Unitary Symmetry and Leptonic Decays Nicola Cabibbo CERN, Geneva, Switzerland April 1963

1. Unitary Symmetry and Leptonic Decays Nicola Cabibbo CERN, Geneva, Switzerland April 1963 Presented by: Stephen Bello, October 2013

2. My presentation today is on the analysis of leptonic decays based on the unitary symmetry for strong interactions in what is known as “eightfold way” and the V-A theory for weak interactions. • To start off we need to make some basic assumptions about the behavior of Jμ(the weak current of strong interacting particles): • Assumption the First: Jμ transforms according to the eightfold representation of SU3 • For this to be true we will need to neglect currents that have ΔS = - ΔQas well asΔI = 3/2 (Isospin Selection Rules). • This neglecting severely limits the scope of our analysis because we are now unable to look at the complex of K0leptonic decays or Σ+ => n + e+ + ν where the ΔS = - ΔQ currents have a non-zero role. • For all other processes we assume the hypothesis that the main contributions are from that part of Jμ which is in the eightfold representation. • Assumption the Second: The vector part of Jμ is in the same octet as the electromagnetic current. • If this is true than the vector contribution can be deduced solely from the electromagnetic properties of the strong interacting particles. For the case of ΔS = 0, this assumption is the same as vector-current conservation.

3. Along with the octet of vector currents, jμ, there is also the matter of the axial currents, gμ, which is also an octet. Each of these octets has two currents. Current #1 : ΔS = 0, ΔQ = 1, jμ(0), and gμ(0) Current #2 : ΔS = ΔQ = 1, jμ(1), and gμ(1) Of course there isospin selection rules are, respectively ΔI = 1 and ΔI = ½. Now looking back at our First Assumption we see that: Setting a = b = 1 will not ensure universality (equal coupling for all currents) because if Jμ is coupled then we can build a current which is not. What we do want is to keep a weaker form of universality by requiring the following Third Assumption: Assumption the Third: Jμ will have “unit length” or, in other words, a2 + b2 = 1. Now with this new assumption we can rewrite Jμ with a and b replaced with cos(θ) and sin(θ) respectively:

4. Determining the Value of θ Since Jμ is in the octet representation we have relations (θ is now a parameter) between the processes with ΔS = 0 and ΔS = 1. In order to determine θ we need to compare the decay rates of K+ =>μ++ ν with π+ =>μ++ ν. This, along with the experimental data, gives a value for θ of 0.257. We can also find this value by an independent investigation of K+ => π0 + μ+ + ν. The matrix element for this process can be connected to the one for π+ => π0 + μ+ + ν (which is from the Second Assumption). From the rate for K+ => π0 + μ+ + ν that was given to us by Sinclair, Brown and Glaser, we get a value for θ of 0.26. This is ~1.2% difference from the experimental value but due to two values coinciding within experimental errors we can use the simpler value of 0.26.

5. Examining the Leptonic Decays Now let us take a look at the leptonic decays of the baryons of the type A => B + e + ν. The matrix element for any member of an octet of currents that is one of two baryonic states (also octet members) can be easily represented in terms of two reduced matrix elements. Here jμ and gμ are once again the vector current octet and axial current octet respectively. The f and d components are from Gell-Mann’s 1961 paper. It turns out that is is sufficient to just consider only the allowed contributions and rewrite the reduced matrix elements as: Looking at the electromagnetic current connection we get the vector coefficients FO = 1 and FE = 0. From neutron decay we see that HO + HE = 5/4 = 1.25.

6. Examining the Leptonic Decays Now that we have values for all four vector coefficients we only have one parameter left. We can find this by examining the rate for Σ-=>Λ+ e-+ νbar. The relevant matrix element for this decay is: W. Willis reported in 1963 for this branching ratio to be 0.9 x 10-4. Using this we get a value for HEto be ± 0.95. We can ignore the negative solution for HEbecause using it will produce a larger branching ratio for the decay of the order of 1%. The positive value gives HE = 0.95 and HO = 0.30. This produces a cancellation of the axial contribution. The positive answer is also good because it explains why the experimental results in this modearemore depressed than the Λ => p + e- + νbar when it comes to Feynman and Gell-Mann’s predictions.

7. Conclusions Table 1 gives a summary of the predictions for the electron modes when ΔS = 1. There are five different decay modes with the branching ratio for each of them coming from Feynman and Gell-Mann’s 1958 paper. The Present Work values are derived from a paper published in 1962 by Gidal, Oswald, and Singleton. Note that the branching ratios for Λ => p + e- + νbar and Σ- => n + e- + νbar are in agreement with the experimental data from Gidal and associates, who discovered the branching ratios to be: Λ = (0.85 ± 0.3) x 10-3and Σ-= (1.9 ± 0.9) x 10-3.