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Hyperons offer a unique tool to study some of the most challenging problems in contemporary physics. In this review, we will in particular demonstrate how polarized and entangled hyperon-antihyperon pairs can be exploited as a precision instrument to probe various aspects of the Standard Model (SM) of particle physics. Though immensely successful in describing the elementary particles and their fundamental interactions, the SM falls short in describing the properties of complex systems such as nucleons, nuclei or even the evolution and composition of our universe. Many puzzles in modern physics are related to the limitations of the SM and they fall into two main categories: those whose solution should be sought within the SM, and those where one needs to go beyond.
Within the SM, it remains a challenge to quantitatively describe the main building block of the visible universe: the nucleon. Its mass, size and structure emerge from the complex dynamics of the strong interaction. The theory of Quantum ChromoDynamics (QCD), an inherent part of the SM, describes these interactions from first principles, but its non-Abelian nature makes it difficult to make quantitative predictions at distances where quarks form hadrons. Replacing one or several light quarks in the nucleon with heavier strange or charm, gives deeper insights into these interactions.
Various experimental observations, such as neutrino oscillations, the need for dark matter to explain cosmological observations and the matter-antimatter asymmetry of the universe, raise the question of whether there is a more fundamental theory beyond the SM (BSM) that encompasses these phenomena. Intensive searches for BSM physics are ongoing at the high energy frontier, in a quest to find hitherto unknown, heavy particles manifesting a new fundamental theory. A complementary approach is the precision frontier: new aspects of particles whose existence is established since long. By studying the properties of these particles, e.g. measuring quantities that should either be zero, or that can be calculated with immense precision, physics beyond the SM would reveal itself by significant deviations from these predictions.
Hyperons are well suited for precision physics through their weak, parity violating decays, giving straightforward experimental access to their spin properties. This is in contrast to e.g. protons, for which dedicated polarimeter detectors are required for this purpose. In hyperon decays, the daughter particles are emitted according to the direction of the spin of the mother hyperon. For example, consider a two-body decay
$ Y \to B M $ , where Y is a spin 1/2 hyperon, B a spin 1/2 baryon and M a pseudoscalar meson as illustrated in Fig. 1. The angular distribution of B in the rest system of Y with respect to some reference axis$ \vec y $ is given by [1, 2]$ W(\cos\theta_B)=\frac{1}{4\pi}(1+\alpha P_y \cos\theta_B). $
(1) The polarization
$ P_y = P_y(\cos\theta_{Y}) $ carries information about the production mechanism, while the decay asymmetry α only depends on the decay process. The parameter α is proportional to the real part of the product of the interfering parity violating and parity conserving decay amplitudes [3]. The imaginary part is accessible in sequential hyperon decays, i.e. hyperons that decay weakly into other hyperons [4, 5]. Eq. (1) demonstrates an example of how physical parameters, such as α and$ P_y $ , can be retrieved from measurable quantities such as$ \cos\theta_B $ and$ \cos\theta_Y $ . This feature is crucial in the work presented in this review. -
Consider the production of a pair of a spin 1/2 hyperon and antihyperon in either the
$ e^+e^- \to \gamma^* \to Y\bar{Y} $ (Fig. 2, left panel) or$ e^+e^- \to \Psi \to Y\bar{Y} $ (Fig. 2, right panel), where Ψ is any vector charmonium resonance:$ J/\Psi $ ,$ \Psi(3686) $ ,$ \Psi(3770) $ etc. In$ e^+e^- $ annihilations, processes with spin-parity$ J^P = 1^- $ dominate completely which means that the spin-parity of the final state is considered to be known. In the case of an intermediate virtual photon$ \gamma^* $ , the process can be parameterised in terms of two complex electromagnetic form factors$ G_E(q^2) $ and$ G_M(q^2) $ [15, 31–33]. The form factors are functions of the momentum transfer$ q^2 $ carried by the virtual photon. The Born cross section$ \sigma_B $ is related to the effective form factor$ G_{\rm eff}(q^2) $ , a linear combination of$ G_E(q^2) $ and$ G_M(q^2) $ :Figure 2. (color online) Illustration of the process
$ e^+ e^- \to \gamma^* \to Y \overline{Y} $ , where$ \gamma^* $ denotes a virtual photon (left) and the$ e^+ e^- \to \Psi \to Y \overline{Y} $ , where Ψ a vector charmonium resonance (right).$\begin{aligned}[b] G_{\rm eff}(q^2) =& \sqrt{\frac{2\tau|G_M(q^2)|^2+|G_E(q^2)|^2}{1+2\tau}}\\=&\sqrt{\frac{\sigma_B \cdot 3q^2}{1+\dfrac{1}{2\tau}(4\pi\alpha^2\beta)}}, \end{aligned} $
(2) where
$ \tau=q^2/(4M_Y^2) $ , α is the electromagnetic fine structure constant and β is the velocity.The modulus of their ratio
$ R=|G_E(q^2)/G_M(q^2)| $ governs the scattering angle θ between the produced hyperon and the positron beam:$ \begin{aligned}[b] \frac{{\rm d}\sigma_B}{{\rm d} \Omega}=&\frac{2\pi\alpha^2\beta}{4q^2}\left(1+\cos^2\theta+\frac{R^2}{\tau}\sin^2\theta\right) \\=& \frac{2\pi\alpha^2\beta}{4q^2}(1+\eta\cos^2\theta), \end{aligned} $
(3) where
$ \eta = (\tau-R^2)/(\tau+R^2) $ . The relative phase$ \Delta \Phi $ of$ G_E(q^2) $ and$ G_M(q^2) $ governs the vector polarization and tensor polarization (i.e. spin correlations) of the produced hyperon-antihyperon pair [4]. If the beams are unpolarized, then a vector polarization of the final state hyperons is only allowed in the direction normal to the plane spanned by the incoming beam and the outgoing hyperon:$ P = \frac{\sqrt{(1-\eta^2)}\sin\theta\cos\theta}{1+\eta\cos^2\theta}\sin\Delta\Phi. $
(4) In addition, there are five non-zero spin correlations. In the case of an intermediate vector charmonium, the electromagnetic form factors are replaced by the so-called psionic form factors
$ G_{E}^{\Psi} $ and$ G_M^{\Psi} $ [4]. The formalism is the same as for an intermediate$ \gamma^* $ , but by convention, the η in Eq. (3) is denoted$ \alpha_{\Psi} $ and the phase$ \Delta\Phi_{\Psi} $ .If a spin 1/2-spin 3/2 hyperon-antihyperon pair is produced in an initial
$ J^P = 1^- $ state, an additional Coulomb quadrupole form factor$ G_C(q^2) $ [34] enters the parameterization. An equivalent description is to use three helicity parameters as shown in Ref. [35]. In a similar way, the production of a hyperon-antihyperon pair which are both spin 3/2 can be parameterized by$ G_E(q^2) $ ,$ G_M(q^2) $ ,$ G_C(q^2) $ and the octupole form factor$ G_O(q^2) $ , or equivalently by four helicity amplitudes [35, 36]. -
The self-analysing hyperon decays, mentioned in the introduction, provide a powerful diagnostic tool to study
● Structure functions, such as electromagnetic form factors.
● The spin of the mother hyperon.
● Parity violating and parity conserving decay amplitudes.
In the following, we will look into two-body hadronic decays of spin 1/2 or spin 3/2 hyperons decaying into a baryon and a pseudoscalar mesons.
