Hyperon Physics with BESIII

Hyperons constitute a unique diagnostic tool to shed light on various unresolved puzzles in contemporary physics. Prominent examples are the matter–antimatter asymmetry of the universe and how the strong interaction confines quarks into hadrons. The weak, parity violating decay of hyperons make their spin properties experimentally accessible. This can be exploited e.g. in searches for CP violation and in the decomposition of the inner, electromagnetic structure of hyperons. The BESIII experiment at BEPC-II, Beijing, China is excellently suited for hyperon physics. Recently collected large data samples have been analysed and a plethora of new results have emerged. In these proceedings, we discuss the virtues of polarised and entangled hyperons, and present a collection of recent results from BESIII.

The Y → B M decay, with the spin direction of Y along the y-axis where Y is a spin 1/2 hyperon, B a spin 1/2 baryon and M a pseudoscalar meson. The angular distribution of B in the rest system of Y with respect to some reference axis y is given by [10,11] W (cos θ B ) = 1 4π (1 + α P y cos θ B ).
The polarisation P y = P y (cos θ Y ) carries information about the production mechanism and the YȲ final state interaction. The decay asymmetry α is independent of the production mechanism. It is proportional to the real part of the product of the interfering parity violating and parity conserving decay amplitudes [12]. The imaginary part of the product is denoted β and is accessible in sequential hyperon decays, i.e. hyperons that decay weakly into other hyperons [13,14]. This is also true for the phase angle φ between the amplitudes. Equation 1 demonstrates an example of how parameters with physical meaning, such as α and P y , can be retrieved from measurable quantities like cos θ B and cos θ Y . This feature makes hyperons a powerful diagnostic tool.

The BESIII Experiment
The BEijing Spectrometer (BESIII) [15] at the Beijing Electron Positron Collider (BEPC-II) offers unique opportunities to explore strange and single-charm hyperons. Hyperon-antihyperon pairs can be produced either in one-photon exchange processes e + e − → γ * → YȲ by off-resonance energy scans, or from vector charmonium decays, e.g. e + e − → J/Ψ → YȲ . The BESIII detector covers 93% of the 4π solid angle. A small-cell, helium-based main drift chamber (MDC) surrounding the e + e − collision point provides precise tracking of charged particles. The BESIII detector also comprises a time-of-flight system (TOF) based on plastic scintillators, an electromagnetic calorimeter (EMC) made of CsI(Tl) crystals, a muon counter (MUC) made of resistive plate chambers, and a superconducting solenoid magnet with a central field of 1.0 Tesla.
The work presented here is based on a low-energy scan from 2015, including a large sample at 2.396 GeV, J/Ψ data from 2009 and 2012, Ψ (3686) data from 2009 and 2012 and a high-energy scan from the + c¯ − c threshold near 4.6 GeV, collected during 2014.

Hyperon Structure
The strong interaction dynamics between quarks in a composite system can be quantified by various structure observables. Among these, the electromagnetic form factors (EMFFs) have the advantage that they are exper-imentally accessible for both protons, neutrons and hyperons. Space-like EMFFs, probed in elastic electronbaryon (e − B → e − B ) scattering, are straight-forward to access for the stable protons. The space-like electric G E and magnetic G M form factors are related to the charge-and magnetisation densities, respectively [16]. For the unstable hyperons on the other hand, space-like EMFFs are hard to access experimentally. Instead, the time-like EMFFs constitute the most viable structure observables for hyperons [17]. These can be probed in e + e − → γ * → B 1B2 reactions, provided the squared momentum transfer q 2 carried by the virtual photon γ * is larger than (M B1 + M B2 ) 2 . The experimentally accessible time-like and the intuitive space-like EMFFs are related via dispersion relations [18]. Whereas space-like EMFFs are real functions of q 2 , the time-like ones are complex. For spin 1/2 baryons, this means that the electric and the magnetic form factor have a relative phase ΔΦ [19]. As |q 2 | approaches a scale q 2 asy , the time-like and the space-like EMFFs should converge to the same real value. Hence, their phase approaches an integer multiple of π [20,21]. From the asymptotic behaviour of the EMFF phase, we therefore get independent information about the space-like region. But how do we measure this phase?
In the e + e − → YȲ reaction, a non-zero phase manifests itself in a polarised final state, even if the initial e + e − state is unpolarised [19]. The polarisation has a well-defined dependence on the hyperon scattering angle and depends on sin ΔΦ [22]. In an experiment, we can access the polarisation from the angular distribution of the decay products, according to Eq. 1. In Sect. 5.1, we will demonstrate how this has been exploited in a recent BESIII measurement.

