Testing discrete symmetries at a super τ -charm factory

Tests of discrete symmetry violation have played an important role in understand the structure of weak interactions in the Standard Model of particle physics. Historically these measurements have been extensively performed at experiments with large samples of K and B mesons. A high luminosity τ -charm facility presents physicists with the opportunity to comprehensively explore discrete symmetry violation and test the Standard Model using τ leptons, charm mesons and charmed baryons. This paper discusses several possible measurements for a future τ -charm factory.


Introduction
The Standard Model of particle physics (SM) describes weak, strong and electromagnetic interactions. The weak interaction is known to violate the discrete symmetries C, P , T and the combination CP . The combination CP T is observed to be conserved. Strong and electromagnetic interactions conserve these symmetries. Parity violation was discovered in 1957 [1] and CP violation a few years later in 1964 [2]. Following these discoveries there was an interest in trying to validate T independently of CP T . Whilst it was recognised that CP T conservation was desirable given the prior evidence available, it was noted that testing T independently of CP T was important [3]. It is possible to test the full set of discrete symmetries using triple product asymmetries and using entangled pairs of neutral mesons. This paper discusses the potential for a Super τ -charm facility in terms of testing discrete symmetries using τ leptons, charm mesons and charm baryons produced near threshold. A number of routes toward CP violation measurements in charm decays are under study in the literature and have been discussed at length elsewhere, for example Refs [4][5][6]; here we review additional possibilities to probe discrete symmetries that complement the traditional routes.
The remainder of this paper discusses the use of triple product asymmetries with four body decays to test C, P and CP (Section 2), followed by the use of entangled pairs of D mesons produced in the decay of ψ(3770) mesons to test CP , T and CP T (Section 3). Finally Section 4 presents a summary of this paper. The data sample assumed for a Super τ -charm facility is 1ab −1 , which corresponds to 10 9 ψ(3770) (10 8 ψ(4040)) mesons for D 0,± (D ± s ) pair production. Facilities capable of producing these kinds of sample sizes are under investigation, for example the proposed High Intensity Electron Positron Accelerator (HIEPA) in China.

