80. Baryon Decay Parameters

180. Baryon Decay Parameters
Revised August 2019 by E.D. Commins (UC Berkeley).

80.1 Baryon semileptonic decays
The typical spin-1/2 baryon semileptonic decay is described by a matrix element, the hadronicpart of which may be written as:hi

B f f1 (q 2 )γλ + i f2 (q 2 )σλµ q µ + g1 (q 2 )γλ γ5 + g3 (q 2 )γ5 qλ Bi .

(80.1)

Here Bi and B f are spinors describing the initial and final baryons, and q = pi − pf , while theterms in f1 , f2 , g1 , and g3 account for vector, induced tensor (“weak magnetism”), axial vector, andinduced pseudoscalar contributions [1]. Second-class current contributions are ignored here. In thelimit of zero momentum transfer, f1 reduces to the vector coupling constant gV , and g1 reduces tothe axial-vector coupling constant gA . The latter coefficients are related by Cabibbo’s theory [2],generalized to six quarks (and three mixing angles) by Kobayashi and Maskawa [3]. The g3 term
is negligible for transitions in which an e± is emitted, and gives a very small correction, which canbe estimated by PCAC [4], for µ± modes. Recoil effects include weak magnetism, and are taken
into account adequately by considering terms of first order inδ=mi − mf,
mi + mf(80.2)

where mi and mf are the masses of the initial and final baryons.
The experimental quantities of interest are the total decay rate, the lepton-neutrino angularcorrelation, the asymmetry coefficients in the decay of a polarized initial baryon, and the polarization of the decay baryon in its own rest frame for an unpolarized initial baryon. Formulae forthese quantities are derived by standard means [5] and are analogous to formulae for nuclear beta
decay [6]. We use the notation of Ref. [6] in the Listings for neutron beta decay. For comparison
with experiments at higher q 2 , it is necessary to modify the form factors at q 2 = 0 by a “dipole”
q 2 dependence, and for high-precision comparisons to apply appropriate radiative corrections [7].
The ratio gA /gV may be written asgA /gV = | gA /gV | eiφAV .

(80.3)The presence of a “triple correlation” term in the transition probability, proportional to Im(gA /gV )and of the form
σi · (p` × pν )(80.4)
for initial baryon polarization or
σf · (p` × pν )(80.5)

for final baryon polarization, would indicate failure of time-reversal invariance. The phase angle
φ has been measured precisely only in neutron decay (and in 19 Ne nuclear beta decay), and theresults are consistent with T invariance.

80.2 Hyperon nonleptonic decays
The amplitude for a spin-1/2 hyperon decaying into a spin-1/2 baryon and a spin-0 meson may
be written in the form
M = GF m2π · B f (A − Bγ5 ) Bi ,(80.6)
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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80. Baryon Decay Parameters2

where A and B are constants [1]. The transition rate is proportional tobf · ωb i + (1 − γ)(ωbf · nbi · nb )(ωb)
R=1+γω
bf · nbi · nbf × ωb i) ,b+ωb) + βnb · (ω+ α(ω(80.7)

b i and ωb f are unit
b is a unit vector in the direction of the final baryon momentum, and ω
where nvectors in the directions of the initial and final baryon spins. (The sign of the last term in the above
equation was incorrect in our 1988 and 1990 editions.) The parameters α, β, and γ are defined as

α = 2 Re(s∗ p)/( | s |2 + | p |2 ) ,β = 2 Im(s∗ p)/( | s |2 + | p |2 ) ,γ = ( | s |2 − | p |2 )/( | s |2 + | p |2 ) ,(80.8)

where s = A and p = | pf | B/(Ef + mf ); here Ef and pf are the energy and momentum of thefinal baryon. The parameters α, β, and γ satisfy
α2 + β 2 + γ 2 = 1(80.9)

If the hyperon polarization is PY , the polarization PB of the decay baryons isPB =b )nb + β(PY × nb) + γnb × (PY × nb)(α + PY · n.
b
1 + αPY · n(80.10)bHere PB is defined in the rest system of the baryon, obtained by a Lorentz transformation along nb and PY are defined.
from the hyperon rest frame, in which nAn additional useful parameter φ is defined byβ = (1 − α2 )1/2 sinφ .

(80.11)In the Listings, we compile α and φ for each decay, since these quantities are most closely related
to experiment and are essentially uncorrelated. When necessary, we have changed the signs ofreported values to agree with our sign conventions. In the Baryon Summary Table, we give α, φ,and ∆ (defined below) with errors, and also give the value of γ without error.
Time-reversal invariance requires, in the absence of final-state interactions, that s and p be
relatively real, and therefore that β = 0. However, for the decays discussed here, the final-stateinteraction is strong. Thuss = | s | eiδs and p = | p | eiδp ,(80.12)

where δs and δp are the pion-baryon s- and p-wave strong interaction phase shifts. We then have
β=−2 | s | | p |sin(δs − δp ) .
| s |2 + | p |2(80.13)

One also defines ∆ = −tan−1 (β/α). If T invariance holds, ∆ = δs − δp . For Λ → pπ − decay, thevalue of ∆ may be compared with the s- and p-wave phase shifts in low-energy π − p scattering, andthe results are consistent with T invariance.
See also the note on “Radiative Hyperon Decays” in this Review.
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380. Baryon Decay Parameters

References[1] E. Commins and P. Bucksbaum, Weak Interactions of Leptons and Quarks (1983), ISBN 9780-521-27370-1, (Cambridge, USA: University Press, 473p.).
[2] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).
[3] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[4] M. Goldberger and S. Treiman, Phys. Rev. 111, 354 (1958).
[5] P. Frampton and W.-K. Tung, Phys. Rev. D 3, 1114 (1971).
[6] J. Jackson, S. Treiman and H. Wyld, Nucl. Phys. 4, 206 (1957).
[7] Y. Yokoo, S. Suzuki and M. Morita, Prog. Theor. Phys. 50, 1894 (1973).

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