Time-reversal asymmetries in

: We study the decays of with . We examine the full angular distributions with polarized , where the T -odd observables are identified. We discuss the possible effects of new physics (NP) and find that the T -odd observables are sensitive to them as they vanish in the standard model. Special attention is given to the interference of (pseudo)scalar operators with (axial)vector operators in polarized , which are studied for the first time. Their effects are proportional to the lepton masses and therefore may evade the constraint from at the LHCb naturally. As is uncontaminated by the charmonia resonance, it provides a clean background to probe NP. In addition, we show that the experimental central value of in at the LHCb can be explained by the NP case, which couples to the right-handed quarks and leptons. The polarization fraction of at the LHCb is found to be consistent with zero regardless of the NP scenarios.


I. INTRODUCTION
The CP violating observables in with play important roles in probing new physics (NP) as they are highly suppressed in the standard model (SM) [1−7].In recent years, special attention has been given to and [8−10].Precise measurements of angular observables are now accessible owing to experimental developments [11−19].They are useful in disentangling helicities, providing reliable methods to probe the Lorentz structure of NP [20−27].Besides, the ratios of were measured, where discrepancies against the SM were found.In particular, 3.1σ and 2.5σ deviations have been found in and [28,29], showing that the lepton universality may be violated by NP.Very recently, a global fit of with B meson experiments was performed [30], and the experimental data permitted the large complex Wilson coefficients beyond the SM.
The baryonic decays of are interesting for several reasons.For polarized , the decays provide a couple dozen angular observables, which are three times more than those in .The polarization fraction of is reported as % at the center of mass energy of 7 TeV of collisions [31].The full angular distribution of has been measured at the LHCb [19].Notably, the experiment obtains that K 10 = −0.045± 0.037 ± 0.006 , deviating from the SM prediction of by .It is reasonable to expect that the precision will be improved in the forthcoming update.In this study, we explicitly show that is a T-odd quantity, which can be sizable in the presence of NP.
Theoretically, the angular distributions of have been studied intensively [26,27,32].In particular, an analysis of NP with real Wilson coefficients has been performed in Ref. [33], where they found % at confidence level.In this study, we focus on the time-reversal (T) violating observables induced by the complex NP Wilson coefficients.Unlike the direct CP This paper is organized as follows.In Sec.II, we decompose into products of two-body decays.In Sec.III, we construct the T-odd observables.In Sec.IV, we briefly review the angular distributions of and identify the T-odd observables.In Sec.V, we estimate the effects of the (pseudo)scalar operators on the T-odd observables.We conclude the study in Sec.VI.

II. HELICITY AMPLITUDES
The amplitudes of , induced by the transitions at the quark level, are given as [34]: where is the Fermi constant, are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, represents the (effective) Wilson coefficients, with being the four-momentum of , , , and is for the quark mass.We only consider the NP operators with the right-handed strange quark in , as the heavy quark mass would suppress the effects of the left-handed ones.
The first (second) term in Eq. ( 2) can be interpreted as followed by , where is an effective off-shell (axial) vector boson, conserving the parity in its cascade decays, and are the couplings of .Alternatively, the interpretation can be rephrased as by eff where and clearly couple only to the right-handed (left-handed) leptons.As parity is conserved in , it is easier to obtain the angular distributions with the interpretation.In this study, the angular distributions are obtained using , while for NP, is used.Similarly, the third (fourth) term describes , with , the (pseudo)scalar boson, decaying to subsequently.
By decomposing the Minkowski metric as we arrive at where is the helicity of , with t indicating spin-0 off-shell contributions, and are the polarization vectors of , given as [36] Chao-Qiang Geng, Chia-Wei Liu, Zheng-Yi Wei Chin.Phys.C 48, 033101 (2024) 033101-2 and in the center of mass (CM) frames of and , respectively.Note that , as they do not contain the spacelike component.In Eq. ( 8), the amplitudes are decomposed as the products of Lorentz scalars, where and describe and , respectively, reducing the three-body problems to two-body ones.
To deal with the spins, we adopt the helicity approach.The projection operators in the rotational group are given by where N and M are the angular momenta toward the direction, the Wigner-D matrices are defined by and are the rotation operators pointing toward .Notably, it is important for Eq. ( 12) to be a linear superposition of , which commutes with scalar operators.In the following, we take the shorthand of .The simplest two-particle state with a nonzero momentum is defined by where are the helicities, the subscript denotes the particles, and is the Lorentz boost, which brings the first (second) particle to .As commutes with , the state defined by Eq. ( 14) is an eigenstate of . Plugging Eq. ( 12) into Eq.( 14) with , we arrive at which expresses the angular momentum eigenstate as the linear superpositions of the three-momentum eigenstate.
Conversely, we have Note that the identities of Eqs. ( 15) and ( 16) are purely obtained from the mathematical consideration.The simplification occurs when the angular momentum conservation is considered.At the CM frames of and , it is clear that only and need to be considered for the and systems, respectively.Utilizing Eq. ( 16), we find that where is an arbitrary scalar operator, and stands for an arbitrary initial state.In Eq. ( 17), the final state on the left side possesses a definite angular momentum, which is irreducible in the group, i.e., it contains only the dynamical details.On the contrary, the one on the right side is a three-momentum eigenstate, containing fewer physical insights but providing a way to compute the helicity amplitude.
Let us return to and .We take the uppercase and lowercase of H and h for the helicity amplitudes of and , respectively.To be explicit, we have where corresponds to the angular momentum (helicities) of , and ( ) is the three-momentum of Λ ( in the CM frame of .Theoretically, the dynamical parts of the amplitudes are extracted by Eq. ( 17), whereas the kinematic dependencies are governed by .
For compactness, we take the abbreviations eff where is the transition operator responsible for , and is not written down explicitly.The artificial is needed to interpret as products of two-body ones.For the interpretation, the helicity amplitudes are

