Probing for new physics with rare charm baryon (Λc, Ξc, Ωc) decays

We analyze rare charm baryon decays within the standard model and beyond. We identify all null test observables in unpolarized Λc→ pℓ+ℓ−, ℓ = e, μ decays, and study the new physics sensitivities. We find that the longitudinal dilepton polarization fraction FL is sensitive to electromagnetic dipole couplings C7′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {C}_7^{\left(\prime \right)} $$\end{document}, and to the right-handed 4-fermion vector coupling C9′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {C}_9^{\left(\prime \right)} $$\end{document}. The forward-backward asymmetry, AFB, due to the GIM-suppression a standard model null test already, probes the left-handed axialvector 4-fermion coupling C10; its CP–asymmetry, AFBCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {A}_{\mathrm{FB}}^{\mathrm{CP}} $$\end{document} probes CP-violating phases in C10. Physics beyond the standard model can induce branching ratios of dineutrino modes Λc→ pνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\nu} $$\end{document} up to a few times 10−5, and one order of magnitude smaller if lepton flavor universality is assumed, while standard model rates are negligible. Charged lepton flavor violation allows for striking signals into e±μ∓ final states, up to 10−6 branching ratios model-independently, and up to order 10−8 in leptoquark models. Related three-body baryon decays Ξc→ Σℓℓ, Ξc→ Λℓℓ and Ωc→ Ξℓℓ offer similar opportunities to test the standard model with |∆c| = |∆u| = 1 transitions.


JHEP09(2021)208
2 Λ c → p + − theory We give general expressions for exclusive rare charm baryon decays in the SM and beyond. In section 2.1 we briefly review the perturbative SM contributions and introduce the weak Hamiltonian at the charm mass scale. Effects from intermediate resonances are discussed in section 2.2. We comment on charmed baryon decays other than Λ c → p + − in section 2.3. The fully differential distribution for unpolarized Λ c → p + − decays is given in section 2.4.

An effective field theory approach to charm physics
For the phenomenological analysis of semileptonic c → u + − and c → uν i ν j processes we use the following effective Hamiltonian, 9,10 where the dimension six operators O k for dilepton and Q ij L/R for dineutrino modes are given as with the electromagnetic field strength tensor F µν and σ µν = i 2 [γ µ , γ ν ]. In the dineutrino operators flavor indices ij are made explicit, while we omit them for brevity in the charged dilepton operators O ( ) 9,10 unless necessary, that is, in the discussion of experimental constraints, lepton universality violation and lepton flavor violation. The primed operators O k are obtained from the O k by interchanging left-handed (L) and right-handed (R) chiral quark fields, L ↔ R. SM contributions to the effective coefficients of O 7 and O 9 are generated by the charged-current four-quark operators at the W mass scale and induced by renormalization group running and mixing, as well as the two-step matching procedure, first at the W mass scale and then at the b mass scale [5,18,19]. Details are provided in appendix A. The SM contributions, including two-loop virtual corrections [18], depend on the dilepton invariant mass squared, q 2 . As shown in figure 9 in appendix A the non-vanishing SM contributions relevant to c → u + − are of the order of few permille for |C eff 7 | and few percent for |C eff 9 | above q 2 > 0.1 GeV 2 . Notably, the GIM-mechanism dictates C SM 10 = 0. Contributions of primed operators are suppressed, and we neglect corresponding SM contributions.

Resonance contributions
The Λ c → p + − decay rate is dominated by intermediateqq-resonances M via Λ c → p M and M → + − . These contributions can be parameterized phenomenologically by Breit-Wigner distributions. For the spin-1 resonances M = ρ(770), ω(782), φ(1020) this corresponds to an effective coefficient C R 9 (q 2 ) of the operator O 9 as Here, an isospin relation is employed to relate the ρ and ω contributions to reduce the theoretical uncertainties. This approach is also necessary due to the lack of experimental information on B(Λ c → pρ), and supported from charmed meson data [8]. The remaining two moduli of the resonance parameters a ω and a φ are fixed by experiment using Taking the respective branching fractions, masses and decay widths from the PDG [20] we obtain a ω = 0.065 ± 0.016 , a φ = 0.110 ± 0.008 , (2.6) in excellent agreement with ref. [9]. As in semileptonic rare b-decays, a dispersion relation plus e + e − → hadrons-data [21] can provide an alternative resonance parameterization to (2.4). It also invokes factorization and fudge factors a M , however, avoids double-counting by extracting the sum of long-and short-distance contributions experimentally, see also [22] and [23] for application to D → ρ and D → π decays, respectively. Since in charm, unlike in beauty, C R 9 dominates over the SM short-distance contribution, see figure 9, and the parameterizations yield similar distributions in D-decays within their sizable, notably, phase-induced uncertainties [8,23], we employ the significantly simpler ansatz (2.5) for null test analyses in charmed baryons.
We also consider the contribution of pseudoscalars η, η to the Λ c → pµ + µ − branching ratio. While leptonically decaying η, η contribute to c → u + − transitions, they are strongly localized [8], and outside of target q 2 -regions (2.14). Neglecting lepton masses we find, in agreement with ref. [24], no interference terms between (axial-)vector contributions and those of the additional pseudoscalar operator O P = (u L c R )( γ 5 ) in (2.1). We model the effects from the η, η -resonances via an effective coefficient assuming the resonance parameter of the η and the η of similar size, a η ∼ a η . Using masses, widths and branching fractions from the PDG [20] and we obtain a η ∼ 5.8 · 10 −4 , much smaller than the corresponding terms for vector mesons (2.6). The impact of η, η on the differential and integrated Λ c → pµ + µ − branching ratio is discussed in section 3.1. Table 1. a ω , a φ parameters for various rare charm baryon modes, see text for details.