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By direct decay, we consider the case when a hyperon Y either decays into a stable baryon B and a pseudoscalar meson M, or the case when the hyperon decays into another hyperon but where the subsequent decay of the daughter hyperon is not measured or where the daughter is a stable baryon. This process is schematically shown in Fig. 3. The production plane of the
$ e^+e^- \to Y\overline{Y} $ process is shown in the middle and the antihyperon decaying into a two-body hadronic state is shown to the left; to the mid-left in the centre-of-mass of the reaction and to the very left in the rest system of the decaying antihyperon.Figure 3. (color online) The reaction
$ e^+ e^- \to Y \overline{Y} $ which decays either hadronically ($ B M $ , to the left) or semi-leptonically ($ B W^{-} $ , to the right). Theplanes illustrate different reference frames, where those where particles are emitted back-to-back represent the rest frame, or the helicity frame, of the decaying particle.The decay of a spin
$ 1/2 $ baryon is described by a parity conserving (P-wave) and a parity violating (S-wave) amplitude. A spin 3/2 hyperon decay can be described in a similar way, but with the S-wave replaced by a D-wave. The real part of their products is the aforementioned$ \alpha_Y $ decay parameter from Eq. (1), while the imaginary part is denoted$ \beta_Y $ [3]. A third parameter,$ \gamma_Y $ , is defined by the difference of the squares of the parity conserving and parity violating amplitudes. These three so-called Lee-Yang parameters fulfil$ \alpha_Y^2 + \beta_Y^2 + \gamma_Y^2 = 1 $ . Furthermore, a parameter$ \phi_Y $ can be defined according to$ \begin{array}{*{20}{l}} \beta_Y = \sqrt{1-\alpha_Y^2}\sin\phi_Y, \; \; \; \gamma_Y = \sqrt{1-\alpha_Y^2}\cos\phi_Y. \end{array} $
(5) Hence, the decay is completely described by two independent parameters
$ \alpha_Y $ and either$ \beta_Y $ ,$ \gamma_Y $ or$ \phi_Y $ . The parameter$ \alpha_Y $ can be understood more intuitively as the longitudinal polarization induced by the decay, as illustrated in Fig. 4. The longitudinal polarization of the daughter will hence be$ \alpha_Y $ plus the transferred polarization from the mother. The latter depends on the decay angle of the daughter. Similarly,$ \beta_Y $ and$ \gamma_Y $ quantify the transversal polarization transfer.Figure 4. (color online) The daughter hyperon
$ Y_2 $ receives longitudinal [$ \alpha_{Y_1} + P_{Y_{1}}\cos\theta_{Y_2} $ ]$ \hat{\bf{z}} $ and transversal polarization components$ \beta_{Y_1}\hat{\bf{x}} $ and$ \gamma_{Y_1}\hat{\bf{y}} $ from the mother hyperon$ Y_1 $ .The formalism of the full process
$e^+e^- \to \gamma^{*}/\Psi \to Y\overline{Y}, Y \to BM, \overline{Y} \to \overline{BM}$ has been outlined in Refs. [19, 33] and demonstrates how the form factors and the decay asymmetry parameters$ \alpha_Y $ and$ \alpha_{\bar{Y}} $ can be accessed from the joint angular distribution of the decay products$ B, M, \overline{B} $ and$ \overline{M} $ . By so-called single-tag analysis, where either the hyperon or the antihyperon decay is measured while the undetected part is reconstructed from missing kinematics, the quantity$ \sin\Delta\Phi $ can be measured [33, 37]. Although the phase$ \Delta\Phi $ can only be determined with a$ \Delta\Phi/180^{\circ}-\Delta\Phi $ ambiguity, it still provides useful information about the structure [18].From the decay asymmetry parameters, we can construct a CP test: Exact CP symmetry implies equal decay patterns for particles and antiparticles, i.e.
$ \alpha_{Y} = -\alpha_{\bar{Y}} $ . Hence, the asymmetry$ A_{CP} = \frac{\alpha_Y+\alpha_{\bar{Y}}}{\alpha_Y-\alpha_{\bar{Y}}} $
(6) quantifies the degree of CP violation. The asymmetry
$ A_{CP} $ is straightforward to access for all weakly decaying ground-state hyperons. Strange hyperons have the advantage that they decay with an appreciable fraction into two-body hadronic states, hence it is feasible to collect large exclusive$ Y\overline{Y} $ data samples where the decays of the hyperon and the antihyperon can be studied simultaneously.The disadvantage of the
$ A_{CP} $ is that its sensitivity to CP violation is somewhat limited. This is because hadronic decays involve strong, CP conserving amplitudes as well as weak, possibly CP violating, amplitudes. For spin 1/2 hyperon decays, the former are quantified by strong phase shifts$ \delta_S $ and$ \delta_P $ in the final state$ BM $ interaction. Here, S and P denote S-waves and P-waves, respectively. The strangeness ($ |\Delta S|=1 $ ) and isospin ($ \Delta I = 1/2 $ ) changing transition in the$ Y \to BM $ decay induce weak, CP violating phases$ \xi_S $ and$ \xi_P $ . Hence, the phase difference$ \xi_P- \xi_S $ arises from the interference between S-waves and P-waves and if different from zero, it indicates CP violation. This is similar to kaon decays, where CP violation is induced by an interference between the isospin transition amplitudes$ \Delta I = 1/2 $ and$ \Delta I = 3/2 $ . Hyperon decays also involve both transitions but the$ \Delta I = 1/2 $ is believed to dominate [38]. Assuming this is the case, the weak, CP violating phase$ (\xi_P- \xi_S) $ can be extracted from the measurable asymmetry$ A_{CP} $ by [38–40]$ \begin{array}{*{20}{l}} A_{CP} \approx -\tan(\delta_P-\delta_S)\tan(\xi_P-\xi_s). \end{array} $
(7) However, we see that if there is no
$ BM $ final state interaction, i.e. if the difference$ (\delta_P - \delta_S) $ is close to zero, the$ A_{CP} $ would vanish even if$ (\xi_P- \xi_S) $ is non-zero. A separation of strong and weak contributions can hence increase the sensitivity to CP violation. As we will discuss next, this is possible through sequential hyperon decays. -
Multi-strange and charmed hyperons decay sequentially, i.e. into other unstable hyperons. These decay chains give access to the additional decay parameters β and ϕ [3]. A formalism for sequentially decaying hyperon-antihyperon pairs from
$ e^+e^- $ annihilations has been outlined for single-tag measurements in Ref. [4] and for exclusive measurement in Ref. [35]. The latter includes the decay chains of spin 3/2 as well as spin 1/2 hyperons and demonstrates how helicity amplitudes and decay parameters can be extracted from the joint angular distribution of the hyperons and their decay products.From the decay parameters β and ϕ, an additional CP test can be constructed:
$ \Delta \phi_{CP} = \frac{\phi_Y+\phi_{\bar{Y}}}{2}. $
(8) This test has the advantage that it, at first order, only depends on the weak phase difference whereas strong phases do not appear:
$ \Delta \phi_{CP} \approx \frac{\left< \alpha_Y \right>(\xi_P-\xi_S)_{LO}}{\sqrt{1-\left<\alpha_Y \right>^2}}, $
(9) where
$ \left<\alpha_Y \right> = \frac{\alpha_Y-\alpha_{\bar{Y}}}{2}. $
(10) This separation of the weak phase difference increases the sensitivity to CP violation by at least an order of magnitude [38].
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In hyperon beam experiments, where a high-energy hyperon is produced in
$ pp $ or$ pA $ interactions, one can integrate out most angles in the multi-step hyperon decay chain. In other words, one obtains a projection of the angular distribution on the final state baryon angle. This allows for the extraction of the product$ \alpha_{Y1}\alpha_{Y2} $ from the final state baryon in the rest frame of the intermediate hyperon$ Y_2 $ :$ \frac{{\rm d}n}{{\rm d}\cos\theta_{B}} = \frac{1}{2}(1 + \alpha_{Y1}\alpha_{Y2}\cos\theta_{B}). $
(11) The CP odd quantity
$A_{CP}^{\rm eff}$ can then be defined$ A_{CP}^{\rm eff} = \frac{\alpha_{Y1}\alpha_{Y2}+\alpha_{\bar{Y1}}\alpha_{\bar{Y1}}}{\alpha_{Y1}\alpha_{Y2}-\alpha_{\bar{Y1}}\alpha_{\bar{Y2}}} \approx A_{CP}^{Y1} + A_{CP}^{Y2}. $
(12) The advantage of
$A_{CP}^{\rm eff}$ is its simplicity, but it is also less sensitive to CP violation: It is possible that non-zero CP violations of hyperon$ Y_1 $ and$ Y_2 $ have opposite signs and therefore cancel out completely or partly. -
Radiative decays of hyperons, i.e.
