at the LHC for scalar LQs decaying fully into a quark and an electron (muon), which has excluded the mass region below 1.8 (1.7) TeV [32].
Accordingly, we can suppose that the only nonvanishing couplings are k eµ,τ µ and demand that they satisfy |k eµ | < 2.0 × 10 −3 and |k τ µ | < 6.4 × 10 −3 . This can be realized with a Yukawa matrix having the texture which can satisfy the limitations from other LFV searches [36] and escapes the D-mixing restraint.
To see the effects of R 2 on c → u / E explicitly, we incorporate the above |k eµ,τ µ | into the expressions displayed in Eqs. (A11) and (A16) in the Appendix for the branching fractions of various FCNC decays of charmed mesons and baryons with missing energy. Thus, we obtain where each entry is a sum of branching fractions of the modes with ν eνµ and ν τνµ in the final states, the former making up merely about 10% of the total.
We find that including k eτ,τ e would barely increase the results above because the associated Y rl 2 elements would have to fulfill other significant requirements. The aforesaid D-mixing requisite |k ℓxℓx | 2.4 × 10 −4 in the lepton-flavor conserving case would translate into numbers that are roughly three orders of magnitude lower.

IV.R 2
From theR 2 part of Eq. (2), in the mass basis of the down-type quarks we have where V ≡ V ckm designates the Cabibbo-Kobayashi-Maskawa mixing matrix and D 1,2,3 = d, s, b refer to the mass eigenstates. At low energies, from LR 2 proceed the |∆C| = 1 effective fourfermion interactions specified by where j, k, x, y = 1, 2, 3 are implicitly summed over, As in the last section, this brings about the FCNC decays of charmed hadrons with missing energy, but now it is the right-handed neutrinos that act as the invisibles. Moreover, L qq ′ NN ′ can induce analogous transitions in the down-type quark sector.
From Eq. (11), one can calculate box diagrams, withR 2 and N y running around the loops, which affects D 0 -D 0 mixing, similarly to the R 2 scenario, via the effective Hamiltonian However, unlike before, there are additionally contributions to its kaon and b-meson (B d and B s ) counterparts, described by and analogous formulas for the B d,s -mixing cases. Since it is unlikely to avoid all the mixing constraints at the same time, we can opt instead to do so in the down-type sector alone. It is evident from Eq. (15) that this as attainable if the nonzero elements of the first and second rows ofỹ lr 2 lie in separate columns. Accordingly, for simplicity we can pick and so we have up to O(λ 2 ), where λ and A are Wolfenstein parameters. To this order in λ, the nonvanishing coefficients in Eq. (13) are Using D 0 |uγ κ P L cuγ κ P L c|D 0 = 0.0805(57) GeV 4 from lattice QCD work [33] and demanding again that theR 2 contribution saturate ∆m exp D , we get x Vỹ lr 2 1x Vỹ lr 2 * 2x at the 2σ level. For mR 2 = 2 TeV, this translates into With this coupling value, the charmed-hadron decay channels with missing energy listed in Eqs. (9) and (10) would have branching fractions about two orders of magnitude smaller than the corresponding numbers displayed therein. We remark that having more nonzero elements inỹ lr 2 would produce little change because they would be subject mainly to the meson-mixing requisites and/or stringent K → π / E limitations.

V.S 1
From theS 1 portion of Eq. (2), we derive where This again gives rise to c → uN xNy , with N xNy emitted invisibly, and affects ∆m D , the latter via Hence the mixing constraint is escapable if the contributing elements of the first and second rows ofȳ rr 1 belong to different columns, as in this simple example: With x = y in kS 1 NxNy , the remaining consequential restriction on theȳ rr 1 elements is from the perturbativity requirement: ȳ rr 1,ix < √ 4π. As for the allowed range of theS 1 mass, the latest quest by the CMS Collaboration [37] for scalar LQs decaying fully into a quark and neutrino has ruled out masses up to 1.1 TeV at 95% CL. Since this is applicable to right-handed neutrinos, we can set mS 1 > 1.2 TeV. These parameters also enter loop diagrams involving the LQ and quarks and modifying the invisible partial width of the Z boson, but we have checked that their impact is negligible. Incorporating these numbers into Eq. (23) yields, for x = y, To illustrate the implications for the aforementioned charmed-hadron decays, we adopt the Yukawa matrix in Eq. (25), in which case the nonzero couplings are kS 1 N 2 N 1 and kS 1 N 2 N 3 . Putting these together with Eqs. (A11) and (A16), both of the parameters having the maximal value in Eq. (26), then leads to where each entry is a combination of branching fractions of the modes with N 2N1 and N 2N3 in the final states. These numbers are considerably higher than their counterparts in the models containing R 2 andR 2 . This is attributable to the fact thatS 1 does not have any direct couplings to the SM lepton and quark doublets.