Charmed baryons other than Λ c
Three-body spin 1/2 → 1/2 + , rare charm baryon decays other than Λ c → p can be studied in a similar manner, specifically Ξ 0 and Ω 0 c → Ξ 0 decays. We use flavor symmetries and employ the form factors from Λ c → p, with obvious modifications in kinematic factors in (B.1)-(B.4). Note that theūc-type currents conserve U -spin and violate isospin by |∆I| = 1/2. 1 Specifically, Ξ + c → Σ + is related by U-spin to Λ c → p, that is, the complete exchange of a d-quark with an s-quark. On the other hand, the Ξ 0 c → Σ 0 form factors receive a factor 1/ √ 2 relative to the Ξ + c → Σ + ones due to isospin [25]. Then, U-spin relates Ξ 0 c → Σ 0 form factors to Ξ 0 c → Λ 0 ones [26,27] with a relative factor of √ 3.
For Ω 0 c decays no relation with the Λ c exists since the Ω 0 c sits in another SU(3) F -multiplet [20]. Due to the lack of other input we use the same form factors as for Λ c → p. To summarize, for any of the ten form factors commonly denoted here as [25], and use form factors from [9]. Predictions for f Ξ 0 c →Σ 0 from table 9 of [11] are consistent within 30% flavor breaking.

Fully differential distribution for m = 0
For the calculation of SM and BSM contributions to the two-fold differential decay rate of Λ c → p + − we utilize the helicity formalism [28,29]. Details on the calculation of the relevant helicity amplitudes are collected in appendix C. The fully differential angular distribution for unpolarized Λ c can be written as

JHEP09(2021)208
Here, θ is the angle of the + with respect to the negative direction of flight of the Λ c in the dilepton rest frame. The q 2 -dependent coefficents are given as [30] K 1ss = q 2 v 2 1 2 in agreement with our own computation and [31].
The q 2 -dependent terms U, L, S, P denote quadratic expressions of helicity amplitudes and correspond to unpolarized transverse, longitudinal, scalar and parity violating contributions, respectively. For further details we refer to appendix C. Here we give the respective expressions in terms of Wilson coefficients and hadronic form factors, f i (q 2 ), g i (q 2 ), i = +, ⊥, 0 and h j (q 2 ),h j (q 2 ), j = +, ⊥, defined in appendix B. For brevity we omit their q 2 -dependence in the following: ·s + + C 10 −C 10 (m Λc +m p ) g 0 2 ·s − , (2.11) The factor N 2 is a global normalization and is given by Λc with the Källén function λ(a, b, c) = a 2 + b 2 + c 2 − 2 (ab + ac + bc) and s ± = (m Λc ± m p ) 2 − q 2 .

JHEP09(2021)208
Note, in the SM all terms vanish except for U 11 and L 11 . Therefore, K SM 1c = 0. At the kinematic endpoint of zero hadronic recoil, where q 2 is maximal, q 2 = (m Λc − m p ) 2 , vanishes s − and P 12 /N 2 = 0 also in the presence of NP. At both endpoints, the zero recoil one and the maximal recoil one, where the dilepton invariant mass is minimal, q 2 = 4m 2 , further holds K 1ss = K 1cc model-independently.
The observables discussed in section 4 can be expressed in terms of K 1ss , K 1cc , K 1c and their CP-conjugated counterpartsK 1ss ,K 1cc ,K 1c . For instance, the fractions of longitudinal and transverse polarization of the dilepton system are, respectively, given by The q 2 -differential decay rate is obtained by integrating (2.9) over the angle θ within The Λ c → p + − decay rate follows as Γ = where the phase space is limited by q 2 min = 4m 2 and q 2 max = (m Λc − m p ) 2 , but we will also employ other q 2 -cuts to enhance the NP sensitivity. Specifically, we introduce, targeting muons, (2.14) Here, the low q 2 region corresponds to the region below the η, whereas the high q 2 region is above the φ. We discuss F L , dB/dq 2 = τ dΓ/dq 2 , with Λ c -lifetime τ , and the resulting branching ratios in the next section 3.

