Volume 47B, number 1 PHYSICS LETTERS 15 October 1973 PION MASS AND THE CABIBBO ANGLE J.L. CHKAREULI and I.V. PAZIASHVILI Institute of Physics, Academy of Sciences of the GeorgianSSR, Tbilisi, USSR Received 7 August 1973 The relation between the Cabibbo angle and the pion mass is analysed in the framework of the Glashow-Weinberg type theory with explicit SU(2) and strangeness breakings, in which the strangeness conservationlimit corresponds to massless pions (without any restrictions on the intraisotopic mass splittings of mesons). In this way we find sin 2 0 = 0r+F~+/K°FKo)(Z~r+/ZKo)-I/2 in good agreement with the experimental value for Z~ = Z K. In accordance with the view of Gell-Mann, Oakes and Renner (GOR) [ 1] the hadron Hamiltonian is approximately invariant under chiral SU(2) × SU(2) with the conservation of the vector-(V) and partial conservation of the axial-vector(A) currents (PCAC). The presence of an explicit non-electromagnetic SU(2) breaking has recently attracted much attention in connection with the determination of the Cabibbo angle [2-4] and isotopic mass splittings of 0 mesons [5], which as it turned out have corrections of the order of 20%, arising from SU(2) noninvariance of the vacuum [5]. In this connection it could be imag- ined that the violation of the isotopic symmetry also revises the partial conservation law * and the whole approximated SU(2) X SU(2) has two independent and of the same order symmetry breaking parameters proportional to the pion mass squared and to the iso- topic mass squared splittings of mesons (e.g., kaons), respectively. Oakes [4], following the GOR philosophy, sup- posed that the violation of the exact SU(2) × SU(2) in the PCAC manner induces the strangeness noncon- servation in weak interactions. His SU(2) X SU(2) limit determination of the zero Cabibbo angle 0 is in fact a two-limits determination. Instead, we suggest on the basis of the Glashow-Weinberg (GW) [7] type theory with explicit SU(2) and strangeness breakings, a more detailed analysis with a one-limit determina- tion, viz., the zero-pion-mass determination for the zero Cabibbo-angle without any restrictions on the isotopic mass splittings of mesons. Our analysis leads to * Some interesting results in this direction are offered by non- linear and algebraic realization of the SU(2) symmetry [6]. ~'+Fu+ [ Zlr+ t -1/2 sin20 - / -~---/ , (l) K°FK o \/-,K o where F+ and FKO are the lepton decay constants of zr + and K ° mesons, respectively; Z a = (01 pal 0), (a = zr +, K +) are the positive wave function renorma- lization constants, and ~r +, K° denote the squares of the particle masses. In conclusion we also discuss the alternative zero isotopic-mass splitting determination. Our Lagrangian is assumed to take the form £=£o+£ns+£s+s~mA +(J~SW;+h.c.), (2) where £o is the SU(3) × SU(3) invariant; £ns and £s are breakings proportional to local scalar and pseudo- scalar fields S i, Pi (_/= 0 ..... 8) transforming under the representation (3,3) + (5,3); -ulem' " -lnsu= v~ns + A~S are the electromagnetic and weak strangeness conserving currents, andA u and W u stand for photon and 14/- boson fields. Usually the nonconservation of strange- ness in weak interactions can be thought to arise from a rotation of the strangeness conserving weak currentsJ~ s in eq. (2) through 20 about the "7" vector direction (with generator F7) in the SU(3) × SU(3) space. It leads to the standard Cabibbo form V and A currents, separately: Vns ~--2- s v~Scos0 + V~ sin0 (3) A~ s ~AuS cos0 + A~ sin0. Wewould like to note, that for the (V+A) combi- nations of the weak currents (with GA =Gv) an anal- ogous form of the total currentY u may be obtained by means of its rotation round the "7" axial direction (with generator F~) too (or generally under a rotation 43