Search for CP Violation in Hyperon Decays
A long-standing explanation of the matter-antimatter asymmetry is that it has been generated dynamically, through Baryogenesis [23]. This, however, requires the existence of processes that violate charge conjugation and parity (CP) conservation. Such processes have been incorporated in the SM by the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [24,25], and they are experimentally established in the meson sector [10]. However, these violations are so small that the resulting matter-antimatter asymmetry would be eight orders of magnitude smaller than the observed one [26,27]. This raises the question whether physics beyond the SM is at play. Spin-carrying baryons provide an additional angle to CP violation since spin behaves differently than momentum under parity flip [28]. Nevertheless, no indication of CP violation has been observed for baryons.

Hyperon Structure
At energies that do not coincide with a vector charmonium resonance, one-photon exchange (e + e − → γ * → YȲ ) dominates the hyperon-antihyperon production. The cross section of this process is then related to the effective form factor G e f f in the following way: where , α E M is the electromagnetic coupling constant and β the Lorentz factor. Furthermore, the parameter η from Eq. 2 is related to the modulus of the form factor ratio, R = G E /G M : The phase ΔΦ then represents the relative phase between the electric and the magnetic form factor. The BESIII Collaboration has applied this formalism on a data set corresponding to an integrated luminosity of 66.9 pb −1 collected at an e + e − CMS energy of q = 2.396 GeV. We have performed an exclusive selection of ¯ final states, which resulted in a sample of 555 events. From this, we were able to extract the form factors at q = 2.396 GeV, as summarised in Table 1. In particular, the phase ΔΦ was measured for the first time, and found to be 37 o ± 12 o ± 6 o . The first uncertainty is statistical and the second systematic. This is the first measurement of its kind in the time-like region for any baryon and corresponds to a "snapshot" of a ¯ pair in the making [30].

Search for CP Violation in Hyperon Decays
CP symmetry implies that the decay patterns of hyperons and antihyperons are the same, but with inverted spatial coordinates. Quantitatively, this means that the decay parameter of the hyperon, α Y equals that of the antihyperon, αȲ , but with opposite sign. Hence, we can construct a CP observable The large amount of exclusively reconstructed hyperon-antihyperon pairs from J/Ψ → YȲ pairs allow for precise CP tests. For instance, the 1.3 · 10 9 J/Ψ events collected by BESIII during 2009 and 2012 resulted in ≈420 000 exclusively reconstructed ¯ events and ≈88,000 +¯ − events. The decay parameters α Y and αȲ can be obtained by applying the formalism of Eq. 2 on these samples. In this way, the BESIII Collaboration has performed the most precise CP tests so far for the [31] and the + [32] hyperons. The resulting asymmetries were found to be A C P = −0.006 ± 0.012 ± 0.007 and A C P = −0.004 ± 0.037 ± 0.010, consistent with CP conservation. The results on decay parameters and the so-called psionic form factor phase ΔΦ Ψ [22] are shown in Table 2. It is noteworthy that ΔΦ Ψ is negative for J/Ψ → +¯ − while positive for Ψ (3686) → +¯ − .

How to interpret psionic form factors in terms of physics is an open question.
A remarkable finding was that the decay parameter of the hyperon was found to be 17% larger than the previous world average, based on proton polarimeter experiments in the 1970s. A re-analysis of CLAS data [33] revealed a value more consistent with that obtained by BESIII than the old world average. The Particle Data Group [10] has now abandoned the old measurements in favour of a world-average based on recent measurement using the decay distribution method similar to the one in Ref. [31].
Neither for the + nor for the hyperon revealed any indication of CP violation. However, the SM as well as BSM models predict deviations from CP symmetry that are smaller than the precision that can be achieved by BESIII. The future PANDA experiment at FAIR could therefore be important in future searches for CP violation [34,35].

Sequentially Decaying Hyperons
Multi-strange and charmed hyperons decay through a multi-step, or sequential, process with lighter hyperons in the intermediate steps. This gives access not only to the decay parameter α but also to β and φ. These parameters, as well as other spin properties, can be studied also when it is only possible to reconstruct either the hyperon or the antihyperon decay chain. This is referred to as the single-tag approach and is particularly suitable for studies of charm hyperons. The latter can decay into many different final states and hence, the branching fraction of each channel is small.
In a recent BESIII measurement, based on 0.5 fb −1 at a e + e − CMS energy of 4.6 GeV, we presented a proof-of-concept for measuring the decay parameters of the + c [36]. Furthermore, we performed the first experimental determination of the + c spin, and found that the spin 1/2 hypothesis was favoured over the spin 3/2 hypothesis by more than 6σ [37].
The triple-strange − hyperon has been studied via the decay chain − → K − , → pπ − and the corresponding chain for the¯ + . For this purpose, a sample of ≈ 450·10 6 Ψ (3686) events, containing 2507 − and 2238¯ + candidates, was used. Here, the general formalism of Ref. [38] was applied and two hypotheses were tested: − having spin 1/2 and 3/2, respectively. The spin 3/2 turned out to be favoured over the spin 1/2 hypothesis by 14σ [39], which is the most precise, model-independent spin test for the − hyperon so far. In addition, the helicity amplitudes were measured for the first time. The helicity amplitudes are the spin 3/2 equivalent of the polarisation.