Triple product asymmetry measurements
If one considers the decay of some particle M to a four body final state abcd and the CP conjugate process M → abcd, then it is possible to use the decay planes defined by the four vectors (or spins) of pairs of final state particles to construct a scalar triple product that allows us to probe the symmetry violating nature of the decay, for example see Refs [7,8]. The scalar triple product can be written as ψ = p c · ( p a × p b ), where the p i , i = a, b, c are particle momentum vectors computed in the rest frame of M . We can study data in terms of the sign of ψ, or as a function of the angle between the decay planes formed by ab and cd in the reference frame of the decaying particle; φ. The angle φ is used when the underlying amplitudes in the decay are known sufficiently well to allow experimenters to understand if the interesting asymmetries are functions of sin φ or sin 2φ. A number of measurements have been made in terms of the sign of the triple product. Following this generic approach we define Γ ± to be the rate at which M decays to a state with ψ > 0 (+) or < 0 (−). The corresponding rates for antiparticles are given by Γ ± .
Twelve asymmetries can be constructed by considering Γ ± and Γ ± [8]. The first six are derived by considering the P , C and CP operators acting on the four Γs. These yield: Here the subscript indicates the symmetry being tested. One can construct an additional six asymmetries considering the remaining permutations, where the superscript denotes the original symmetry considered and the subscript denotes the subsequent permutation: The symmetry being tested by these last six asymmetries can be determined by multiplying the sub-and superscripts together. There are three types of decay that we can consider measuring, the most general case has been considered so far, however we can consider the simplification when abcd = abcd. In this limit the twelve asymmetries remain non-trivial. In the case that we further simplify to also require that M = M we obtain only a single unique and non-trivial asymmetry given by where the average rates are indicated to highlight that M is indistinguishable from M . Before discussing charm mesons it is useful to proceed via an interlude (Section 2.1) that reviews triple product asymmetry measurements in neutral kaon decays. Following this we discuss applications to charm mesons and baryons (Section 2.2) and τ leptons (Section 2.3). We discuss a model-based interpretation of these asymmetries in Section 2.4.
The decay K L → π + π − e + e − has been studied both theoretically and experimentally. Reflection on these results provides useful insight into how to address measurements of triple product asymmetries in the charm sector. It was noted by Heiliger and Sehgal that this mode proceeds via four amplitudes; K L → π + π − γ photon conversion; bremsstrahlung from the CP violating decay K L → π + π − ; a CP conserving magnetic dipole component; and finally a short distance component related to sd → e + e − . The radiative K L → π + π − decay is CP violating and it is the interference between this amplitude and the remaining CP conserving ones that gives rise to a non-zero CP asymmetry. Heiliger and Sehgal predicted that the level of CP violation manifest in this decay is of the order of 14% [9]. Shortly after this prediction was made the KTeV experiment at the Fermi National Accelerator Laboratory measured this triple product asymmetry and confirmed the existence of a large effect [10]. Subsequently the NA48 experiment measured the triple product asymmetry of both the K L and K S meson decaying into π + π − e + e − [11]. These results were found to be consistent with KTeV for the K L mode, and consistent with CP conservation for the K S decay as expected (given that K S → π + π − is CP conserving). This highlights an important issue with regard to CP asymmetries; first one needs to identify a CP violating amplitude, and only then the interference of that amplitude with other contributions may manifest effects that will be non-zero. This is a well known statement of fact and is far from being profound. This factor should be taken into consideration during the following discussion with regard to possible measurements that one can make, and measurements that have been made. Thus far D decays to four body final states studied require amplitude analyses for a complete interpretation of results. These are complicated and have not yet been attempted; a priori it is not clear from inclusive measurements if nonzero triple product asymmetries are driven (in part) by a non-zero weak phase difference between pairs of amplitudes or not. One has to understand the dominant 2 amplitude contributions to the decay model and from that model one can evaluate what the expected outcome might be. At the time of writing model dependent analyses have not been performed, however a simple example is discussed below in Section 2.4. It is hoped that measurement (and theoretical considerations) will evolve to permit model dependent studies of these decays over the coming decade.

Testing charm mesons and baryons
In section 2.2.1 we start by considering tests using charm mesons with a brief summary of the current state of the art in terms of measurements, then move on to discuss possible measurements for the future. In doing so we link back to reflect on the work done in kaon decays to draw analogies and highlight several modes that have been ignored thus far. Having discussed measurements with mesons we move to consider baryon (Section 2.2.2) and τ lepton (Section 2.3) decays.