$ Y_1 \to Y_2 \gamma $ , are either purely electromagnetic (if$ Y_1 $ and$ Y_2 $ have the same flavour and hence do not involve transitions between quark generations) or weak (if$ Y_1 $ and$ Y_2 $ have different flavour and hence involve transitions such as$ s \to u $ ). Radiative decays into nucleons, i.e.$ Y \to N\gamma $ are always weak. Electromagnetic decays, e.g.$ \Sigma^0 \to \Lambda \gamma $ , is expected to be parity-conserving, given that i) the$ \Sigma^0\Lambda\gamma $ coupling is related to the$ nn\gamma $ coupling by SU(3) flavour symmetry and ii) the upper limit of the neutron EDM is found to be extremely small [41]. Hence, the decay asymmetry$\alpha_{\rm EM}$ , defined by the real part of the product of the electric and magnetic transition dipole moments, should be very close to zero. The$\alpha_{\rm EM}$ is experimentally accessible through the angle between the photon from the$ \Sigma^0 \to \Lambda \gamma $ decay and the proton from the subsequent$ \Lambda \to p \pi^- $ decay. In Ref. [26], it was shown that the asymmetries$ \begin{array}{*{20}{l}} A^{\rm EM}_{CP} = \alpha^{\rm EM}_{Y}+\alpha^{\rm EM}_{\bar{Y}} \end{array} $
(13) indicates strong CP violation, while
$ \begin{array}{*{20}{l}} A^{\rm EM}_{C} = \alpha^{\rm EM}_{Y}-\alpha^{\rm EM}_{\bar{Y}} \end{array} $
(14) indicates C violation. Predictions from the SM of
$A^{\rm EM}_{CP}$ are of the order$ 10^{-14} $ , i.e. far below any foreseeable experimental resolution. However, this means that even tiny signals would indicate physics beyond the SM.Weak radiative decays, involving transitions between quark generations
$ s \to d\gamma $ , can be parameterized in a similar way as the weak hadronic decays, i.e. in terms of a decay parameter$ \alpha_{\gamma} $ . However, since the photon pseudoscalar meson spin are different, the polarization transfer has a different sign. This means that for unpolarized hyperons, the longitudinal polarization of the daughter baryon is$ -\alpha_{\gamma} $ in contrast to$ +\alpha_Y $ for a hadronic decay, recalling Fig. 4. Weak radiative decays contain additional complexity due to the involvement of strong, possibly parity violating processes. According to Hara's theorem , parity violating amplitudes are predicted to vanish in the limit of SU(3) flavour symmetry, under the assumption that the weak interaction is CP conserving and only includes left-handed currents. This implies that the weak radiative decay asymmetry$ \alpha_{\gamma} $ would be zero. This is in contrast to SU(3) breaking where we instead have$ \alpha_{\gamma}=\frac{m_{s}^{2}-m_{d}^{2}}{m_{s}^{2}+m_{d}^{2}}, $
(15) $ m_{s} $ and$ m_{d} $ being the mass of the strange and down quark, respectively. Constituent quark masses ($ m_{s}\sim 500 $ MeV and$ m_{d}\sim330 $ MeV) yields$ \alpha_{\gamma} $ around +0.4 and +0.5. This is similar to the$ b\to s\gamma $ transition, where one replaces$ m_{s} $ by$ m_{b} $ . Since$ m_{b} \gg m_{d} $ , the asymmetry parameter will then be close to +1. The$ b\to s\gamma $ transition has put constraints on physics beyond the SM that remove the chirality suppression. In a similar way, it is possible to investigate beyond SM effects in$ s\to d\gamma $ . -
Semi-leptonic decays of SU(3) octet hyperons can be parameterized in terms of four weak form factors, which in the limit of zero momentum transfer reduces to three: the vector
$ g_V(q^2) $ , the axial vector$ g_A(q^2) $ and the induced tensor$ f_2(q^2) $ where$ q^2 $ is the momentum transfer squared from the mother to the daughter hyperon [42, 43]. Since the momentum transfer in semi-leptonic hyperon decays is small compared to the baryon masses ($ q^2 \to 0 $ ), a systematic expansion can be performed in terms of the symmetry breaking parameter δ, defined by the mass difference of the mother and daughter hyperon [44]:$\delta = (M_{Y1}- M_{Y2})/M_{Y1}$ . This simplifies significantly the dependence of the decay rate on the form factors. The ratio between the vector and axial vector form factors can be written$ g_{AV} = \left|\frac{g_A(0)}{g_V(0)}\right|\cdot {\rm e}^{{\rm i}\phi_{AV}}. $
(16) The phase
$ \phi_{AV} $ quantifies the presence of a triple correlation term in the transition probability and a non-zero value would imply violation of time-reversal invariance [1]. The phase is accessible for polarized mother and/or daughter hyperons from the joint angular distribution of the decay [45]. The form factors from different channels are connected through the approximate SU(3) flavour symmetry.The semi-leptonic hyperon decays from hyperons produced in
$ e^+e^- $ annihilations can be analysed analogous to hadronic hyperon decays (see Fig. 3) where the scalar particle is replaced by the off-shell gauge boson$ W^- $ decaying into an electron-neutrino pair. The study of joint angular distributions typically require large data samples ($ N > 1000 $ ). Since semi-leptonic hyperon decays are typically rare compared to hadronic ones, it is challenging to collect the samples necessary for polarization studies. However, also for the modest samples it is possible to gain valuable insights, for example by constructing CP tests from the semi-leptonic decay rates of hyperons and antihyperons:$ A_{CP}^{\rm SLW} = \frac{\Gamma(Y_1 \to Y_2 l^-\bar{\nu}_l)}{\Gamma(\bar{Y}_1 \to \bar{Y}_2 l^+\nu_l)} $
(17) $ {\text{or }} $
$ A_{CP}^{\rm SLW} = \frac{{\rm BR}(Y_1 \to Y_2 l^-\bar{\nu}_l)-{\rm BR}(\bar{Y}_1 \to \bar{Y}_2 l^+\nu_l)}{{\rm BR}(Y_1 \to Y_2 l^-\bar{\nu}_l)+{\rm BR}(\bar{Y}_1 \to \bar{Y}_2 l^+\nu_l)}. $
(18) Furthermore, it is straight-forward to construct a test on lepton universality from the ratio
$ R_{\mu e} = \frac{\Gamma(Y_1 \to Y_2 \mu^-\bar{\nu}_{\mu})}{\Gamma(Y_1 \to Y_2 e^-\bar{\nu}_{e})}$
(19) of different lepton decay channels.
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Electromagnetic properties of hadrons have been studied in electron-hadron scattering ever since the 1950s [46] which has led to important insights on the structure and size of the proton [12, 13, 47, 48] and to some extent also the neutron [9, 8]. Due to experimental challenges, corresponding progress for hyperons is yet to come: so far, the only existing charge radius measurement is for the
$ \Sigma^- $ hyperon [10]. However, the advent of modern, high-intensity$ e^+e^- $ colliders, has ignited a veritable revolution for hyperon structure measurements. -
The BaBar experiment carried out a pioneering measurement of the Λ and
$ \Sigma^0 $ form factors and the$ \Sigma^0\Lambda $ transition form factor by exploiting Initial State Radiation (ISR) on a 230 fb$ ^{-1} $ sample collected at the bottomium resonance$ \Upsilon(4S) $ [49]. Though the resulting hyperon-antihyperon samples were small, they were sufficient to extract the energy dependence of the cross section and the corresponding effective form factor. The CLEO-c experiment studied Λ, Σ, Ξ and Ω production in the$ e^+e^- \to Y\bar{Y} $ reaction [50, 51] at energies corresponding to masses of the charmonium resonances$ \Psi(3686) $ ,$ \Psi(3770) $ and$ \Psi(4170) $ . The results indicated the importance of diquark correlations [52]. However, the interpretation in terms of electromagnetic form factors rely on the assumption that the$ Y\bar{Y} $ pair production is dominated by one-photon exchange ($ e^+e^- \to \gamma^* \to Y\bar{Y} $ ) with a negligible contribution from charmonium decay ($ e^+e^- \to \Psi \to Y\bar{Y} $ ). This is in contrast to a recent measurement from the BESIII collaboration [53], that shows that the branching fraction of the decay$ \Psi(3770) \to \Lambda\bar{\Lambda} $ is at least an order of magnitude larger than the prediction used in the CLEO-c papers [50, 51]. Furthermore, the BESIII results indicate that also other charmonium resonances have a significant impact on the$ Y\bar{Y} $ production. This, however, implies that hyperon-antihyperon production in the vicinity of vector charmonium resonances can be utilized for getting deeper insights into the properties of these resonances [54].Dedicated energy scan campaigns by BESIII have enabled increased precision in effective form factor measurements for the full spin
$ 1/2 $ octet. A low-energy scan near the$ \Lambda\bar{\Lambda} $ threshold revealed an unexpected enhancement, resembling a cusp or a peak [55]. Vector meson dominance, in particular with strong influence by the$ \phi(2170) $ , has been suggested as an explanation [56, 57]. Pioneering measurements have also been achieved for$ \Sigma^{\pm} $ [58],$ \Sigma^0 $ [59],$ \Xi^- $ [60, 61] and$ \Xi^0 $ [62]. The production cross sections of these hyperons in the$ e^+e^- \to Y\bar{Y} $ process from all BESIII measurements, are summarized in the left panel of Fig. 5. These results allow for a systematic analysis of the$ G_{\rm eff} $ behaviour, similarly to what has recently been done for protons [63–65] and neutrons [9]. It has been found [66, 67] that for nucleons, the effective form factor can be parameterised according toFigure 5. (color online) Left: The