SM phenomenology
We discuss the SM prediction for the q 2 -differential and integrated branching ratios in section 3.1, and F L in section 3.2.

Branching ratio
The differential SM branching ratio of Λ c → pµ + µ − decays is shown in figure 1 based on the non-resonant (blue) and resonant (orange) contributions. Uncertainties arise from the form factors, taken from [9], and for the non-resonant contribution mainly due to the scale uncertainty of the Wilson coefficients. The main source of uncertainty for the resonant contributions are the strong phases δ ω and δ φ in eq. (2.4). The central value curve of the resonance band is obtained by averaging after varying both phases from −π to π. Clearly, SM contributions from the effective coefficients C eff 7, 9 , see figure 9, are negligible in the branching ratio in the full q 2 range. Therefore, we only consider resonant contributions for the SM contributions in the subsequent analysis, unless otherwise stated. In figure 1 additional contributions of η, η -resonances decaying leptonically to the differential branching fraction are displayed using (2.7) and [24] dB/dq 2 | η, η = 6τ q 2 N 2 g 2 Although the η-peak reaches values as large as O(10 −3 ), its contribution to B(Λ c → pµ + µ − ) is very small, O(10 −9 ), due to its narrow width, and negligible in the branching ratio. Similar results are obtained for the η . In the following we therefore do not consider the η, η -contributions any further, as they are quite located and outside the primary q 2 -bins of interest (2.14).
The LHCb collaboration has obtained the 90 % CL upper limit [13] Bounds on the electron modes are two orders of magnitude weaker, B(Λ c → pe + e − ) < 5.5 × 10 −6 at 90 % CL (BaBar) [12]. The upper limit in eq. (3.2) is based on a search for events outside the ω and φ windows -±40 MeV around the resonance masses -and subsequent use of a phase space model to "fill the gaps". As neither the latter nor the corresponding correction factor to the event rate is publicly available a straightforward comparison with theory predictions is not possible. We therefore suggest to avoid extrapolations in future experimental analyses and instead quote plain rates with the cuts they are based on.

JHEP09(2021)208
Using eqs. (2.4) and (2.6) we compute the Λ c → pµ + µ − branching ratio in the full q 2 -region minus the ±40 MeV windows around the ω and φ resonances and obtain For the decays other than the Λ c we employ the resonance parameters in table 1. The branching ratios for , which is mainly driven by the different lifetimes. For the same reason and the suppression factors in the form factors the branching ratios of In the remainder of this work we analyse null tests of the SM and present our results in terms of contributions to the decay Λ c → p . However, all observables can also be studied for the other modes. We recall that dominant decay modes are

F L
The fraction of longitudinally polarized dileptons, F L , defined in (2.12), is mostly dictated by helicity. As is apparent from the expressions given in section 2.4, at the zero recoil endpoint, holds model-independently F L (s − = 0) = 1/3, as in B → K * + − , as this kinematic point has higher symmetry [32], also observed for baryons [33], also note [34]. In particular, the four-momenta of the decaying hadron, the final hadron, and the dilepton system, q µ , are proportional to each other at zero recoil, allowing for fewer Lorentz structures, hence relations between helicity amplitudes. At the maximum hadronic recoil endpoint, we also obtain F L (4m 2 ) = 1/3, in agreement with [10]. 2 This is no coincidence, as at maximum recoil the four-momenta of the leptons are the same, causing also here the angular distribution (2.9) to become flat, dΓ/d cos θ = const. 3 In between these kinematic endpoint constraints F L rises due to the dominant longitudinal contributions with 1/q 2enhancement. This can be seen in figure 2, where F L is shown in the SM. The structures around q 2 ∼ 0.8 GeV 2 are caused by interference of C R 9 , which has significant uncertainties in the region between the ω and the φ, see figure 1, with the perturbative effective dipole coefficient C eff 7 . Due to the dominance of C R 9 , numerically the impact of C eff 9 is negligible. The shape of F L can change dramatically with BSM physics, however, its behavior can still be understood by helicity arguments, see section 5.2 for details.
As we defined F L with normalization to the differential decay rate (2.12), rather than the integrated rate, the cancellation of resonance effects between numerator and denominator is highly efficient. Note, this cancellation hinges on the fudge factors a M being 2 This is in conflict with figure 7 of [9]. We thank Stefan Meinel for confirming an error in his plot. 3 The q 2 → q 2 min = 4m 2 limit depends crucially on the treatment of lepton masses and corresponding degrees of freedom: for finite m the maximum recoil endpoint is dominated by the longitudinal polarization, whereas for m = 0 the photon poles ∝ C ( ) 7 /q 2 in the transverse amplitudes dominate. This also happens in B → K * + − decays, where in order to use (2.12), we identify K1ss = J1s + J1c/2 − (J2s + J2c/2), K1cc = J1s + J1c/2 + J2s + J2c/2 from distributions in [35]. For m = 0 one recovers, using appendix A and B of [35], the well-known result that FL equals ( , which vanishes towards q 2 → 0, while for m = 0 we verify the symmetry limit FL(4m 2 ) = 1/3. R only. The orange band includes in addition C eff 7,9 (q 2 ). The pale bands illustrate a 10% splitting between the longitudinal and the transverse fudge factors, which is forbidden at zero recoil, see text. At both zero and maximum recoil holds F L = 1/3. polarization-independent. In principle, polarization-specific fudge factors could be taken into account if F L is measured in Λ c → pM decays. We simulate effects of the order 10% (pale bands). We stress, however, that endpoint constraints prohibit polarizationdependent fudge factors at zero recoil. We therefore expect the SM prediction from the phenomenological parameterization to be most robust towards maximum q 2 .
We provide an integrated polarization fraction in the SM, integrating full and high q 2 regions, as in (2.14), respectively, Normalizing to the integrated decay rate, with cuts corresponding to (3.3), cutting ±40 MeV around the ω, φ masses, we obtain instead 0 ≤F L ≤ 1. As expected, here uncertainties do not cancel and remain sizable, and cover the full physical range.