VI. CONCLUSIONS
We have explored the FCNC decays of charmed hadrons into a lighter hadron plus missing energy, / E, carried away by a pair of either SM or right-handed sterile neutrinos in LQ scenarios, concentrating on the influence of the R 2 ,R 2 , andS 1 scalar LQs. We take into account various relevant constraints and learn that the meson-mixing ones and those inferred from LHC searches are especially important. Nevertheless, we point out that the meson-mixing restrictions may be evaded in certain situations. Additionally, we demonstrate that the contributions of these LQs to the branching fractions of D → M / E, M = π 0 , η (′) , ρ 0 , ω, and D + s → K ( * )+ / E can be evaluated without knowing the details of the mesonic form factors associated with the quark currents if the invisibles have vanishing masses, by making use of the data on the corresponding semileptonic modes and assuming isospin symmetry. As a consequence, the calculated D (s) rates are free from the uncertainties attendant in form-factor estimation.
Our numerical work indicates that the branching fractions of these charmed-meson decays and of their baryon counterparts Λ + c → p / E and Ξ +,0 c → Σ +,0 / E are currently permitted to reach the 10 −7 -10 −6 levels if they are induced by R 2 , whereas the effects ofR 2 are comparatively much smaller. On the other hand, the contributions ofS 1 , which has fermionic interactions exclusively with the SM right-handed up-type quarks and the sterile neutrinos, could yield branching fractions reaching a few times 10 −3 . These substantially greater results than those in the R 2 andR 2 models are understandable because these two LQs are coupled to the SM left-handed lepton and quark doublets, respectively, implying relatively stronger restraints on the LQ Yukawa parameters. Based on the outcomes of this study, we conclude that these charmed hadron decays are potentially promising as probes of LQ interactions involving sterile neutrinos. Lastly, some of our predictions, notably in theS 1 scenario, may be large enough to be testable in the ongoing Belle II and BESIII experiments.
where f and f ′ are either SM leptons or SM-gauge-singlet fermions. This gives rise to D → Pff ′ and D → Vff ′ , where D stands for a charmed pseudoscalar meson and P and V designate charmless pseudoscalar and vector mesons, respectively. The amplitudes M D→Pff ′ and M D→Vff ′ for these decays depend on the mesonic matrix elements [7,38] P|uγ where m X and p X are the mass and momentum of X, respectively, ε denotes the polarization vector of V, and F ± , V , and A 0,1,2 symbolize form factors which are functions ofq 2 . In this paper, we focus on the possibility that the f and f ′ masses, m f and m f ′ , are sufficiently small to be negligible. 2 It follows that where u f and vf′ represent the fermions' Dirac spinors and the contributions of the terms with q α = p α f +p α f ′ in Eq. (A2) have dropped out upon contraction with the ff ′ current due to m f,f ′ ≃ 0. For f = f ′ , these amplitudes translate into the differential rates For the LQ-mediated operators in Eqs. (4), (12), and (22), Now, the semileptonic transitions D 0 → M − νe + with M = π, ρ receive SM contributions described by L sm dcνe = − √ 8 G F V * cd dγ α P L c ν e γ α P L e + H.c. Comparing this to L ucff ′ in Eq. (A1) and neglecting the lepton masses, we can see that the expressions for the differential rates of D 0 → M − νe + with M = π, ρ in the SM are equal to those in Eqs. (A6) and (A7), respectively, but with coefficients given by |C v,a νe | = |c v,a νe | = G F |V cd |/ √ 2. For the rate of LQ-induced D + → M + ff ′ , ignoring small isospin-breaking effects we can then arrive at 4Γ lq D + →M + ff ′ |V cd | 2 = Γ sm D 0 →M − νe + |k ff ′ | 2 without having to know how F + , V , and A 1,2 depend onŝ. As the LQ interactions in Eq. (2) do not directly affect D 0 → M − νe + , we can replace Γ sm D 0 →M − νe + with their experimental values. This implies the branching-fraction relation where τ D +(0) is the measured D +(0) lifetime. It is straightforward to write down analogous formulas for other modes, particularly D 0 → Mff ′ with M = π 0 , η (′) , ρ 0 , ω and D + s → K ( * )+ ff ′ . Clearly, the outcomes of this procedure do not suffer from the uncertainties inherent in the estimation of hadronic matrix elements.
To proceed, we need the empirical information on the relevant semileptonic modes [13]: The numbers in Eq. (A11) have relative errors approximately equal to those of the corresponding data in Eq. (A10).
It is worth mentioning that when using Eq. (A11) for the models in Secs. III-V, where f and f ′ are invisible, we ignore the SM contributions, which are highly suppressed [24]. We also note that D + (s) → M + (s) ff ′ with M (s) = π, ρ (K, K * ) and invisible f and f ′ have SM backgrounds from the sequential decays D + (s) → τ + ν and τ + → M + (s) ν [1]. The latter can be removed by implementing kinematical cuts such asŝ min = m 2 D − m 2 τ m 2 τ − m 2 M 2 /m 2 τ [39]. Our D + (s) numbers in Eq. (A11) do not yet incorporate them, and we suppose that they will be taken into account in the experimental searches.
The LQ-induced operators in Eqs. (4), (12), and (22) bring about analogous transitions in the charmed-baryon sector. Here we look at those of the singly charmed baryons Λ + c , Ξ + c , and Ξ 0 c , which have spin parity J P = 1/2 + , make up a flavor SU(3) antitriplet, and decay weakly [13]. Specifically, we examine the modes Λ + c → p / E and Ξ +,0 the meson modes above and must instead rely on theoretical estimates for the baryonic matrix elements. Thus, numerically, for the Λ + c → p form factors we adopt the results of the lattice QCD calculation in Ref. [40], while for Ξ +,0 c → Σ +,0 we employ those computed with light-cone QCD sum rules in Ref. [41] and assume isospin symmetry. Putting things together, we then obtain where the Λ + c and Ξ +,0 c results have uncertainties of order 10% and 30%, respectively [40,41], and the difference between the Ξ +,0 c numbers is ascribable mainly to τ Ξ + c = 3.9 τ Ξ 0 c .