Radiative hyperon decays 1

Radiative Hyperon Decays
Revised July 2011 by J.D. Jackson (LBNL).
The weak radiative decays of spin-1/2 hyperons, Bi → Bf γ, yield information about
matrix elements (form factors) similar to that gained from weak hadronic decays. For apolarized spin-1/2 hyperon decaying radiatively via a ∆Q = 0, ∆S = 1 transition, theangular distribution of the direction p̂ of the final spin-1/2 baryon in the hyperon rest
frame isdN
N=(1 + αγ Pi · p̂) .
(1)dΩ4π
Here Pi is the polarization of the decaying hyperon, and αγ is the asymmetry parameter.
In terms of the form factors F1 (q 2 ), F2 (q 2 ), and G(q 2 ) of the effective hadronic weakelectromagnetic vertex,
F1 (q 2 )γλ + iF2 (q 2 )σλµ q µ + G(q 2 )γλ γ5 ,αγ is
∗ (0)]
2 Re[G(0)FM,(2)|G(0)|2 + |FM (0)|2where FM = (mi −mf )[F2 − F1 /(mi +mf )]. If the decaying hyperon is unpolarized, thedecay baryon has a longitudinal polarization given by Pf = −αγ [1].
The angular distribution for the weak hadronic decay, Bi → Bf π, has the same form
as Eq. (1), but of course with a different asymmetry parameter, απ . Now, however, if thedecaying hyperon is unpolarized, the decay baryon has a longitudinal polarization given
by Pf = +απ [2,3]. The difference of sign is because the spins of the pion and photon
are different.

αγ =Ξ0 → Λγ decay— The radiative decay Ξ0 → Λγ of an unpolarized Ξ0 uses the hadronicdecay Λ → pπ − as the analyzer. As noted above, the longitudinal polarization of the Λwill be PΛ = −αΞΛγ . Let α− be the Λ → pπ − asymmetry parameter and θΛp be theangle, as seen in the Λ rest frame, between the Λ line of flight and the proton momentum.
Then the hadronic version of Eq. (1) applied to the Λ → pπ − decay gives
NdN=(1 − αΞΛγ α− cos θΛp )(3)d cos θΛp2
for the angular distribution of the proton in the Λ frame. Our current value, from theCERN NA48/1 experiment [4], is αΞΛγ = −0.704 ± 0.019 ± 0.064.
Ξ0 → Σ0 γ decay— The asymmetry parameter here, αΞΣγ , is measured by following
the decay chain Ξ0 → Σ0 γ, Σ0 → Λγ, Λ → pπ − . Again, for an unpolarized Ξ0 , thelongitudinal polarization of the Σ0 will be PΣ = −αΞΣγ . In the Σ0 → Λγ decay, aparity-conserving magnetic-dipole transition, the polarization of the Σ0 is transferredto the Λ, as may be seen as follows. Let θΣΛ be the angle seen in the Σ0 rest frame
between the Σ0 line of flight and the Λ momentum. For Σ0 helicity +1/2, the probabilityamplitudes for positive and negative spin states of the Σ0 along the Λ momentum arecos(θΣΛ /2) and sin(θΣΛ /2). Then the amplitude for a negative helicity photon and anegative helicity Λ is cos(θΣΛ /2), while the amplitude for positive helicities for the photon
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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Radiative hyperon decaysand Λ is sin(θΣΛ /2). For Σ0 helicity −1/2, the amplitudes are interchanged. If the Σ0has longitudinal polarization PΣ , the probabilities for Λ helicities ±1/2 are thereforep(±1/2) =11(1 ∓ PΣ ) cos2 (θΣΛ /2) + (1 ± PΣ ) sin2 (θΣΛ /2) ,22(4)and the longitudinal polarization of the Λ isPΛ = −PΣ cos θΣΛ = +αΞΣγ cos θΣΛ .

(5)Using Eq. (1) for the Λ → pπ − decay again, we get for the joint angular distributionof the Σ0 → Λγ, Λ → pπ − chain,Nd2 N=
1 + αΞΣγ cos θΣΛ α− cos θΛp .
d cos θΣΛ d cos θΛp4(6)Our current average for αΞΣγ is −0.69 ± 0.06 [4,5].
References:
1. R.E. Behrends, Phys. Rev. 111, 1691 (1958); see Eq. (7) or (8).
2. In ancient times, the signs of the asymmetry term in the angular distributions ofradiative and hadronic decays of polarized hyperons were sometimes opposite. For
roughly 50 years, however, the overwhelming convention has been to make them thesame. The aim, not always achieved, is to remove ambiguities.
3. For the definition of απ , see the note on “Baryon Decay Parameters” in the NeutronListings.
4. J.R. Batley et al., Phys. Lett. B693, 241 (2010).
5. A. Alavi-Harati et al., Phys. Rev. Lett. 86, 3239 (2001).

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