Charm meson decays
The most studied triple product asymmetries for four body D decays are for the channel D 0 → K + K − π + π − . This has been studied by FOCUS, BaBar and LHCb and provides an interesting window of opportunity given a relatively large branching ratio; (2.43 ± 0.12) × 10 −3 . Experimentally the symmetry in the final state results in cancellation of a number of systematic uncertainties. The FOCUS measurements were insufficient to establish any non-zero triple product asymmetry, but laid the foundations for subsequent work by BaBar; this B factory initially repeated the FOCUS measurement but with a larger data sample. BaBar found non-zero values for A P and A P , but the CP asymmetry A P C was consistent with zero. BaBar has recently performed a measurement of all twelve asymmetries [12,13]. LHCb with its large sample of data has provided an interesting insight into these decays, as they have been studied in bins of K + K − and π + π − invariant masses as well as performing phase space integrated measurements [14]. The distributions for these invariant mass distributions indicate a rich resonant structure in the final state. In addition to the K + K − and π + π − combination studied one should investigate the K ± π ∓ combinations to facilitate building a robust amplitude model to further study of the data. Interpretation of these results is complicated by the lack of a detailed amplitude model, however by considering the results of the simple model discussed below one can conclude that there is no evidence for a non-zero weak phase difference in this decay. All of the non-zero asymmetries measured by BaBar and LHCb can be driven by strong phase differences (see Section 2.4).
The channel D + → K S K + π + π − has been studied by BaBar, where all measured asymmetries are found to be consistent with zero (integrating over phase space) [15]. The branching fraction for this channel is (1.75 ± 0.18) × 10 −3 . It remains to be seen if there is a more complex picture that averages out to this null result when integrating over phase space. The corresponding D + s decay has a branching fraction of (1.03 ± 0.10) × 10 −3 and also been studied [15]. Here the pattern observed for D 0 → K + K − π + π − is repeated; the asymmetries driven by a non-zero weak phase difference are all zero, but those that can be driven by strong phase differences are not. The decays D + (s) → K L K + π + π − have not been studied by LHCb or the B factories. Given the presence of the K L in the final state one would expect a small residual level of CP violation from kaon decays to be present. These modes would have significant amounts of background at a B factory and would be difficult to attempt to reconstruct in a hadronic environment like the LHC. A significant virtue of a τ -charm factory is the ability to infer the missing energy and effectively reconstruct the K L four momentum. With 1ab −1 of data one could make precision measurements of triple product asymmetries in D + → K S,L K + π + π − . A data sample of 100fb −1 collected at D s threshold would provide about 32 × 10 6 D ± s to perform similar measurements. This would be sufficient to provide several tens of thousands of D + s → K S,L K + π + π − decays to study. A statistical precision on triple product asymmetries better than a percent would be achievable with such a sample.
A τ -charm factory is well placed to perform precision measurements of these, and many other decay channels. For final states with one or more neutral meson there are obvious advantages in using data from an e + e − environment compared with pp collisions at the LHC. Kang and Li have studied the prospects for a variety of D decays to V V final states [16] (here V is a vector particle with J P = 1 − ). Sub-percent level precisions are attainable with modest data samples (∼ 20fb −1 ) by BES II for the modes studied: neutral D meson decays to ρ 0 ρ 0 , K * 0 ρ 0 , ρ 0 φ, ρ + ρ − , K * + K * − and K * 0 K * 0 , and charged D decays to K * 0 ρ + . The statistical precision of the charged D decay is at the per mille level with this sample size.
A super τ -charm factory would be expected to accumulate significantly larger samples of data than this. For example a factory accumulating 1ab −1 of data at charm threshold could achieve statistical uncertainties on triple product asymmetry measurements for all of these decays at or below the per mille level. If we consider the kaon measurements discussed in 2.1, these are triple product asymmetries from four body decays derived from CP violating and CP conserving two body decays of the kaon. The equivalent possibility for investigation in charm has been ignored thus far; i.e. the search for CP violation in D decays to h + h − + − final states, where h = K, π and = e, µ. Assuming that D → K + K + and/or π + π + would exhibit CP violation at some level, one could use the interference between amplitudes generated in an analogous way to generate an asymmetry in these decays. The PDG reports upper limits on these modes ranging between 3.1 × 10 −4 and 3.0 × 10 −5 [17]. Some of these limits are just above naive expectations of the branching fractions based on the known two body final state branching fractions. The first step would be to search for these data at a B, τ -charm factory or the LHC and subsequently explore the triple product asymmetry structure of the decays to search for symmetry violation. It is worth noting that an advantage of these modes is that they are unambiguous; the hadronic and di-lepton systems can be treated as ab and cd, respectively; unlike the current measurements where there are two possible pairing combinations to consider when probing amplitudes. The corresponding set of measurements for D + (s) decays would involve h + h 0 + − final states, where h = π, K. The Cabibbo suppressed decays would allow us to search for CP violation, and the Cabibbo favoured states would provide useful control samples; however K L,S π + + − states would ultimately have a small CP violating effect resulting from the kaon CP violation intrinsic to the final state. A number of four body D 0 and D ± (s) decays are yet to be studied; it is worth noting that the modes measured so far all have large branching fractions. Rare decays are more suitable for searches for physics beyond the SM as small SM amplitudes can generate large effects when beating against any hypothetical new physics amplitude of a comparable size. The one thing that we do know about new physics amplitudes is that they are at best small for the energy scale being probed. Thus multi-body rare charm decays may provide an interesting test bed for CP violation; In addition to obtaining a more complete understanding of the copious decays, experimentalists should study the available data for the rarer processes. It remains to be seen if one can generate large effects in the SM in analogy with the K L → π + π − e + e − case.