Null test observables
We give the complete set of observables in three-body rare charm baryon decays and identify null tests of the SM. All observables are based on the angular coefficients K 1ss , K 1cc and K 1c (and their CP-conjugates), which are given in terms of Wilson coefficients and form factors in eq. (2.10) and eq. (2.11) and further details in appendix C. As already discussed in section 3 and illustrated in figure 1 the q 2 -differential decay rate (2.13) has little sensitivity to BSM physics, due to resonance contributions (orange band), however, it can be used to form null tests, such as ratios of dimuon to dielectron rates, or CP-asymmetries, discussed below. An observable which is also not a null test -9 -JHEP09(2021)208 yet sensitive to NP is the fraction of longitudinal polarization of the dilepton system, F L , defined in (2.12).
In this work we consider in addition the following null test observables: 1. The forward-backward asymmetry of the leptonic scattering angle, is defined as The angular observable K 1c is proportional to linear combinations of C 10 and C 10 and thus A FB constitutes a clean, charm-specific null test of the SM. Note that at higher order in the fine structure constant loop-induced contribution to the matrix element of O 10 arise, at the level of 10 −4 [7,19], that are entirely negligible for the present and foreseeable uncertainties.
2. Lepton universality can be probed analogously to the ratios of charmed meson decays [4,7,8] with the ratio where q 2 min and q 2 max can be chosen in bins or the full kinematically allowed region, but necessarily need to be equal for the muon and the electron mode. Due to the universality of the SM interactions R Λc p is expected to be close to 1 in the SM, such that R Λc p − 1 is an excellent null test.
3. The CP-asymmetry in the decay rate is defined as where the bar indicates CP-conjugation. Since CP-violating effects in charm are strongly CKM-suppressed in the SM, parametrically by V * cb V ub /(V * cs V us ), A CP constitutes a null test of the SM which is sensitive to NP with a CP-violating phase. 4. It is also possible to study the CP-asymmetry of an angular observable. Here, we study the CP-asymmetry of the forward-backward asymmetry, defined as studied previously in other decay modes [3,36]. In the SM it is protected by GIM and small CP-phases. In the CP-conserving limit holds A FB = −Ā FB , that is, A FB is a CP-odd observable. The reason for this is that we keep for both Λ + c and Λ − c decays the lepton angle θ defined with respect to the same, positively charged lepton, while CP-conjugation would exchange + ↔ − . The CP-conjugated angular distribution for Λ − c is thus obtained by exchanging θ ↔ π−θ , which, using cos (π − θ ) = − cos θ results in an extra minus sign in (2.9) in front of K 1c .
. Due to the negligibe SM contribution, suppressed by the GIM mechanism, any signal is NP. In addition, branching ratios can reach upper limits depending on charged lepton flavor. This allows to test lepton universality and probe for cLFV [14,37].
We comment on different conventions to normalize A FB and A CP used in the literature. While in eqs. (4.1) and (4.3), and also eq. (2.12), the normalization is to the q 2differential decay rate, one may consider normalizing to the rate Γ integrated over different bins [q 2 min , q 2 max ] as in refs. [5,7,8] (4.5) Normalizing to the rate (4.5) is advantageous as it avoids artificially large asymmetries from a suppressed denominator, such as dΓ/dq 2 , see figure 1, corresponding to low event rates. It also allows to probe the q 2 -shape of the numerator. On the other hand, cancellations are more efficient when normalizing to dΓ/dq 2 . Due to the different pros and cons we present results for the CP-asymmetry and the forward-backward asymmetry in both normalizations. Note, for sufficiently small bins, or sufficiently flat distributions, the integrals Ã CP dq 2 and A CP dq 2 , and similarly for the forward-backward asymmetry, of course, coincide.
In the case of the CP-violating forward-backward asymmetry in eq. (4.4), we stress that in experiment it might be convenient to measure the sum and the difference of an angular observable for the decay of particle and antiparticle, and then form the ratio, eq. (4.4).
In the following section 5 we investigate in detail the NP sensitivity of the aforementioned observables.