$ e^+e^- \to Y\bar{Y} $ cross sections, measured by the BESIII experiment for different ground-state hyperons, as a function of the velocity β. Right: The effective form factor, after subtracting the dipole term, for neutral baryons studied by BaBar, BESIII and SND experiments, as a function of the square root of the$ e^+e^- $ CMS energy. The solid curve represents the fit to BESIII neutron data and the dashed a ft to BESIII and SND data.$ \begin{array}{*{20}{l}} G_{\rm eff}(q^2) = G_D(q^2) + G_{\rm osc}(q^2), \end{array} $
(20) where
$ G_D(q^2) $ is a dipole function and$ G_{\rm osc}(q^2) $ describes oscillations that are periodic in either$ q^2 $ or the relative momentum between the nucleon and antinucleon, p. The oscillations of the proton and neutron form factors have different amplitude and phase, but the same period [9] and have been attributed to nucleon-antinucleon rescattering into an intermediate hadron-antihadron state. In a recent paper [68], the$ G_{\rm eff} $ was analysed for neutral octet baryons$ n, \Lambda, \Sigma^0 $ and$ \Xi^0 $ using all available data. The study indicated a universal behaviour where$ G_{\rm osc}(q^2) $ can be parameterized with a common phase and amplitude, as indicated with a red dashed line in the right panel of Fig. 5. However, their conclusion is not supported by new low-$ q^2 $ neutron form factor from the SND experiment [69]. Fitting the oscillation model to both BESIII and SND neutron data results in a significantly lower oscillation frequency than that of the proton. The question remains if the observed oscillations in baryon form factors follow a pattern that can be explained within e.g. SU(3) symmetry and hence have a common origin, or if they are accidental.The structure of the charm baryon
$ \Lambda_c^+ $ has been studied by the Belle experiment using the ISR method [70] and by BESIII in an energy scan [71]. The Belle data revealed a peaking cross section near threshold, in contrast to BESIII where a plateau was seen. However, it should be pointed out that the differences between Belle and BESIII are within the experimental uncertainties. Attempts have been made to understand the threshold behaviour of the cross section in terms of vector charmonium resonances [72, 73] and final state interactions [74].The cross section of
$ \Omega^- $ production in$ e^+e^- \to \Omega^- \bar{\Omega}^+ $ has been studied in a recent analysis of energy scan data from BESIII [75]. Having spin 3/2, the$ \Omega^- $ effective form factor is interpreted as a linear combination of the electric, magnetic, quadrupole and octupole form factors [76]. The energy dependence is, within the uncertainties, consistent with the prediction from perturbative QCD and from Ref. [76]. -
The energy scan campaigns carried out by BESIII included particularly large data samples at 2.396 GeV, 2.64 GeV, 2.9 GeV, 4.57 GeV and 4.59 GeV. Using either double-tag (
$ e^+e^- \to Y\bar{Y} $ ) or single-tag ($ e^+e^- \to Y\bar{X} $ or$ e^+e^- \to \bar{Y}X $ , where X refers to an unmeasured hyperon) selection, sufficiently many events were found to reconstruct the hyperon scattering angle distribution. This enabled a separation of the electric and the magnetic form factors for Λ [77],$ \Sigma^+ $ [58] and$ \Lambda_c^+ $ [71], as summarized in Fig. 6. For Λ and$ \Lambda_c^+ $ , the$ G_E/G_M $ ratio was found to be close to unity near threshold, whereas for the$ \Sigma^+ $ , it was found to be significantly larger near threshold while approaching unity at larger energies.Figure 6. (color online) The
$ |G_{E}/G_{M}| $ of Λ,$ \Sigma^+ $ and$ \Lambda_c^+ $ hyperons as a function of the velocity β. The filled dots represent data from BESIII and the open dots BaBar.The large sample at 2.396 GeV, in combination with the relatively large
$ e^+e^- \to \Lambda\bar{\Lambda} $ cross section at this energy, resulted in more than 500 exclusively reconstructed$ \Lambda \to p\pi^-, \bar{\Lambda} \to \bar{p}\pi^+ $ events. This gave access to the joint angular distribution of the production and decays from which the relative phase$ \Delta\Phi $ between$ G_E $ amd$ G_M $ could be extracted. Hence, the time-like form factors could be unambiguously determined for the first time for any baryon – a complete snapshot from its time evolution [77]. This measurement gave rise to theory activity world-wide [54, 68, 78–84,]. In particular, a new method was outlined which allows for the extraction of the charge radius squared$ \left< r^2 \right> $ from polarization measurements [18]. Combining the R measurement from BESIII and BaBar with the$ \Delta\Phi $ measurement from BESIII, a first calculation of$ \left< r^2 \right> $ was carried out. However, since$ \Delta\Phi $ is only measured at one energy, the result turned out to be ambiguous and six different solutions were presented. Phase measurements in a broader energy range will be helpful in pinpointing the charge radius squared for the first time for a neutral hyperon. -
The large data samples collected by BESIII at charmonium resonances such as
$ J/\Psi $ ,$ \Psi(3686) $ and$ \Psi(3770) $ allow for a very precise determination of the psionic form factor phase$ \Delta\Phi_{\Psi} $ and the angular distribution parameter$ \alpha_{\Psi} $ , as shown in Table 1. It is found that$ \Delta\Phi_{\Psi} $ varies substantially between resonances and hyperon types. Summarising the results, we note thatFinal state $ {\cal B}(\times 10^{-4}) $ $ \alpha_\psi $ $ \Delta\Phi $ /radRef. $ J/\psi\to\Lambda\overline{\Lambda} $ $ 19.43 \pm 0.03 \pm 0.33 $ $ \phantom{-}0.4748 \pm 0.0022 \pm 0.0031 $ $ \phantom{-}0.7521 \pm 0.0042 \pm 0.0066 $ [90,91] $ \psi(3686)\to\Lambda\overline{\Lambda} $ $ 3.97 \pm 0.02 \pm 0.12 $ $ \phantom{-}0.82 \pm 0.08 \pm 0.02 $ $ \phantom{-} $ –[90] $ \psi(3770)\to\Lambda\overline{\Lambda} $ $>0.024 $ (90% CL)$ \phantom{-}0.85^{+0.12}_{-0.20}\pm 0.02 $ $ \phantom{-}1.24^{+1.15}_{-0.80} \pm 0.09 $ [53,92] $ J/\psi\to\Sigma^+\overline\Sigma\vphantom{X}^- $ $ 10.61 \pm 0.04 \pm 0.36 $ $ -0.508 \pm 0.006 \pm 0.004 $ $ -0.270 \pm 0.012 \pm 0.009 $ [93,85] $ \psi(3686)\to\Sigma^+\overline\Sigma\vphantom{X}^- $ $ 2.52\pm 0.04 \pm 0.09 $ $ \phantom{-}0.682 \pm 0.030 \pm 0.011 $ $ \phantom{-}0.379 \pm 0.070 \pm 0.014 $ [93,85] $ \psi(3686)\to\Lambda\overline\Sigma\vphantom{X}^0 $ $ 0.0160 \pm 0.0031 \pm 0.0059 $ – – [94] $ J/\psi\to\Sigma^-\overline\Sigma\vphantom{X}^+ $ – – – – $ \psi(3686)\to\Sigma^-\overline\Sigma\vphantom{X}^+ $ $ 2.82\pm 0.04 \pm 0.08 $ $ \phantom{-}0.96 \pm 0.09 \pm 0.03 $ – [89] $ J/\psi\to\Sigma^0\overline\Sigma^0 $ $ 11.64\pm 0.04 \pm 0.23 $ $ -0.449 \pm 0.020 \pm 0.008 $ – [90] $ \psi(3686)\to\Sigma^0\overline\Sigma^0 $ $ 2.44 \pm 0.03 \pm 0.11 $ $ \phantom{-}0.71 \pm 0.11 \pm 0.04 $ – [90] $ J/\psi\to\Xi^0\overline\Xi\vphantom{X}^0 $ $ 11.65 \pm 0.04 \pm 0.43 $ $ \phantom{-}0.66 \pm 0.03 \pm 0.05 $ – [95] $ \psi(3686)\to\Xi^0\overline\Xi\vphantom{X}^0 $ $ 2.73 \pm 0.03 \pm 0.13 $ $ \phantom{-}0.65 \pm 0.09 \pm 0.14 $ – [95] $ J/\psi\to\Xi^-\overline\Xi\vphantom{X}^+ $ $ 10.40 \pm 0.06 \pm 0.74 $ $ \phantom{-}0.586 \pm 0.012 \pm 0.010 $ $ \phantom{-}1.213 \pm 0.046 \pm 0.016 $ [96,87] $ \psi(3686)\to\Xi^-\overline\Xi\vphantom{X}^+ $ $ 2.78 \pm 0.05 \pm 0.14 $ $ \phantom{-}0.693 \pm 0.048 \pm 0.049 $ $ \phantom{-}0.667 \pm 0.111 \pm 0.058 $ [96,88] Table 1. Properties of the
$ e^+e^-\to J/\psi\to B\overline B $ decays to the pairs of ground-state octet hyperons.● The
$ \Sigma^+ $ phase is negative at the$ J/\Psi $ resonance [85] while positive for Λ [86] and$ \Xi^- $ [87].● The
$ \Sigma^+ $ phase changes sign when going from the$ J/\Psi $ to the$ \Psi(3686) $ [85].● The phase of
$ \Xi^- $ is significantly larger at$ J/\Psi $ [87] compared to at$ \Psi(3686) $ [88].● The angular distribution parameter
$ \alpha_{\Psi} $ of the$ \Psi(3686) \to \Sigma\bar{\Sigma} $ decay is significantly larger for$ \Sigma^- $ compared to$ \Sigma^+ $ and$ \Sigma^0 $ [89].The interpretation of psionic form factors is still a matter of discussion, but the possibility to perform precision measurements calls for a theory framework in which the relation between psionic form factors and charmonium decay is outlined. Hopefully, this could provide a helpful tool to understand the properties of vector charmonia and their coupling to baryon-antibaryon states.