Charm baryon decays
While measurements so far have focused on mesons, there is also a rich area of study in the decay of charm baryons. These systems are accessible using data from Belle II, BES III, the LHC, and a super τ charm facility. The prospects for Λ c decays to final states including baryons, pseudoscalars and vector particles have been studied in Ref. [18]. This paper assumes one year of data taking corresponding to an integrated luminosity of 5 fb −1 at the X(4630) peak with BES-III. This data sample corresponds to 2.5 × 10 6 Λ + c Λ − c pairs. The estimated precisions attainable for triple product asymmetries with such a data sample are typically at the level of a few percent. A high luminosity τ -charm facility would provide the opportunity to reach the sub-percent level in all modes studied with a data sample of about 80 fb −1 . If one considers the mode Λ c → Λ(pπ − )ρ + (π + π 0 ), one could reach a per mille level statistical precision on the triple product asymmetries with data samples as small as 100 fb −1 . As CP violation is expected to be small in the charm sector these decays provide an excellent set of laboratories to search for physics beyond the SM. With 100 fb −1 of data one would have about 5 × 10 7 Λ + c Λ − c pairs which would enable searches for rare decays of the Λ + c .

τ decays
Searches for CP violation in τ decay have concentrated on the channel τ → K S πν [19,20]. The level of experimental sensitivity is approaching that of the intrinsic effect of CP violation in neutral kaons, which is a SM background to the search for new physics. One of the problems with performing a triple product asymmetry measurement for a tau decay such as τ → hh h ν, where h ( [ ]) = K, π, η, is that the center of mass frame needs to be determined. Here a τ -charm factory has an advantage over other experimental facilities; while running on τ + τ − threshold the leptons are created at rest in the laboratory frame, and hence the kinematics are fully constrained by the observed four momenta of the reconstructed particles. Energy-momentum conservation allows one to infer the neutrino and hence fully reconstruct the event. In doing so it becomes possible to compute the full set of triple product asymmetries outlined at the start of this section in the search for new physics. Decays with odd numbers of charged kaons in the final state suffer from detection asymmetry effects which are well known, but provide additional systematic uncertainties. Those with neutral kaons suffer from regeneration and interference effects, which again provide additional uncertainties which come into play when interpreting results. Higher energy sys-4 tems may be able to perform triple product asymmetry measurements, however those are affected by the fact that it is not possible to fully reconstruct the decay for energies above threshold. The decays τ → π − π 0 K 0 ν, K − π 0 K 0 ν, and π − K 0 ην are all expected to manifest CP violation, resulting from the neutral kaon in the final state, and provide an interesting complement to the τ → K S πν mode already studied. Any large CP violation effect observed in τ decay would be a clear sign of new physics. This is a largely unexplored experimental area that can be studied extensively at a τ -charm facility such as BES III, or at a super τ -charm factory.

A simple model
We can increase our understanding of the twelve triple product asymmetries introduced in Ref. [8] by considering a simple model of two interfering scalar amplitudes divided into + and − parts according to the sign of the scalar triple product: A + = a 1 e i(−φ1+δ1,+) + a 2 e i(−φ2+δ2,+) , where δ represents a strong phase and φ a weak phase.
Here the coefficients a 1 and a 2 are just the magnitudes of the interfering amplitudes. In this case, as shown in Ref. [8], the six asymmetries A P C , A C , A C , A C P , A C CP , and A CP C can only be non-zero if the difference between the weak phases is non-zero. The remaining asymmetries can be non-zero even if φ 1 − φ 2 = 0. This simple model can be extended from the interfering (pseudo)scalar amplitude case to a more general scenario amplitudes with higher spins following the procedure outlined in [21].