III. T-ODD OBSERVABLES
From Eq. ( 3), we see that the NP contributions are absorbed into the couplings of , whereas the Lorentz structures of are plain.Thus, to discuss the NP effects, it is sufficient to study .
The most simple T-odd operator in is defined as [37] are the spin operators of Λ and , respectively, and is the unit vector of .The spin operators can only be defined for the massive objects, given as where M is the particle mass, and , , , and are the time translation, space translation, rotation, and Lorentz boost generators, respectively.Note that and are T-odd, whereas is T-odd.In addition, satisfies the relations ⃗ ω T T with arbitrary .The key to solving the eigenstates of relies on being a scalar operator.Thus, we have and resulting in the eigenstates where and are the eigenvalues of and , respectively.They are also the eigenstates of , as commutes with both and .Note that are not involved since they are contributed by spinless .
Because and are T-odd and T-even, respectively, we have where is the time-reversal (space-inversion) operator, and depends on the convention.On the other hand, would interchange and , given as S eff I t which vanishes if is invariant under .Explicitly, we find that I t which are proportional to the relative complex phase.They are called T-odd quantities, as interchanges the final states of the two terms in Eq. ( 31).
The operator of contains , which is difficult to measure directly.To probe the spin of Λ, it is plausible to study the cascade decays of .Subsequently, the final states involve four particles , containing three independent three-momenta.It is then possible to observe the triple product, given by where α is the polarization asymmetry in , and is the three momentum of the proton.Notice that α is a necessary component in Eq. (33), as does not affect if .As the product in Eq. ( 33) is P-even, we have to construct P-even observables from Eq. (32).From the transformation rules, it is easy to see that are both T-odd and P-even.