Sensitivity to physics beyond the standard model
We analyze the NP sensitivity of c → u + − transitions in Λ c → p + − decays. In section 5.1 we briefly summarize existing constraints on BSM Wilson coefficients. Null tests from the angular distribution are discussed in section 5.2. We consider lepton universality tests in section 5.3. CP-asymmetries and lepton flavor violating decays are discussed in section 5.4 and 5.5, respectively. Section 5.6 provides information on baryon modes with dineutrinos in the final state.

Constraints on Wilson coefficients
Most recent results for lepton flavor conserving and lepton flavor violating Wilson coefficients are presented in [6] and based on latest results from the LHCb experiment searching  for rare purely leptonic and semileptonic charm meson decays [38]. Here we use, barring cancellations, Bounds from high-p T searches in dilepton tails [39,40] are of the same order of magnitude; they are also able to constrain couplings involving τ 's which cannot be kinematically accessed in semileptonic rare charm decays.

A FB andÃ FB
The forward-backward asymmetry in Λ c → p + − decays can exhibit sizable NP effects, see row) and with normalization to the decay rate,Ã FB (4.5) (bottom row) in the full q 2 region (plots to the left). A FB is markedly suppressed around the resonances since here the q 2 -differential decay rate, which acts as a normalization (4.1), is peaking. Outside the resonances we find that C 10 even as small as a percent can induce values of A FB of up to O(0.1). While this is a clean signal of NP, hadronic uncertainties are sizable and challenge an extraction of the Wilson coefficients. The inflation of uncertainties is caused by large cancellations in the differential decay rate. This situation improves in the high q 2 region above the φ-peak (plots to the right), where we show A FB for C 10  In the lower plots of figure 3 we show the forward-backward asymmetry normalized to the integrated decay rate,Ã FB , defined in (4.5). Note the resonance-catalyzed, NP-induced peaks around m ρ , m ω and m φ . This structure resembles for real-valued NP coefficients the one of Re C R 9 , shown in figure 9.Ã FB has sensitivity to C 10 as low as a few percent, somewhat weaker than A FB , and in general a more modest uncertainty from varying strong phases.
Both A FB andÃ FB vanish for Re C R 9 C * 10 = 0, and are invariant under a simultaneous sign-flip of C 10 and shift in the resonance phases δ M → δ M + π. Note that C 7 (C 7 ) can only be probed together with a non-zero C 10 (C 10 ).

F L
In contrast to the forward-backward asymmetry the fraction of longitudinally polarized dimuons, F L , is non-zero in the SM. The SM prediction of F L , shown in figure 2, is not distinguishable from a NP scenario with only left-handed C 9 , C 10 contributions, or only the right-handed coupling C 10 switched on. In figure 4 we show the NP sensitivity of F L to C 9 and to dipole contributions C ( ) 7 . For the former, shown by the red band, the overall q 2 -shape of F L follows the resonant-one of the SM (orange line) modulated with resonance interference contributions and corresponding increased uncertainties from strong phases. On the other hand, significant deviations from the SM are possible for NP contributions to dipole operators C ( ) 7 = 0. Note that the huge uncertainty bands are due to limited knowledge of the strong phases of the resonant contributions, which cancel in the SM case, where predominantly resonance contributions are present, as shown in figure 2.
To understand the different behavior of NP contributions from C 9 and C ( ) 7 in F L , one can use an argument similar to appendix D in ref. [32]. Using eqs. (2.10) and (2.11) and considering terms without m 2 -suppression, which is suitable for q 2 4m 2 , F L can be written as Contributions of C 9 , 10 scale with m 2 Λc /q 2 in L 11+22 , whereas contributions in U 11+22 have an extra factor two. Thus, the leading q 2 -dependence from C 9 , 10 -induced contributions to  increasing from F L 1 towards the zero recoil endpoint where F L = 1/3. To understand the behavior for small q 2 the m 2 -terms matter which dictate F L = 1/3 at maximum recoil. Together with interference effects and the q 2 -dependent size of resonant contributions as in eq. (2.4) one obtains the shapes shown in figure 4.