Tests using entangled states
John Bell resolved the EPR conundrum in 1961, and in doing so invented the concept of entangled quantum states [22]. to write the wave function: The subscripts ± denote the CP eigenvalue of the D decay as even or odd, respectively. The Roman numeral subscripts refer to the time ordering of decaying mesons; either the first (1) or second (2) meson to decay. The second set written down indicates the final state reconstructed in for the flavour basis, or CP eigen value for the CP basis. The filter decays to a lepton +X are an accurate way to determine the quark flavour in a charm decay, the mis-tag probability at an e + e − machine running at charm threshold is small. The set of CP filter decays to complement these include η CP = +1 (−1) We can consider the possible combinations of decay to occur via either the flavour or CP filters described above, which gives rise to three possible measurements of interest. However, it is useful to note that in addition to filtering using only flavour or only CP states, we can also filter using a combination flavour then CP filters or CP then flavour filters. This results in a total of 15 distinct asymmetries [23] as listed in Table 1. The two flavour filter only asymmetries have been studied for many decades. The CP filter only asymmetry has not been studied before for any neutral meson system. The remaining twelve asymmetries are derived using the approach described in Refs [24,25], and measured by BaBar for neutral B decays [26]. It is worth noting that when using only a single filter basis pair it is not possible to construct an unambiguous test of a single symmetry, however the constructed asymmetry can be used to simultaneously test a pair of symmetries. When using two filter basis pairs it is possible to resolve the remaining ambiguity to obtain a set of tests of only one symmetry.
In general one should perform these measurements as a function of the proper time difference between the first and second D meson decays in the event (usually denoted as ∆t in the literature, for example see Refs [4,5] for details of time-dependent analyses). However the mixing frequency and lifetime difference between D 0 and D 0 is small in the charm system; x = ∆m/Γ ∼ 0.5% and y = ∆Γ/2Γ ∼ 0.7%. Hence initially time-integrated measurements of the asymmetries outlined below would be of direct interest; and a small correction would be required when interpreting precision measurements in or-5 der to take into account the fact that x and y are nonzero. Table 1: The fifteen possible pairings of reference and symmetry conjugated transitions used to study CP , T and CP T for pairs of neutral D mesons.

Symmetry
Reference Conjugate Section 3.1 discusses measurements of asymmetries constructed from the flavour filter basis pair, Section 3.2 discusses possible measurements of the asymmetry constructed from CP filter basis pairs, and Section 3.3 discusses the remaining measurements using a combination of CP and flavour filter basis pairs.

Using flavour filters
It is possible to construct tests of CP and T and of CP and CP T using flavour filter states. These measurements require studies as a function of lifetime difference between opposite and same sign tagged final states. The asymmetries that one measures are The former measurement is usually referred to as a measurement of CP in mixing, however it is worth noting that this is also simultaneously testing T , c.f. the Kabir asymmetry measured by CPLEAR in kaon decays [27]. The typical experimental signature that one would pursue for this would be to reconstruct both D mesons via a semi-leptonic decay and search for same sign di-leptons; one being from each decay. A non-zero value of the resulting asymmetry A CP,T as a function of proper time difference between the decaying D mesons would indicate a violation of both CP and T . The corresponding test for A CP,CP T requires opposite sign dilepton final state, and a non-zero value of this asymmetry would indicate a violation of both CP and CP T . This could only be manifest by physics beyond the SM. It is worth noting that while these tests are performed using an entangled state prepared in the decay of a ψ(3770), it is also possible to use a hadronic production environment with associated production of charm to flavour tag the neutral D meson at the point of production, and reconstruct the semileptonic decay at a later time. A second route that is viable at the LHCb experiment is to use semileptonic B decays to tag the flavour of the decaying neutral D meson at the point of production, and the leptonic charge at the point of decay to provide the required rates to compute A CP,T and A CP,CP T .
Over the past few years there has been a lot of interest in the like-sign semileptonic asymmetry measurement made by the D0 experiment for B s mesons [28]. This is a measurement of A CP,T using B s decays. The reported D0 result is A CP,T = −0.787 ± 0.172 ± 0.093, which deviates from the SM expectation of zero by 3.9σ. All corresponding measurements made by the B factories for this asymmetry in B d mesons are consistent with zero (See [4] and references therein). If the anomalous like-sign di-muon asymmetry in D0 is the result of some kind of new physics then that may also be manifest in the charm sector. Hence, it is important to study charm decays in order to search for evidence of CP and T violation. As noted in [29] it is possible for systems with ∆Γ 0 to result in a zero asymmetry measurement for A CP,T even when the symmetry is violated. For neutral charm (like B s ) mesons ∆Γ = 0; hence such a measurement for D 0 mesons is an important test to complement the studies performed thus far.
A recent review of semi-leptonic (SL) decays by Lui outlines experimental issues related to reconstructing these states [30]. The branching fraction of SL decays is large, and so precision measurements of A CP,T and A CP,CP T are in principle achievable assuming that systematic uncertainties may be kept under control. 6

