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2m~ 1  G~ =   2 GA ng2 rrNN in .NET framework
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+ BSUS
(11143) with Uo transforming as a scalar under SU(3) and Us as the eighth component of an octet. A phenomenological analysis shows that Bs/Bo is close to .)2 rather than zero. This indicates that SU(2) x SU(2) rather than SU(3) would be a better approximate symmetry. 114 THE
MODEL
We shall now describe a field theoretic model originally introduced by GellMann and Levy (1960) as an example realizing chiral symmetry and partial conservation of the axial current. The name (J model originates in one of their notations. We take this opportunity to study the interplay of renormalization and symmetries. Following the ideas of Lee and Symanzik this enables us to introduce the machinery of Ward identities which will be useful in the study of gauge fields. SYMMETRIES
1141 Description of the Model
The a model involves a fermionic isodoublet field ljJ of zero bare mass, a triplet of pseudoscalar pions, and a scalar field a. The corresponding lagrangian is written !l' !l's
= !l's + ca
il/[i
+ g(a + in rys)]ljJ + H(on)2 + (oa)2] (11144) /1 A _  (a 2 + n 2) __ (a 2 + n 2
It is usually referred to as the linear model for a reason to be discussed later on. The part !l's(the indexS is for symmetric) is invariant under an SU(2) x SU(2) chiral group acting as follows. The right and left combinations ljJR = i(1 + Ys)ljJ, ljJL = i(1  ys)ljJ transform respectively according to the representations (t, 0) and (0, i) while the set (a, n) belongs to the (i, i) representation. To see this, we rewrite the coupling term
ljJ(a
+ in 'rYs)ljJ =
ljJL(a
+ in r)ljJR + ljJR(a  in r)ljJL
If (U, V) stands for an element of SU(2) x SU(2) with U and V varying independently, the transformation (a + in r) + V(a + in r)U 1 is an allowed one, leading to real a' and n' fields. Therefore if we perform simultaneously the isospinor rotations ljJ R+ U ljJ R, ljJ L + V ljJ L the interaction term is obviously invariant. So is the quantity a 2 + n2 proportional to the determinant of (a + in r). Finally, the kinetic term il/i ljJ is equal to il/Ri ljJ R + il/Li ljJ L, showing the in variance of the full lagrangian. The corresponding infinitesimal variations are generated by the chiral charges Q~.L with a running from one to three: [Q~, ljJR] = haljJR
[Q~, ljJL] = [Ql, ljJL] [Ql, ljJ R] haljJL
(11145) [Q~, a] = ~ n
[Ql, a] A compact notation for the (a, n) transformations is
(11146) a form which makes the in variance properties of the lagrangian easy to check. The Lie algebra of SU(2) x SU(2) is isomorphic to the one of its factor group 0(4) = SU(2) x SU(2)jZ2 and (a, n) transforms as a vector under 0(4). QUANTUM FIELD THEORY
In the absence of the breaking term conserved. They read
C(J, the vector and axial currents are
A~ =
Ii!Y/lYs
~IX 1/1 + (O'a/lnlX  (11147) nlXa/lO') The breaking term cO' leaves the diagonal SU(2) group unbroken. The vector current remains conserved and the axial current, the expression of which is not modified, acquires a non vanishing divergence (11148) This model therefore incorporates all the desirable features and the divergence of the axial current appears naturally as proportional to the pion field. The linear breaking term implies that the quantum a field has a nonvanishing vacuum expectation value <Or a [0) = v, meaning that a perturbation expansion must be performed taking into account the fluctuations of this field around the value v instead of zero. This is achieved by shifting the field as