IV. ANGULAR DISTRIBUTIONS
The lepton helicity amplitudes are calculated as where with being the lepton mass.In contrast, the baryon matrix elements are conventionally parameterized by the form factors, given by where and are the Dirac spinor and mass of , respectively.In turn, we find that H Am where , , and Time-reversal asymmetries in Chin.Phys.C 48, 033101 (2024) 033101-5 with Combining the relations the evaluations of H are completed once the form factors are given.
The angular distributions of , related to the kinematic part, are given by piling , read as where , , , , and for .The angles are defined in Fig. 1, where and are defined in the CM frames of and , respectively, and are the azimuthal angles between the decay planes.The breakdown of the physical meaning of Eq. ( 53) is as follows: • is responsible for , where H and D describe the dynamical and kinematic parts of the amplitudes.
• The kinematic part of is described by and the dynamical part by .
• and describe the dynamical and kinematic parts of , respectively.
The derivation is similar to those in the appendices in Ref. [36].We cross-check our result of with Ref. [32] and find that they match.For practical purposes, is expanded as follows [19]: with where the definitions of and can be found in Appendix A and Appendix B, respectively.We note that are proportional to , imposing difficulties in extracting physical meanings since depends on their production.Interestingly, in the SM, and are found to be After identifying the T-odd observables, we are ready to estimate the NP contributions.If (pseudo)scalar operators from NP are involved, their contributions are divided into two categories: one is from the interference between NP, which scales as , and the other arises from the interference of (pseudo)scalar operators with the SM, scaling as .We focus on the latter, as it is expec- ted to be larger.As the contributions are proportional to the lepton masses, our main concern lies in the decay channel of .Furthermore, this channel is contaminated a little by the charmonia resonance, providing a clean background to probe NP.We take in and notice that would not be influenced by the (pseudo)scalar operators.However, is sensitive to and .
In this work, we evaluated the form factors of from the homogeneous bag model, the details of which are given in Appendix C and good accordance to the experiments is found in .The results are plotted in Figs. 2 to 5 with error bands, where when .In Figs. 2 and 3, are set to be purely real, whereas in Figs. 4 and 5, they are purely imaginary.
We find that is more sensitive to the imaginary part of .In the region of , the interference between (axial)vector and (pseudo)scalar operators can be significantly enhanced.To estimate the experimental results, we consider the integrated related to Figs. 2 to 5 in Table 1.The integrated is defined as where . From the table, we find that the contributions from the imaginary part of are primary and larger than others by one order of magnitude.If we take at the LHCb Run3&Run4, a reconstruction efficiency , and , we have , which can be measured at the LHCb Run3&Run4.
To consider the NP contributions in , we may ignore the contributions from (pseudo)scalar operators, as they are suppressed by the  2 with four different scenarios 1) .To illustrate this, we calculate with and in different scenarios given in Table 2.We fit from the experimental and find that is consistent with zero regardless of the presence of NP.
In the absence of relative complex phases in the SM, are found to be less than .Therefore, they provide excellent opportunities to test the SM.Although are proportional to the imaginary parts of the NP Wilson coefficients, which have not yet been determined, their signs remain unknown.However, nonzero values in the experiments would be a smoking gun in NP, regardless of the signs.Scenario #1 affects little in , and the results are not listed.We find that is very small in all scenarios, which is consistent with the experiments.Remarkably, the experimental data of can be explained by Scenario #4.On the other hand, and are highly suppressed by .
Since the CP-conserving phases are absent, the T-odd observables are proportional to the imaginary parts of the Wilson coefficients.Noting that , we deduce that in are equal to those in .Here, represents the decay asymmetry of .033101-8

VI. CONCLUSION
We derived the angular distributions of based on the effective schemes of , and the results were found to be consistent with those in the literature.By studying the effective two-body decays of , we identified the T-odd correlations in the form of .We found that and are related to , and is sensitive to the complex phases generated by NP.For , we found that can ex- plain the puzzle.We recommend revisiting the experiment, considering for a stringent constraint.In , we focused on the effects of the interference of (pseudo)scalar operators with (axial)vector operators and found that the effects of the (pseudo)scalar operators can be largely enhanced in the high region.

K i
Here, all are real.They are given by Notice that the region of and in units of GeV are contaminated largely by the charmonium resonance and are not considered.The computed results within the HBM are given in Table C2, along with those from the literature and the experimental data [18,19].The computed values of and have little uncertainties as are correlated in the model calculations.In the literature, Ref. [47] employs the lattice QCD, and Ref. [35]
to the complex phases of NP, as they are proportional to the imaginary parts of the helicity amplitudes.V. NUMERICAL RESULTSO(C2 S ,P ) O(C S ,P )
From the global fit in the B meson decays[30], the permitted imaginary parts of the NP Wilson coefficients are found in Table

Table 2 .
The Wilson coefficients and in units of in four NP scenarios.

Table C2 .
includes the contributions from the charmonium resonances.We see that the angu-FB lar observables in the literature and this study are basically consistent.Our results of and are slightly larger than the others owing to the updated α1).Notably, the experimental values of are nearly twice larger than the theoretical predictions.Decay observables, where and are in units of GeV and GeV , respectively.