Towards a global fit
Unlike in the SM strong phase uncertainties do not cancel in F L in the presence of NP, and hamper an extraction of Wilson coefficients. On the other hand, the differential branching fraction has high sensitivity to the strong phases. A global fit of future data can therefore benefit from fitting Wilson coefficients and hadronic phases together. Note that dB/dq 2 and the angular observables A FB and F L remain invariant under simultaneously flipping of all strong phases in C R 9 and the sign of a NP coefficient. In figure 5 we illustrate the sensitivities and complementarities between dB/dq 2 (plot to the left) and F L (plot to the right). Shown is the resonant SM in orange and three NP scenarios. Strong phases are fixed to δ ω = δ φ = 0 (solid curves) and, partly flipped only for clarity, δ ω = 0, δ φ = π (dashed curves). Even with this partly flipped strong phases, it can be noticed from F L that the solid (dashed) curve with C 9 = 0.5 (red) has similar shape as the dashed (solid) one for C 9 = −0.5 (dashed). We recall that F L is sensitive to right-handed currents, whereas A FB probes C 10 , and therefore both are complementary and important to pin down the type of possible NP. The benchmark C 7 = 0.3, δ ω = 0, δ φ = π (blue dashed curve) is excluded by the experimental upper limit (3.2). However, since the strong phases remain unknown, C 7 = 0.3 cannot be excluded at this point. This highlights the necessity to measure the angular observables and differential branching ratio simultaneously in the same q 2 bins, as only then simultaneous fits to Wilson coefficients and resonance parameters can be performed.

Lepton universality ratios
The observable R Λc p (4.2) probes electron-muon flavor universality. We study R Λc p in the SM and in BSM scenarios for the full, low and high q 2 regions. The main source of uncertainty in the branching fraction stems from the resonance contribution, parameterized by C R 9 (q 2 ). We stress that the latter is universal for the muon and the electron mode, and corresponding uncertainties cancel efficiently in the ratio of electron to muon branching fractions.
Predictions for R Λc p are given in table 2. We find near-universality, 1 ± O(%) for the SM, as lepton mass effects are O(m 2 /m 2 Λc )-suppressed. A study of higher order QEDcorrections is beyond the scope of this work, but may become relevant once data around this level are available. We consider NP scenarios with NP in the muons only to break lepton universality. Integrating the differential decay rates over the full q 2 region we find SM-like values, since the lepton universal resonance contributions dominate. On the other hand, deviations from the SM can be very large in the low and high q 2 regions, hence, binning is necessary. Hadronic uncertainties are more significant in the high q 2 region, -15 -
where we find values of R Λc p of up to O(100). The theoretical uncertainties of R Λc p originate from the resonance parameterization and form factors and can be reduced in the future, however, any significant deviation from 1 signals NP.

CP-asymmetries
(T-even) CP-asymmetries can be measured in rare charm decays at the resonances [41]. If there is NP with a CP-violating phase, interference with the resonance contribution leads to a non-vanishing CP-asymmetry. Due to the smallness of CKM-induced CP-violation in c → u transitions CP-asymmetries are SM null tests. We study CP-asymmetries in the decay rate, and in the forward-backward asymmetry.

A CP andÃ CP
In figure 6 we show the CP-asymmetry (4.3) for C 9 = 0.5e iπ/4 and different strong phases δ φ = 0, ± π 2 , π, while varying δ ω . A CP vanishes around the resonance peaks, such that smaller values of A CP coincide with regions with larger Λ c → p + − event rates. In figure 6 (left) the uncertainties are huge around the ρ, ω resonances, because their strong phase is varied in the plot. On the other hand, the fixed values of δ φ lead to smaller uncertainties around the φ resonance. Improving knowledge on strong phases therefore is important to reduce uncertainties. This can also be seen in the plot to the right in figure 6, which is the same as the plot to the left zoomed into the q 2 -region around the φ.
On the other hand,Ã CP with the integrated decay rate as normalization (4.5) illustrates the behavior of the resonance enhanced CP-asymmetry best [5,8,41], see figure 6 (lower plot). The q 2 -integrated denominator prohibits localized, large cancellations and therefore uncertainty bands are reduced compared to A CP . Binning is necessary as, for instance, CP-asymmetries with δ φ = ±π/2 (green and red) would vanish if integrated symmetrically around q 2 = m 2 φ .

A CP FB
The CP-asymmetry of the forward-backward asymmetry is defined in eq. (4.4). Since the forward-backward asymmetry is sensitive to C 10 , one obtains sensitivity to the imaginary part of C 10 , see [36] for related discussion in B-decays. In fact, keeping only terms with C 10 and resonance contributions C R 9 in eq. (2.10), one obtains K 1c ∝ P 12 ∝ Re(C R 9 C * 10 ).  Figure 6. The CP-asymmetry of Λ c → pµ + µ − decays for various strong phases in the C 9 = 0.5e iπ/4 scenario with normalization as in eq. (4.3) in the full q 2 range (left) and the region around the φ resonance (right). Uncertainties are due to form factors and the strong phase variation in δ ω , whereas δ φ is fixed to 0, π, ± π 2 in yellow, blue, green and red, respectively. We also displayÃ CP with alternative normalization (4.5) for q 2 ∈ (m φ − 5 Γ φ ) 2 , (m φ + 5 Γ φ ) 2 (bottom).
In addition, dropping for the moment for simplicity all resonances except the φ this implies Since the strong phase δ φ is unknown, the position of the singularity, induced by the zero in the denominator of eq. (5.6), cannot be predicted. Further singularities arise when all resonance contributions are taken into account. However, the singularities coincide with the zeros in A FB and joint analysis can help to pin down the strong phases. In the following it is useful to define the CP-violating difference and the CP-average individually, respectively, as where we recall that A FB is CP-odd, and then eq. binned measurements are crucial to disentangle not only the NP contributions but also the size of the strong phases. Although the singularities make it impossible to give predictions for integrated values of A CP FB , binned measurements are highly sensitive to the size of strong phases. Note that there is no sensitivity in A CP FB to C 10 , as its contribution does not involve requisite strong phases.