-
Searches for CP violation in the hadronic
$ \Lambda \to p\pi^- $ and$ \bar{\Lambda} \to \bar{p}\pi^+ $ have been carried out since the mid-1980s. The first was performed at the CERN Intersecting Storage Rings in the reactions$ pp \to \Lambda X $ and$ \bar{p}p \to \bar{\Lambda}X $ [97]. From the measured$ \alpha_{\Lambda} P_{\Lambda} $ and$ \alpha_{\bar{\Lambda}} P_{\bar{\Lambda}} $ , where P is the polarization, the quantity$ \alpha_{\Lambda} P_{\Lambda} / \alpha_{\bar{\Lambda}} P_{\bar{\Lambda}} $ was defined. Assuming$ P_{\Lambda} = P_{\bar{\Lambda}} $ , this quantity should be 1 when CP is conserved. However, it is straight-forward to construct a CP test similar to Eq. (6)$ A_{CP}^{\alpha P} = \frac{\alpha_{\Lambda}P_{\Lambda} + \alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}} {\alpha_{\Lambda}P_{\Lambda} - \alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}} \approx \frac{\alpha_{\Lambda} + \alpha_{\bar{\Lambda}}} {\alpha_{\Lambda} - \alpha_{\bar{\Lambda}}}. $
(21) The approximately
$ 10^4 $ events yielded the result$ A_{CP} = 0.02 \pm 0.14 $ . The DM2 collaboration measured the$ A_{CP} $ using the$ J/\Psi \to \Lambda\bar{\Lambda} $ process, however overlooking the possibility that the Λ may be polarized. As a result, terms in the joint angular distribution that are sensitive to$ \alpha_{\Lambda}/\alpha_{\bar{\Lambda}} $ differences cancel when being integrated over [98]. The PS185 collaboration at LEAR, CERN, provided the first essentially model-independent CP test by using the$ \bar{p}p \to \bar{\Lambda}\Lambda $ reaction and studying the quantity defined in Eq. (21) as a function of the$ \bar{\Lambda} $ scattering angle. This, in combination with the large data sample,$ 96\cdot10^3 $ exclusively reconstructed$ \Lambda\bar{\Lambda} $ events, resulted in a significantly improved precision [99] that remained unchallenged for more than 20 years.The large
$ J/\Psi $ samples from the BESIII collaboration –$ 1.3\cdot10^{9} J/\Psi $ during 2009 and 2012 and$ 10^{10} J/\Psi $ during 2018 and 2019 – started a new era in the search for CP violation in baryon decays. First, the 2009 and 2012 data resulted in$ \approx42\cdot10^4 $ exclusively reconstructed$\Lambda\bar{\Lambda} \to p\pi^-\bar{p}\pi^+$ events [86] and$\approx88\cdot10^3\; \Sigma^+\bar{\Sigma}^- \to p\pi^0\bar{p}\pi^0$ events [85]. This enabled the most precise Λ CP test so far and the first CP test for$ \Sigma^+ $ . In addition, the asymmetry parameter of the$ \bar{\Lambda} \to \bar{n}\pi^0 $ was measured for the first time [86]. A remarkable finding in the$ \Lambda \to p\pi^- $ decay study was that the asymmetry parameter was found to be 17% larger than the previous world average, mainly based on proton polarimeter experiments in the 1970s. A re-analysis of CLAS data [100] revealed a value more consistent with that obtained by BESIII than the old world average. The Particle Data Group [1] has now abandoned the old measurements in favour of a world-average based on Refs. [86] and [100]. A recent study of the$ J/\Psi \to \Xi^-\bar{\Xi}^+ \to \Lambda\pi^-\bar{\Lambda}\pi^+ \to p\pi^-\pi^-\bar{p}\pi^+\pi^+ $ provided an independent measurement of$ \alpha_{\Lambda} $ ,$ \alpha_{\bar{\Lambda}} $ and$ A_{CP} $ , consistent with previous results from BESIII. The recent$ \alpha_{\Lambda} $ measurements, as well as the previous PDG world average from older experiments, are summarised in Fig. 7.Figure 7. (color online) Asymmetry parameter values
$ \alpha_{\Lambda} $ from the PDG from 2018 [101] and from new measurements by BESIII [86, 87, 91], CLAS [100] and LHCb [102]. The PDG-2018 value is based on polarimeter measurements from the 1960s and 1970s. In the new BESIII measurements, the average value$ \left< \alpha_{\Lambda}\right> = (\alpha_{\Lambda} - \overline{\alpha}_{\Lambda})/2 $ is displayed. The * and ** denote the 2009-2012 and 2009-2019$ J/\psi $ samples, respectively.The precision of the
$ \alpha_{\Lambda} $ measurement from$ \approx73 \cdot10^3 \Xi^-\bar{\Xi}^+ $ [87] decays is about the same as the one achieved from the almost one order of magnitude larger$ J/\Psi \to \Lambda\bar{\Lambda} $ sample [86]. This is because Λ hyperons from$ \Xi^- $ decays have a larger degree of polarization than Λ hyperons from$ J/\Psi $ decays. Furthermore, Λ hyperons from a sequential decay can be polarized in several directions while those from a decaying$ J/\Psi $ can only be polarized along the production plane.Even more recently, the full
$ 10^{10} J/\Psi $ sample, including$ 3\cdot10^6 $ exclusively reconstructed$ \Lambda\bar{\Lambda} $ decays, has been analyzed for the purpose of achieving a high-precision CP test. The$ \Lambda \to p\pi^- $ decay asymmetry was found to be consistent with the previous measurement and the precision of$ A_{CP} $ is the most precise so far:$0.0025 \pm 0.0046 \pm 0.0012$ . -
Sequential decays of multi-strange and charmed hyperons give, in addition to the decay asymmetry
$ \alpha_Y $ , also access to β and γ or ϕ (see section IV.B.2). The studies of these parameters are not limited to exclusive, double-tag$ Y\bar{Y} $ measurements but can also be studied when it is only possible to reconstruct either the hyperon decay chain or the antihyperon ditto, i.e. using the single-tag approach.The world data bank [1] of the
$ \Xi^0 $ and$ \Xi^- $ hyperons is dominated by experiments of polarized or unpolarized secondary hyperon beams. Many of these apply the formalism in Eq. (11). The most precise measurement of the$ \alpha_{\Xi^0}\alpha_{\Lambda} $ was achieved by the NA48 experiment at CERN [103]. The$ \phi_{\Xi0} $ parameter was measured with large uncertainties at various experiments [104–106].The E756 collaboration at Fermilab achieved the best precision so far of the
$ \alpha_{\Xi^-}\alpha_{\Lambda} $ product and also measured the$A_{CP}^{\rm eff}$ defined in Eq. (12) for the first time [107]. Thanks to a dedicated measurement, the HyperCP collaboration improved the precision of$A_{CP}^{\rm eff}$ and their measurement is the most precise so far:$ (0.0 \pm 5.1 \pm 4.4)\times 10^{-4} $ [108].By studying the joint angular distribution of polarized
$ \Xi^- $ hyperons, the E756 [109] and HyperCP [110] collaborations also measured the$ \phi_{\Xi} $ parameter, from which the strong phase difference ($ \delta_P - \delta_S $ ) was extracted. These two measurements implied a significant improvement of the world data bank [104–106, 111–114]. Indeed, the ($ \delta_P - \delta_S $ ) was found to be very small, showing that even a CP violating weak phase would be difficult to detect from$ A_{CP} $ using Eq. (7).A recent measurement from BESIII demonstrates the merits of polarized, entangled and sequentially decaying hyperon-antihyperon pairs [90]. The full process
$ e^+e^- \to \Xi^-\bar{\Xi}^+, ~ \Xi^- \to \Lambda \pi^-,~ \Lambda \to p \pi^- + c.c. $ was studied using$ \approx73\cdot10^3 $ exclusively reconstructed, or double-tagged,$ \Xi^-\bar{\Xi}^+ $ pairs identified from a sample of$ 1.3\cdot10^{9} J/\Psi $ . The asymmetries$ A_{CP}^{\Xi} $ and$ A_{CP}^{\Lambda} $ were determined separately. From$ \Delta \phi_{\Xi} $ , the weak phase difference$ \xi_P - \xi_S $ has been determined for the first time in a direct measurement. The result,$ (\xi_P - \xi_S) = (1.2 \pm 3.4 \pm 0.8)\cdot10^{-2} $ rad, is consistent with CP conservation. It is also noteworthy that the strong phase difference$ \delta_P - \delta_S $ was measured with a similar precision as the one achieved by HyperCP [110] despite the 2000 times larger data sample from HyperCP. The excellent sensitivity achieved by the BESIII method can be attributed to the fact that the clean, well-known production process and the subsequent decays can be treated in a model-independent way: all angles are measured and no information is integrated out from the differential cross sections. Hence, all quantum correlations between the hyperon and the antihyperons, and their decay products, are taken into account, as well as the polarization transfer in multiple directions from the mother$ \Xi^- $ to the daughter Λ hyperon. Extending this analysis to the eight times larger data sample collected in 2018 and 2019 will further improve the precision and enable comparison with theory calculations of the strong phase from e.g. [115]. -