Using CP filters
The asymmetry constructed using only CP filter states allows us to perform a simultaneous test of both T and CP T . In order to perform this test we need to identify D meson decays into CP even and CP odd final states. For example one can measure the asymmetry between D → K S (ω, φ, ρ 0 ) followed by D → h + h − (or D → K L (ω, φ, ρ 0 )) and D → h + h − (or D → K L (ω, φ, ρ 0 )) followed by D → K S (ω, φ, ρ 0 ) final states. Any combination of +1 and −1 states can be used to test T using this method. The SM expectation is that A T,CP T = 0. Any non-zero value for any of these combinations would indicate violation of both T and CP T , and physics beyond the SM. This type of test complements the flavour filter tests of A CP,T and A CP,CP T described above. The initial CP filter state can be tagged via the decay of a ψ(3770). As a result of incoherent production of charm at a hadron collider or B factory does not permit an obvious route to performing this type of asymmetry measurement via other means. Experimentally the double D → K S,L ω 0 decays should proceed with a rate of the order of 1.2 × 10 −4 . Allowing for the ability to reconstruct these decays with a modest efficiency would set the single event sensitivity at a the level of O(f ew 10 −5 ). A Super τ -charm factory would be able to accumulate about 10000 events with 1ab −1 to perform a measurement of this type. The double decays to D → K S,L ρ 0 and D → K S,L φ have product branching fractions of 3.6 × 10 −5 and 4 × 10 −6 , respectively. Samples of about 1000 and 100 events, respectively could be recorded in order to permit a measurement of A T,CP T for these decays.

Using both flavour and CP filters
The remaining twelve asymmetries can be constructed from Table 1 and these constitute four tests of each of CP , T and CP T . These tests complement the A CP,T , A CP,CP T and A T,CP T asymmetries discussed above as they each unambiguously identify one symmetry to test. These asymmetries have only been measured thus far for neutral B mesons [26], where results consistent with the SM were obtained; namely that CP and T are violated, whilst CP T remains conserved. These measurements provide an important cross check of our understanding of symmetry violation to complement existing routes to search for symmetry violation. The magnitudes of asymmetries determined in these decays are related to unitarity triangle angles in the charm sector (just as the asymmetries measured in Ref. [26] are related to sin 2β from the B d "Unitarity Triangle"). As CP violation is expected to be small in the charm sector, so the angles measurable in the CP and T asymmetries are expected to be small (i.e. compatible with zero within uncertainties). The CP T asymmetries are expected to be zero in the SM, to signify that this symmetry is conserved. Significant deviations from this pattern would be an indication of physics beyond the SM. A discussion of how to relate the angles of the charm unitarity triangle to decays in the charm system can be found in Ref. [5]. Table 2 summarises the fifteen asymmetries in terms of the final states that must be reconstructed for reference and conjugated processes. These clearly highlight the symmetries being tested by "same" and "opposite sign" asymmetry measurements, as well as allowing one to clearly identify the combinations for testing the remaining thirteen quantities. Table 2: Final states reconstructed for the first and second D mesons in an event, along with the conjugate processes to test the symmetries CP , T and CP T . The + X ( − X) state is the flavour filter for a D 0 (D 0 ), and +1 and