Lepton flavor violation
For the discussion of LFV decays with c → u − + ( = ) the following effective Hamiltonian is defined, where the K ( ) i denote Wilson coefficients and the operators read as (5.10) Note that there is no O ( ) 7 contribution since the photon does not couple to different lepton flavors. For = the angular observables in eq. (2.9) contain additional terms due to non-vanishing lepton amplitudes [42]. The modified angular observables in terms of the lepton masses m and m are given in appendix D. We present distributions and branching ratios summing both lepton charge combinations, defining, for instance, Figure 8 shows the differential branching fraction for the decay Λ c → pµ ± e ∓ in different lepton flavor violating BSM scenarios. Branching ratios up to O(10 −8 ) are possible in LQ scenarios, whereas SUSY models only reach few×O(10 −12 ) and Z models are at most at O(10 −13 ). LFV branching ratios are clean SM null tests, as no resonance pollution and no SM background exists. Bounds on LFV Wilson coefficients are also of special interest for combinations with flavor summed dineutrino branching ratios, see section 5.6. The presented NP benchmarks are sufficiently below the 90 % CL upper limits from BaBar [12] which read B Babar Λ c → pµ − e + < 9.9 · 10 −6 , B Babar Λ c → pµ + e − < 19 · 10 −6 .

Dineutrino modes
Null test opportunities for rare charm decays with dineutrinos modes were recently presented [14,37]. The experimental signature includes missing energy and requires a clean environment such as an e + e − -machine. Due to the strong GIM suppression and absence of QCD resonances, c → uνν branching ratios in the SM are well beyond any experimental reach in the foreseeable future. Interestingly, NP can induce sizable rates which are in reach of present flavor facilities Belle II [15], BES III [16], and beyond [17]. Moreover, the achievable upper limits on the branching ratios depend on charged lepton flavor, therefore, a search for rare charm dineutrino modes not only tests the SM, it tests at the same time lepton universality (LU), or charged lepton flavor conservation (cLFC) [14,37]. The latter originates from the SU(2) L -link between left-handed charged leptons and neutrinos. Since the neutrino flavors are not tagged, the upper limits on dineutrino branching ratios are obtained as an incoherent sum of all final state flavors which are constrained by charged dilepton data [14,37]. Note, the PMNS-matrix drops out from this link, and that the upper limits are data-driven, and can be improved. Bounds on dilepton couplings, including τ 's, exist from high-p T -data [39,40]. The strongest upper limit arises in models with LU; since the muon bounds are the strongest they dictate the limit. Relaxing LU but within cLFC branching ratio limits increase. In the most general situation, denoted below by "general", that is, LFV, the maximum limits allowing only for light left-handed neutrinos are obtained.

(5.18)
A measurement above a bound based on a lepton flavor symmetry implies the violation of the latter in the charged lepton sector. For instance, B(Λ c → pνν) > 1.8 · 10 −6 implies lepton universality violation, R Λc p = 1, and B(Λ c → pνν) > 1.1 · 10 −5 implies non-vanishing signals in LFV modes. Upper limits on branching ratios in specific leptoquark models are given in [43].