The triple-strange
$ \Omega^- $ hyperon should, according to the quark model, have spin 3/2. However, this was not empirically well-established until recently; the first measurements from SPS at CERN [116, 117] were based on very small data samples and therefore very simplified angular distributions. In a later measurement by BaBar,$ \Omega^- $ hyperons were studied in$ \Omega_c $ and$ \Xi_c $ decays and the conclusions relied on assumptions of the helicity of the mother baryons [118]. Recently, the BESIII collaboration applied the model-independent formalism outlined in Ref. [35] on single-tagged$ \Omega^- \to \Lambda K^-, \Lambda \to p \pi^- $ and$ \bar{\Omega}^+ \to \bar{\Lambda} K^+, \bar{\Lambda} \to \bar{p} \pi^+ $ from the$ e^+e^- \to \Psi(3686) \to \Omega^-\Omega^+ $ reaction. The sample contained$\approx450\cdot10^6 \;\Psi(3686)$ events, including 2507$ \Omega^- $ and 2238$ \bar{\Omega}^+ $ candidates. Two hypotheses were tested:$ \Omega^- $ having either spin 1/2 or 3/2. The spin 3/2 turned out to be favoured over the spin 1/2 hypothesis by 14σ [119], which is the most precise, model-independent spin test for the$ \Omega^- $ hyperon so far. This validates not only the quark model but also the results based upon its validity. In addition, the decay asymmetry parameter$ \phi_{\Omega} $ was measured for the first time.In the past, the decay asymmetries of the decays
$ \Omega^- \to \Lambda K^- $ ,$ \Omega^- \to \Xi^0 \pi^- $ and$ \Omega^- \to \Xi^- \pi^0 $ have been studied with a secondary hyperon beam at CERN [120]. More recently, the HyperCP experiment measured the decay asymmetry of$ \Omega^- \to \Lambda K^- $ and$ \bar{\Omega}^- \to \bar{\Lambda} K^+ $ and the CP asymmetry$ A_{CP} $ . The HyperCP employed a secondary Ω beam produced in$ p Cu $ collisions and since all Ω hyperons were selected to be collinear with the proton beam, they were unpolarized. This enabled the measurement of the$ \alpha_{\Omega}\alpha_{\Lambda} $ and$ \alpha_{\bar{\Omega}}\alpha_{\bar{\Lambda}} $ products. By combining these quantities with$ \alpha_{\Lambda} $ and$ \alpha_{\bar{\Lambda}} $ from other measurements, it was concluded that the$ \Omega^- \to \Lambda K^- $ decay conserve parity almost completely. Their measured values of$ A_{CP} $ as defined by Eq. (6) and the effective asymmetry$ A_{CP}^{\rm \it eff} $ by Eq. (12) were both found to be consistent with CP conservation. -
Charm hyperons can decay into many different final states and as a consequence, the branching fraction of each channel is small. Therefore, charm hyperon decays are almost exclusively studied using the single-tag approach.
Most measurements of the
$ \Lambda_c^+ $ decay parameters, by e.g. ARGUS [121], CLEO [122, 123] and FOCUS [124] collaborations only consider the angular distribution of the final state hadron, i.e. not the joint decay of the multi-step process. This means that the decay asymmetry$ \alpha_{\Lambda_c} $ can be retrieved, but not β and$ \gamma/\phi $ . In addition, the results rely on a precise knowledge of the decay parameter of the hyperon in the second step, for example Λ or$ \Sigma^+ $ . Nevertheless, the FOCUS collaboration performed the first CP test for a charmed baryon, by measuring$ A_{CP} $ for the$ \Lambda_c^+ \to \Lambda\pi^+ $ decay to be$ 0.07 \pm 0.19 \pm 0.24 $ .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 parameter$ \alpha_{\Lambda_c} $ , and for the first time also$ \beta_{\Lambda_c} $ and$ \gamma_{\Lambda_c} $ , of the$ \Lambda_c^+ $ decay into$ \Lambda\pi^+ $ ,$ \Sigma^+\pi^0 $ and$ \Sigma^0\pi^+ $ [125], all containing hyperons that decay subsequently. In addition,$ \alpha_{\Lambda_c} $ of the$ \Lambda_c^+ \to pK_S $ was measured for the first time, though with large uncertainties. Due to limited sample size, it was assumed that$ \alpha_{\Lambda_c} = - \alpha_{\bar{\Lambda_c}} $ i.e. CP symmetry. The BESIII collaboration has also carried out the first model-independent experimental determination of the$ \Lambda_c^+ $ spin. The analysis combined several sequential two-body decay chains and found that the spin 1/2 hypothesis was favoured over the spin 3/2 hypothesis by more than 6σ [126].The most precise measurements of
$ \alpha_{\Lambda_c} $ for several two-body decays have been achieved by the Belle experiment in the$ e^+e^- \to \Lambda_c^+ X + c.c. $ reaction [127]. The CP asymmetry$ A_{CP} $ has been calculated and though the statistical precision is comparable to that of strange hyperon decays, the total uncertainty is dominated by the systematics. The Belle [128, 129] and CLEO [130] experiments have also measured the$ \alpha_{\Xi_c} $ of several$ \Xi_c^0 $ decays. -
From Hara's theorem [27] and from the breaking of SU(3) symmetry in the quark model, on expects a positive value of the weak radiative decay parameter
$ \alpha_{\gamma} $ for decays involving$ s \to d $ transitions. One example of such a decay is$ \Sigma^+ \to p\gamma $ . Therefore, it came as a surprise when several experiments reported a large and negative value [131–134]. Other experiments reported similar values for the$ \Xi^0 \to \Lambda \gamma $ and$ \Xi^0 \to \Sigma^0 \gamma $ decay [135, 136, 103], in disagreement with older measurements [137, 138]. The discrepancy between theory predictions and experimental results sparked wide interests and various solutions to the puzzle were proposed [139–142]. It has been suggested that the$ \Lambda\to n\gamma $ can provide further insights [143, 144]: if positive, it indicates a violation of Hara's theorem.Polarized and entangled hyperon-antihyperon pairs from vector charmonium decays have the advantage that the polarization of the hyperon (antihyperon) can be measured with high precision if one considers the "leg" of the reaction where a hyperon (antihyperon) decays into a two-body hadronic final state. A simultaneous measurement of the other radiative weak decay "leg" of the antihyperon (hyperon) will then also be precise, thanks to constrained production parameters such as the polarization and spin correlations. This has been demonstrated in a recent study by the world-record data sample of
$ 10^{10} J/\Psi $ decays from BESIII. These data contain an appreciable fraction of$ J/\Psi \to \Lambda\bar{\Lambda}, \Lambda \to n\gamma, \bar{\Lambda} \to \bar{p}\pi^+ + c.c. $ events that can be exclusively reconstructed. The large transverse polarization of Λ [86] enabled the first determination of the decay parameter$ \alpha_{\gamma} $ of$ \Lambda\to n\gamma $ . It was found to be$ -0.16\pm0.10_{\rm stat.}\pm0.05_{\rm syst.} $ [145], indicating that Hara’s theorem still holds for hyperon radiative decays. The results provide essential input to constrain the amplitudes of parity violation contribution in theoretical calculations, and can be used to validate the predictions and constrain the parameters of the various theoretical models [146–148]. Recent calculations based on Chiral Perturbation Theory are in good agreement with the BESIII data [146]. -
