GoKawiilTransition form factors are fundamental hadron properties describing the dynamic behaviour of the transition between two states. The transition matrix element for a semileptonic weak decay of a hyperon is parametrized by six form factors, f i (q2) and g i (q2) (i = 1, 2, 3). These represent the vector and axial vector parts of the weak interaction, respectively, and depend on the squared momentum transfer, q2, of the intermediate W boson. The vector form factors are commonly denoted as the vector f 1 (q2), weak magnetism f 2 (q2) and scalar f 3 (q2) form factors. The axial-vector form factors are further subdivided into the axial g 1 (q2), weak electricity g 2 (q2) and pseudoscalar g 3 (q2) form factors6. Each form factor is associated with a specific current and encodes information about the underlying hadronic structure. Because both the scalar and the pseudoscalar form factors are suppressed by the squared ratio of the lepton to hyperon mass, the electron–antineutrino decay modes effectively depend only on four form factors and V us , as depicted in Fig. 1.
Fig. 1: Schematic representation of the Λ decay to a proton, an electron and an electric antineutrino through virtual (mediating) W− boson. Full size image The pie chart shows the relevant magnitudes of the dominant vector and axial-vector form factors partaking in the transition.
When neglecting the q2 dependence of the form factors, the branching ratio \({\mathcal{B}}\) of the semileptonic decay \(\Lambda \to {{\rm{pe}}}^{-}{\bar{
u }}_{{\rm{e}}}\) is given by6:
$${\mathcal{B}}=\frac{{\tau }_{\Lambda }}{\hbar }{G}_{{\rm{F}}}^{2}|{V}_{{\rm{us}}}{|}^{2}\frac{{\beta }^{5}{{M}_{\Lambda }}^{5}}{60{{\rm{\pi }}}^{3}}\left[\left(1-\frac{3}{2}\beta \right)(\,{f}_{1}^{2}+3{g}_{1}^{2})-4\beta {g}_{1}{g}_{2}+\frac{2}{7}{\beta }^{2}{{\mathcal{F}}}_{2}+{\mathcal{O}}({\beta }^{3})\right],$$ (1)
in which β = (M Λ − M p )/M Λ ≈ 0.159, with M Λ and M p denoting the masses of the Λ and proton, respectively. The weak decay constant G F and the Λ lifetime τ Λ are known with high accuracy4. The form-factor-dependent function \({{\mathcal{F}}}_{2}\) is defined as:
$${{\mathcal{F}}}_{2}=3{f}_{1}^{2}+3{f}_{1}{f}_{2}+2{{f}_{2}}^{2}+6{{g}_{1}}^{2}+6{g}_{2}^{2}+21{g}_{1}{g}_{2}.$$ (2)
The ratios g 1 /f 1 (throughout this paper, f i ≡ f i (q2 = 0) and g i ≡ g i (q2 = 0) are implied, unless explicitly noted), f 2 /f 1 and g 2 /f 1 represent the axial-vector (g av ≡ g 1 /f 1 ), weak-magnetism (g w ≡ f 2 /f 1 ) and weak-electricity (g av2 ≡ g 2 /f 1 ) couplings at zero momentum transfer q2, respectively. Therefore, a precise measurement of the decay branching fraction can be used to determine |V us | if the form factors are known. The relative couplings g av , g w and g av2 can be determined from kinematic angular variables, but a reliable theoretical determination of f 1 requires an understanding of the subtle differences between the d and s quarks7. In this context, approximate flavour SU(3) symmetry serves as a good approximation, as confirmed by previous experimental results8,9,10. This assumption treats the s and d quarks as equivalent, with symmetry breaking arising from their small mass difference relative to both M Λ and M p . For instance, f 1 is protected from leading-order SU(3)-breaking effects11. However, corrections are essential at next-to-leading order. There are various model estimates of f 1 using quark models12,13, large-N c (refs. 14,15,16,17), chiral expansions18,19,20 and QCD sum rules21. A drawback is that these models disagree on the size of the SU(3)-breaking corrections, which affects the precision of |V us | (ref. 22). During the past decade, the lattice QCD community has joined the efforts of determining f 1 using a non-perturbative method5,23,24 and a model-independent approach25 developed for the form factors of meson decays26,27. Still, understanding the quark mass dependence of SU(3)-breaking remains a crucial issue.
Another way to determine the vector form factor is indirectly through the g av coupling and g 1 form factor. The axial-vector coupling has been measured experimentally8,9,10, with the most precise determination, g av = 0.719 ± 0.016 ± 0.012, performed more than 30 years ago by a Fermilab-based fixed-target experiment using a neutral-hyperon beam10, which observed approximately 37 × 103 \(\Lambda \to {{\rm{pe}}}^{-}{\bar{
u }}_{{\rm{e}}}\) candidates. The average value based on the Particle Data Group gives g av = 0.718 ± 0.015 (ref. 4). In contrast to f 1 , the axial-vector form factor is not protected by the Ademollo–Gatto theorem11. Thus, SU(3)-breaking corrections must be considered at leading order. During the past several years, the g 1 of hyperon beta decays have been calculated with high accuracy from first principles using the techniques of lattice QCD28. Although the hyperon axial form factor can be determined from the lattice, there is at present not enough experimental information. The weak-electricity coupling, g av2 , is intimately linked to SU(3) symmetry. In the absence of second-class currents—which are characterized by their G-parity asymmetric transformation—it vanishes in the SU(3) limit29. This quantity has so far only been determined once by the KTeV experiment for the process \({\Xi }^{0}\to {\Sigma }^{+}{{\rm{e}}}^{-}{\bar{
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