Conclusions
We present a comprehensive analysis of rare charm baryon decays mediated by c → u + − , c → uµ ± e ∓ and c → uνν transitions. The three-body decays Λ c → p ( ) and friends, • A FB , shown in figure 3, probes C 10 , leptonic axial-vector couplings, which are protected in the SM by the GIM-mechanism. A FB allows to probe C 10 at order 10 −2 , about two orders below the present bound (5.2). If normalized to the decay rate, shown in the lower left plot in figure 3, NP signals are enhanced on resonance.
• F L probes electromagnetic dipole contributions C ( ) 7 , see figure 4. In addition, F L is sensitive to NP effects from right-handed currents with leptonic vector coupling C 9 . F L is therefore complementary to A FB . The reach in F L is order percent and few percent in C ( ) 7 and C 9 , respectively, allowing to significantly improve existing limits on the Wilson coefficients (5.2).
• CP-violation: the CP-asymmetry in the decay rates shown in figure 6 is sensitive to CP-violation in left-handed C 9 and right-handed currents C 9 , and dipole couplings C ( ) 7 . The forward-backward CP-asymmetry A CP FB , shown in figure 7, probes Im C 10 .
• Lepton flavor universality can be probed in lepton flavor specific modes, using ratios of branching fractions, R Λc p , comparing dimuon to dielectron final states (4.2). NP effects in R Λc p can be very large in the high q 2 region, see table 2. Lepton flavor universality can also be probed with branching ratios into dineutrino final states [14,37], which are null tests of the SM. Data-driven, upper limits assuming universality (LU) are given in (5.14)-(5.18).
• Lepton flavor violation can be searched for in e ± µ ∓ final states, which allow for sizable, model-independent rates (5.12), also in concrete SM extensions, such as leptoquarks, shown in figure 8. Charged lepton flavor violation can also be probed with branching ratios into dineutrino final states [14,37], Upper limits assuming flavor conservation (cLFC) are provided in (5.14)-(5.18).
Even more Wilson coefficients can be probed if more than one receives NP contributions. Such a broad program calls for a fit that should be performed once data are available, where we stress the importance of q 2 -binning and, more general, probing the shapes of distributions. We further suggest to simultaneously fit process-dependent hadronic phases along with the Wilson coefficients. The differential branching ratios within the ω, ρ and the φ peaks are particularly sensitive to such phases, see figure 5 and also [7]. Such a global c → u fit should also include rare D-meson measurements. Recall that angular distributions in D → P + − , P = π, K, modes are SM null tests which probe semileptonic (pseudo-)scalar and tensor operators [8], different from those in (2.2). Rare charm baryon decays also involve both types of coupling combinations C i + C i and C i − C i . The importance of rare charm baryon decays is amplified as the decays into vector mesons, such as D → ρ + − have, for kinematic reasons, essentially no high-q 2 -region above the φ. Rare charm baryon modes therefore complement searches with D-meson decays, and together, they advance our understanding of the up-type flavor sector.

A Standard model Wilson coefficients
For completeness we give the SM Wilson coefficients using refs. [5,18,19]. The calculation is done in a different basis than the one utilized in eq. (2.1). Instead, Contributions to Wilson coefficients are calculated systematically at (partly) NNLL order in QCD, e.g. expanded in powers of α s (µ) as where the most important noteworthy properties of the calculation are listed below.  Table 3. SM Wilson coefficientsC j for j ∈ [1, 10] up to NNLO accuracy. In the first three rows we show values for the charm mass scale µ = m c and for scale variations of µ = √ 2 m c and µ = m c / √ 2, respectively. The last two rows explicitly give the central values and uncertainties, which are used in this work.
•C 1 andC 2 are calculated at the electroweak scale and evaluated down to the b quark mass within renormalization improved perturbation theory.
• At the b-quark mass scale, a five to four flavor matching is performed, as the b quark is integrated out. Due to the matching, operators other than P 1,2 are induced.
• After matching at µ b , all contributions are evolved to the charm mass scale, where µ c ∈ mc √ 2 , √ 2 m c is taken as an scale uncertainty. In addition the W and b mass scales are also varied by a factor 2 to estimate further scale uncertainties.
•C 10 is zero in the SM to all orders.
The calculation is performed in Python with numerical results summarized in table 3, where the first three rows show the complete NNLO result for the three different charm mass scales and the fourth and fifth row the resulting central value and uncertainty, respectively. Our values are in perfect agreement 4 with refs. [5,18,19].
The Wilson coefficientsC j are absorbed into "effective" coefficients C respectively. λ p can therefore take values of ± 1 2 , whereas λ γ takes values of 0, ±1 and we further distinguish λ γ = t in the J γ = 0 case and λ γ = 0 in the J γ = 1 case. The index m distinguishes between leptonic vector (m = 1) and axial vector (m = 2) contributions. The m = 2 amplitudes thus only contain C 10 and C 10 .
Following ref. [30] we introduce (m ( ) = 1, 2) To obtain the helicity amplitudes we sum the contributions from different hadronic matrix elements. To do so we define amplitudes corresponding to one single hadronic matrix element as H a,m λp,λγ , and H m λp,λγ = a H a,m λp,λγ . a = 1 (2) corresponds to matrix elements involving C ( ) 7 with (without γ 5 ). The same notation distinguishes vector and axial vector matrix elements of C ( ) 9 (10) for a = 3, 4. Flipping the helicities will result in a minus sign for the amplitudes a = 2 and a = 4, due to parity H  = − √ 2 C 9(10) − C 9 (10 ) √ s + g ⊥ q 2 . (C.5)

D Lepton flavor violating angular observables
For the lepton flavor violating decays the angular observables in equation (2.9) are given by where we use the expressions from eq. (C.1) for the hadronic helicity amplitudes with the appropriate replacement of Wilson coefficients C -28 -