Effect of Mass Splittings on the Conserved Vector Current

Abstract

In order to explain the apparent lack of renormalization of the vector j3 -decay coupling constant, Feynman and Ge 11 -Mann and Gerstein and Zeldovich have suggested the idea of a conserved vector current. 1 The proposed current is strictly conserved only in the approximation in which one neglects small deviations from isotopic spin conservation such as those originating from multiplet mass splittings and from the usual electromagnetic corrections. At present, the effect of these deviations is of considerable interest in view of the existence of an apparent small discrepancy between the beta and muon decay coupling constants. 2 It is the aim of this note to prove some theorems, valid to all orders in the strong coupling, concerning the effect of these mass splittings on the effective vector £-decay matrix element.These corrections can most easily be calculated by writing an "effective" Lagrangian which consists of the usual free and strongly interacting charge-independent boson-fermion Lagrangian, L 0 , the weak decay interaction L w , plus terms involving operators which represent the multiplet mass splittings (and which, when taken between physical states, give the experimentally observed mass differences). These mass operators may be interpreted as arising from interactions which violate charge independence, such as the electromagnetic coupling, and represent the clothing by such a field of the fermion and boson propagators. 3 The total Lagrangian may be written w ^"^./^^^H-.H-^S 8 **^"^,^^**^-,^^2 3 ' jL 2 =(6m \°, ff ±7rt(T 3 2 "^T 2)ir + ^( 6OT S° S + + 6W V s -) s ( T 3 2 "^r 2)S ' (1) where 6m ab z m t : -m b , (Sm 2 ) ab =m a 2 -m b 2 .It should be noted that in isotopic spin space, L 0 , L 19 and L 2 behave as spherical harmonics Y 0°, Y ±°, and F 2°, respectively. The weak-interaction Lagrangian will be expressed in the form L w =gj^j^9where j^ is the strangeness-conserving part of a conserved charged vector current. For the following theorems, we will assume that the part of j^ constructed from the strongly interacting fields satisfies the condition 4 SiW,where g =ee^7 lT 2. it is not necessary to specify further the form of j^ In momentum space, the most general vector matrix element for /3 decay arising from L is of the form (3) where T^ is the vertex operator F = (ay +bq + co q )r , and p f and p are the four-momenta of the neutron and proton, q=p f -p is the four-momentum transfer, and bm x and 6m 2 denote the various mass differences which appear in L x and L 2 , respectively. The quantities a, 6, and c are invariant functions of q 2 and of 6m x and 6m 2 . The first theorem states that a and c must be even functions and b an odd function of Sm^ In order to 186

Supersymmetric contributions to weak decay correlation coefficients

“…Isospin breaking corrections imply deviations from unity of order a few 10 ÿ4 [29] (for an earlier estimate, see Ref. [30]). A two parameter fit to -decay data yields g A 0 =g V 0 1:272 93 46 [24], as- 3 The quantity is often denoted by in the literature.…”

Fermi matrix element with isospin breaking

“…In Ref. [5] it was shown that the difference between the radius of a neutron and a proton would grow with density and it was then claimed that this would lead to a violation of the Behrends-Sirlin-Ademollo-Gatto [6,7] (BSAG) theorem. This result, which was a consequence of an incorrect application of the constraint equations in the bag model, is wrong.…”

Vector form factor inKl3semileptonic decay with two flavors of dynamical domain-wall quarks

“…The Ademollo-Gatto theorem [14] states that SU (3) breaking effects are second order in (m s − m ud ). From previous theoretical studies, it turned out that the SU (3) breaking effects f + (0)−1 are order of 3 -5%.…”

“…A good theoretical control on f + (0) is provided by SU (3) symmetry and the Ademollo-Gatto theorem [14]. The vector current conservation guarantees f + (0) = 1 at zero momentum transfer [15], and then the Ademollo-Gatto theorem states that the SU (3) breaking effects f + (0)−1 start at second order in (m s −m ud ), where m ud represents the averaged mass of up and down quarks.…”