The small branching ratios and the immense difficulty in detecting neutrinos make the study of semi-leptonic hyperon decays a challenging task. On the one hand, a semi-leptonic decay can be identified also without the neutrino thanks to the special characteristic of a three-body decay compared to the more abundant two-body hadronic decays. On the other hand, the angular analyses from which the physics is revealed, require all other particles in the decay to be accurately reconstructed. Nevertheless, experiments with polarized beams made pioneering measurements of
$ \Lambda \to p e^-\bar{\nu}_{e} $ in the 1970s [149–156], though with relatively modest sample sizes. Later measurements offered larger samples [157–159] and enabled measurements of the axial vector form factor ratio$ g_A/g_V $ for Λ [1]. More recently, the KTeV [160–162] and NA48 [163, 164] collaborations have studied the decay$ \Xi^0 \to \Sigma^+e^-\bar{\nu}_{e} $ , providing the form factor ratios$ g_1(0)/f_1(0) $ and$ f_2(0)/f_1(0) $ . In addition, the KTeV measured$ g_2(0)/f_1(0) $ .Semi-leptonic decays provide an independent way of measuring the CKM matrix element
$ |V_{us}| $ , in addition to kaon and τ decays. The current world-average of$ 0.2243\pm0.0008 $ is essentially calculated from$ K\to\pi l\nu(\gamma) $ and$ K\to\mu\nu(\gamma) $ decays [1]. Combining all available data on semi-leptonic hyperon decays provide comparable precision as for kaon decays and can thereby further improve the precision of the average. The fit of previous measurements of semi-leptonic hyperon decays yields$ |V_{us}|=0.2250\pm0.0027 $ [42]. Further theoretical and experimental progress should improve the precision in the semi-leptonic hyperon decays.The abundant production of polarized hyperons and antihyperons in the BESIII experiment, in combination with the versatile and near 4π detector, enables a new generation of semi-leptonic hyperon decay studies. Recent measurements have been carried out of decays such as
$ \Lambda\to p\mu^-\bar{\nu}_{\mu} $ [165] and$ \Sigma^+\to\Lambda e^+\nu_{e} $ [166]. In the latter study, the CP asymmetry defined in Eq. (17) was measured for the first time. Furthermore, searches have been performed for the expectedly rare$ \Xi^- \to \Xi^0e^-\bar{\nu}_{e} $ [167] and$ \Xi^0 \to \Sigma^-e^+\nu_{e} $ [168] decays. The results from the BESIII semi-leptonic studies are summarised in Table 2.$ Y\bar{Y} $ pairFinal state $ {\cal B}(\times 10^{-4}) $ $ N^{\mathrm{SLW}} $ $ R_{\mu e} $ $A_{CP}^{\rm SLW}$ Ref. $ J/\psi\to\Lambda\overline{\Lambda} $ $ \Lambda\to p\mu^-\bar{\nu}_{\mu} $ $ 1.48\pm0.21\pm0.08 $ $ 64\pm9 $ $ 0.178\pm0.028 $ [165] $ J/\psi\to\Xi^-\overline{\Xi}^+ $ $ \Xi^-\to\Xi^0 e^-\bar{\nu}_e $ $<2.59(90\%) $ $ - $ [167] $ J/\psi\to\Xi^0\overline{\Xi}^0 $ $ \Xi^0\to\Sigma^- e^+\nu_e $ $<1.6(90\%) $ $ - $ [168] $ J/\psi\to\Sigma^+\overline{\Sigma}^- $ $ \Sigma^+\to\Lambda e^+\nu_e $ $ 29.3\pm7.4\pm1.3 $ $ 16\pm4 $ $ 1.06\pm0.28 $ [166] Table 2. BRs measurement of semi-leptonic hyperon decays in the
$ e^+e^-\to J/\psi\to Y\overline Y $ decays at BESIII. The data statistics are$ 10^{10} J/\psi $ events.All aforementioned results are obtained by analyzing only the semi-leptonic decay itself, i.e. not considering the production process. The ability of the BESIII experiment to reconstruct the complete
$ Y\bar{Y} $ pair and its subsequent hadronic and semi-leptonic decays, opens up for a new method where the production is taken into account. The method outlined in Ref. [35] can be generalized to describe the joint angular distributions [169]. It allows for the extraction of spin correlations and polarization of the produced hyperon-antihyperon system which improves the precision of the semi-leptonic decay parameters Eq. (16).A data collection campaign during 2019-2021, where 4.5 fb
$ ^{-1} $ of data were collected between 4.6 GeV and 4.7 GeV, enabled a pioneering study of the differential decay rate and weak form factors of the semi-leptonic charm decay$ \Lambda_c^+\to\Lambda e^+\nu_e $ [170]. The results were compared to predictions from lattice QCD calculations [171]. By combining the measured branching fraction, the life-time$ \tau_{\Lambda_c} $ and the$ q^2 $ -integrated rate predicted by lattice QCD, the matrix element$ |V_{cs}| $ was determined to be$ |V_{cs}|=0.936\pm 0.017_{\mathcal{B}}\pm0.024_{\mathrm{LQCD}}\pm0.007_{\tau_{\Lambda_c}} $ . The semi-leptonic$ \Lambda_c^+ $ decay measurement provides an important consistency test for the SM and a probe for physics beyond the SM, complementary to D meson analyses [172, 173].Searches for violations of lepton flavour universality in hyperon decays have been studied in the numerous hyperon beam experiment in the 1970s [174–177]. The most precise measurement was performed in 1972 [177] and was reported as a relative
$ \Gamma(\Lambda\to p\mu^-\bar{\nu}_{\mu})/\Gamma(\Lambda\to N\pi)= (1.4\pm0.5)\cdot10^{-4} $ based on 14 signal events which were selected from about$ 6\cdot10^5 $ bubble chamber pictures. The current world average of the observable$ R_{\mu e} $ , defined in Eq. (19), is$ 0.189\pm0.041 $ for Λ [1]. The large experimental uncertainty is dominated by the$ \mathcal{B}(\Lambda\to p\mu^-\bar{\nu}_{\mu}) $ . Recently, the BESIII collaboration [165] reported a value of$ 0.178\pm0.028 $ based on 31 signal events obtained from the$ 10^{10} J/\psi $ sample. All results so far are consistent with the SM prediction, obtained at next-to-leading order, of$ 0.153\pm0.008 $ [28]. A more accurate measurement of$ R_{\mu e} $ will provide a more stringent test of lepton flavour universality. -
In the following, we will discuss the existing facilities BESIII and Belle II, as well as the planned PANDA and STCF, where polarized and entangled hyperon-antihyperon pairs can be studied.
-
Data from the energy scan performed by BESIII in 2014-2015 have resulted in several pioneering hyperon structure measurements so far [77, 59, 58], but the story does not end here. Several analyses of these data are still ongoing and the results will be published soon. These studies include Λ and
$ \Sigma^+ $ form factor phase measurements at several energies.A campaign to collect 20 fb
$ ^{-1} $ data at the$ \Psi(3770) $ , would enable precision studies of all the SU(3) octet hyperons using the ISR technique. Furthermore, a dedicated campaign has been carried out between CMS energies of 4.6 GeV and 4.95 GeV, with the purpose of a precision measurement of the$ q^2 $ dependence of the$ \Lambda_c^+ $ form factors.At the time of this review, the most precise Λ CP test so far has emerged from the world-record data sample of
$ 10^{10} $ events. However, data from the same sample are being analysed for high-precision CP tests in$ \Sigma^0 $ ,$ \Sigma^+ $ ,$ \Xi^- $ and$ \Xi^0 $ decays. Since most of these hyperons decay into an intermediate Λ, each of these studies will provide an independent measurements of the$ \alpha_{\Lambda} $ parameter. In addition, this and future BESIII samples will enable searches for rare hyperon decays with unprecedented precision. In addition to the aforementioned radiative and semi-leptonic decays, it is possible to search for rare or forbidden processes, for example those involving flavour changing neutral currents.The physics potential of BESIII will reach a new level with a planned upgrade of the BEPCII collider to a maximum beam energy of 2.8 GeV. This corresponds to a CMS energy of 5.6 GeV where the peak luminosity will be increased by a factor of three compared to what is possible at CMS energies below 4.0 GeV. As a result, BESIII will offer a great opportunity to study the production and structure of various charmed baryons, as well as their decay properties. In the left panel of Fig. 9, the integrated luminosities previous and planned data campaigns are summarized.