Polarised, Entangled and Sequentially Decaying Hyperons
Since hyperon decays occur through an interplay between the strong and weak interactions, the decay amplitudes are quantified in terms of strong and weak phase differences. For spin 1/2 hyperons, these are denoted δ P − δ S and ξ P − ξ S , respectively [40,41]. The strong phase difference is CP symmetric while a non-zero weak phase difference implies CP violation.
The aforementioned CP observable A C P depends on a first order approximation on these phases in the following way: The sensitivity of A C P to CP violation therefore depends on the magnitude of the strong phase difference. In the case of the double-strange Ξ − hyperon, it has been found to be close to zero [42]. As a consequence, the A C P asymmetry would be vanishingly small even in the event of CP violation. However, by measuring the decay parameters β or φ, the strong and weak phase differences can be separated: where Δφ CP = (φ Y + φȲ )/2. Combining polarised, entangled and sequentially decaying hyperon-antihyperon pairs in the analysis, gives straight-forward access to α Y , β Y and φ for both hyperons and antihyperons. This enables a separation of the strong and weak phase difference, which increases the sensitivity to CP violation by several orders of magnitude. The full process e + e − → Ξ −Ξ + , Ξ − → π − , → pπ − + c.c. has been studied using ≈ 73,000 exclusively reconstructed Ξ −Ξ + pairs from BESIII. These Ξ −Ξ + pairs were identified from a sample of 1.3·10 9 J/Ψ collected in 2009 and 2012. Nine angles were measured: the scattering angle of the Ξ − hyperon in the e + e − CMS system, the (¯ ) helicity angles in the Ξ − (Ξ + ) rest system and finally the proton (antiproton) helicity angles in the (¯ ) rest system. The formalism derived in Ref. [38] relates these nine measured angles ξ = (θ, θ , ϕ , θ , ϕ , θ p , ϕ p , θ p , ϕ p ) to the eight physical parameters ω = (α ψ , ΔΦ, α Ξ , φ Ξ , α Ξ , φ Ξ , α , α ) by the following expression: Here, the C μν = C μν (θ ; α ψ , ΔΦ) is a 4×4 spin density matrix of the Ξ −Ξ + production, while the matrices a Y μν represent the propagation of the spin in the sequential decays and are parameterised in terms of the weak decay parameters α Y and φ Y . The eight independent production and decay parameters were estimated using a Maximum Log Likelihood fit applied on Eq. 8. In addition, the polarisation and spin correlations were estimated independently using the Method of Moments. The results from the two approaches agree.
From the results, three independent CP tests could be constructed, based on the asymmetries A Ξ C P , A C P and Δφ Ξ . Most importantly, the weak phase difference ξ P − ξ S could be determined for the first time in a direct measurement. It was found to be (ξ P − ξ S ) = (1.2 ± 3.4 ± 0.8) · 10 −2 radians, which is consistent with CP conservation. The results also include an independent measurement of the strong phase difference δ P − δ S and of the decay parameter α . The results are summarised in Table 3.

Summary
The experimentally accessible spin properties of hyperons make them excellent diagnostic tools for various phenomena, in particular the strong interaction at the scale where quarks form hadrons, as well as fundamental The psionic form factor phase ΔΦ Ψ , the decay parameters α Ξ , φ Ξ and α of the Ξ − → π − , → pπ − decay chain and the corresponding antihyperon decay parameters α Ξ , φ Ξ and (α ). In addition, the C P asymmetries A Ξ CP , Δφ Ξ CP and A CP are presented as well as the average φ Ξ . The first and second uncertainties, are statistical and systematic, respectively symmetries. Polarised and entangled hyperon-antihyperon pairs enable a complete determination of the timelike hyperon structure, by measurement of production parameters such as form factor ratio and phase. This has been done in a recent paper by BESIII, where the form factors were completely determined for the first time. Furthermore, applying the same methods on large samples from the decay of vector charmonia into hyperon-antihyperon pairs, a CP test can be constructed from the decay parameters. Recent studies by BESIII provide the most precise meausurements so far for and + . Sequentially decaying multi-strange and charm hyperons give access to additional decay parameters and offer a model-independent way to determine the hyperon spin. This has been exploited in recent BESIII studies of the triple-strange − and the charm + c . Combining the approach for polarised and entangled hyperon-antihyperon pairs that decay sequentially, provides a powerful tool to separate strong and weak contributions to the decay amplitudes. This increases the sensitivity to CP violation by several orders of magnitude, as demonstrated in a recent BESIII measurement. The result is consistent with CP conservation. However, the world-record sample of 10 10 J/Ψ events collected during 2018 and 2019 opens up new possibilities for future discoveries.