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The goal of the ongoing campaign with the Belle II experiment at SuperKEKB in Japan [184] is to collect in total 50 ab
$ ^{-1} $ of data in the bottomium region, with an intermediate milestone of 10 ab$ ^{-1} $ . This enables high-precision structure studies of strange and charmed hyperons through the ISR technique: although the radiator function in the ISR cross section results in a reduced effective luminosity (see right panel of Fig. 9). The huge integrated luminosity will eventually outperform other off-resonance samples from other$ e^+e^- $ colliders.The Belle II and its predecessor Belle have, with the data already on tape, performed pioneering measurements of charmed baryon decay asymmetry parameters, as well as CP tests. New data have been and are being collected by Belle II and can provide better statistical precision. In addition, larger samples enable studies of more complete decay chains, exploiting the richness of the angular distributions. This could reduce the systematic uncertainties which currently dominate the total uncertainty.
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The methods developed for hyperon-antihyperon pairs provide sensitive CP tests, but are currently limited by statistical precision. Improving the uncertainties by several orders of magnitude is not feasible with currently operating facilities world-wide, but requires next-generation experiments where polarized and entangled/quantum-correlated hyperon-antihyperon pairs are produced in abundance. The PANDA experiment at FAIR, Germany and the proposed STCF at USTC, China are two excellent candidates which will be presented in the following.
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The future experiment antiProton ANihilation at DArmstadt (PANDA) will produce hyperon-antihyperon pairs in abundance, thanks to the expected large production cross section in
$ \bar{p}p $ annihilations and the large luminosities enabled by the stored antiproton beam. The latter will be delivered from the High Energy Storage Ring (HESR) within a momentum range from 1.5 GeV/c up to 15 GeV/c and impinge on a hydrogen cluster or pellet target. During the first phase of operation, Phase One [185], the HESR will provide a luminosity of$ \approx 10^{31} $ cm$ ^2 $ s$ ^{-1} $ whilst the design luminosity of$ \approx 2\cdot10^{32} $ cm$ ^2 $ s$ ^{-1} $ will be achieved during Phase Three. Since PANDA will cover nearly$ 4\pi $ of the solid angle, the prerequisites for exclusive reconstruction of hyperon-antihyperon pairs and their subsequent decays are well fulfilled. This has been demonstrated by simulation studies in for example Refs. [186, 185], summarised in Table 3. We conclude that PANDA, already during the first phase of operation, will be able to collect samples several orders of magnitude larger than what is currently possible with BESIII. What does this mean in terms of the precision of the CP asymmetries?Reaction $ p_{\overline{p}} $ /(GeV/c)S/B Events / day, Phase One Events / day, full luminosity $ \overline{p}p \rightarrow \overline{\Lambda}\Lambda $ 1.64 114 $ 3.8\cdot 10^6 $ $ 7.6\cdot 10^7 $ $ \overline{p}p \rightarrow \overline{\Sigma}^0\Lambda $ 6.0 21 $ 4.3\cdot 10^5 $ $ 4.1\cdot 10^6 $ $ \overline{p}p \rightarrow \overline{\Xi}^+\Xi^- $ 4.6 274 $ 2.6\cdot 10^4 $ $ 5.2\cdot 10^5 $ In
$ \bar{p}p $ annihilations, many partial waves contribute to the production. This is in contrast to the$ e^+e^- $ case, where the spin and parity of the initial state is almost exclusively$ J^P = 1^- $ . Hence, in$ e^+e^- $ production, all production parameters such as polarization and spin correlations are well-defined functions of the hyperon scattering angle$ \theta_Y $ , governed by two global parameters$ \alpha_{\Psi} $ and$ \Delta\Phi $ that depend on the collision energy only. Hyperon-antihyperon pairs produced in$ \bar{p}p \to \bar{Y}Y $ reactions are quantum correlated rather than entangled since several possibilities exist for the initial spin-parity state. Nevertheless, in the case of spin 1/2 hyperons, the reaction can be parameterized in terms of five independent production parameters, each with an unknown$ \theta_Y $ dependence. Decay parameters and CP asymmetries, therefore, need to be extracted from global fits. However, as demonstrated in Ref. [99], the precision for a given sample size is about the same as in$ e^+e^- $ experiments [86]. The future CP tests by PANDA really have the potential to significantly improve the precision of the world data and put SM and BSM predictions to the test. -
The Super Tau-Charm facility (STCF) proposed in China is a next-generation electron-positron collider that will operate within a CMS energy range from 2.0 to 7.0 GeV [187]. The design luminosity is
$ 0.5\cdot10^{35} $ cm$ ^{-2} $ s$ ^{-1} $ or higher at the optimal CMS energy of 4.0 GeV. With this luminosity, it will be possible to collect 1 ab$ ^{-1} $ of data per year. At the$ J/\Psi $ mass, this corresponds to$ 3.4\cdot 10^{12} J/\Psi $ events per year, given a production cross-section of 3400 nb and a beam energy spread that equals that of BESIII. This opens up for high-precision CP tests in decays of hyperon-antihyperon pairs. Realistic projections of the precision can be obtained by scaling the statistical uncertainty achieved with BESIII with at least two orders of magnitude larger data sets expected with the STCF. The results are presented in Table 4 and are discussed in detail in Ref. [183].$ N_{J/\psi} $ $ \sigma(A_{CP}^{\Lambda}) $ $ \sigma(A_{CP}^{\Xi^-}) $ $ \sigma(B_{CP}^{\Xi^-}) $ BESIII $ 1.3\cdot10^{9} $ $ 1.0\cdot10^{-2\ast} $ $ 1.3\cdot10^{-2} $ $ 3.5\cdot10^{-2} $ BESIII $ 1.0\cdot10^{10\ast\ast} $ $ 3.6\cdot10^{-3} $ $ 4.8\cdot10^{-3} $ $ 1.3\cdot10^{-2} $ STCF $ 3.4\cdot10^{12\ast\ast} $ $ 2.0\cdot10^{-4} $ $ 2.6\cdot10^{-4} $ $ 6.8\cdot10^{-4} $ Table 4. Expected statistical uncertainty for the CP asymmetries
$ A_{CP}^{\Lambda} $ ,$ A_{CP}^{\Xi^-} $ and$ B_{CP}^{\Xi^-} $ at BESIII and the proposed STCF. BESIII has already published measurements on$ A_{CP}^{\Lambda} $ [86, 91] and$ A_{CP}^{\Xi^-} $ [87], the former with the full$ 10^{10} J/\Psi $ data sample. The projected uncertainties for future measurements at BESIII and STCF are obtained by scaling the statistical uncertainties with the number of events. The$ \ast $ and$ \ast\ast $ indicate the combination of BESIII measurements [86, 91] and projected precision, respectively.Future upgrades may include even higher luminosities, a longitudinal polarization of the electron beam and a reduced beam energy spread. It is estimated that an electron beam polarization of about
$ 80\% $ at$ J/\psi $ energies can be obtained without reducing the beam current [188]. This polarization is much larger than the observed natural transverse polarization of hyperons and antihyperons (about 30% at its maximum) from$ J/\psi $ decays and will increase the sensitivity of most CP tests in hyperon decays [183].The beam energy spread can be reduced by a collision scheme where electrons (positrons) with higher momenta collide with positrons (electrons) with lower momenta [189–191], similar to the Belle II experiment at SuperKEKB [184]. Reducing the CMS energy spread compared to what is currently possible with BEPC-II can further increase the data quality. In particular, if the spread approaches the natural width of the
$ J/\psi $ ,$ \rm{\Gamma}=90 $ keV, the number of the$ J/\psi $ events produced with a given integrated luminosity will increase. Furthermore, the off-resonance background can be reduced which would improve the sensitivity to rare processes.
Production and decay of polarized hyperon-antihyperon pairs
- Received Date: 2023-03-04
- Available Online: 2023-05-15
Abstract: Polarized hyperon-antihyperon pairs shed light on various unresolved puzzles in contemporary physics: How the strong interaction confines quarks into hadrons, how accurately the Standard Model describes microcosmos and even why our universe consists of so much more matter than antimatter. Thanks to their weak, parity violating decays, hyperons reveal their spin properties. This can be exploited e.g. the decomposition of the electromagnetic structure of hyperons, precision tests of flavour symmetry and searches for CP violation. At the BESIII experiment at BEPC-II, Beijing, China, hyperon-antihyperon pairs can be produced in abundance. Recently collected large data samples have triggered the development of new methods that provide unprecedented precision and a plethora of new results have emerged. When applied at future high-intensity facilities like PANDA and STCF, precision physics will be taken to a new level which can contribute to the solution to the aforementioned puzzles.