O"=O'-v in .NET framework

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and requiring that a' has zero vacuum expectation value. Reexpressed in terms of a' the complete lagrangian reads
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!l' = Ii![i
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+ gv + g(O" + in' -rYs)JI/I + H(an)2 + (aO")2J
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(11-149) The effect of the translation has been threefold. The mass degeneracy of meson fields has disappeared. On inspection these masses now read
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m; = /1 2
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m; =
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+ Av2 /1 2 + 3Av 2
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(11-150)
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Furthermore, the fermion has acquired a mass equal to (11-151) Finally we find a new trilinear coupling O"nn. The vacuum expectation value v will be constrained to satisfy a complicated condition <a') = O. The best which can be done is to implement it perturbatively by requiring that tadpole diagrams for the transition 0"--* vacuum vanish (Fig. 11-12). The Born approximation to this condition is, from (11-149),
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/12V -
Av 3 = 0
SYMMETRIES
Figure 11-12 Tadpole amplitude.
If we return to Eq. (11-117) we are led to the following identification:
j"m;; = - c = - v(p,z
Therefore,
+ Av2) = -
vm;;
j,,= -v
(11-152)
= gj"
which is the Goldberger-Treiman relation to this order where GA/G V = l. When c goes to zero two distinct situations may arise. One possibility is that v also goes to zero, in which case we find the normal mode of symmetry described above with massless nucleons. An alternative possibility arises when p2 < 0, and the limit corresponds to
p,z A
(11-153)
This is the Goldstone phenomenon studied in Sec. 11-2-2, with a vanishing pion mass
m;; = p2
+ Av2 =
In this phase the (J model may be used to derive the low-energy theorems for pion-pion or pion-nucleon scattering.
11-4-2 Renormalization
The lagrangian 2((J', n, !/J) obtained above after translation of the (J field, or its version in the limit c = corresponding to the Goldstone mode, is renormalizable in the sense of power counting. All the monomials in the interaction lagrangian are of dimension smaller or equal to four; the same is true of the possible counterterms. It remains to show that the lagrangian plus its counterterms has a similar form as in (11-149), a remnant of the original structure (11-144). In particular, what will be the fate of the PCAC relation (11-148) We shall show that these properties will be preserved by performing the renormalization in the symmetric normal phase and proving that this is sufficient to treat the cases of explicit (c i= 0) or spontaneous (c = 0, p,z < 0) symmetry breaking. To simplify matters we omit the fermion fields. A complete treatment does not reveal any new difficulty. We also use compact notations with ljJ a multiplet of n fields transforming according to the vector representation of a symmetry group O(n). In the previous instance n was equal to four. We write the lagrangian
QUANTUM FIELD THEORY
5e=5es +c ljJ 5e s = i(oljJ)2 _ J12 ljJ2 _
~ (ljJ2)2
(11-154)
5es is therefore invariant under the transformations
(11-155) where the 'Fj% are the representatives of the infinitesimal generators of the group, in the present case n by n antisymmetric real matrices 'Fj% + 4} = o. It is convenient first to regularize the theory in an invariant manner. This may be done, for instance, by modifying the kinetic term into the form
f d4x (oljJ)2 = f d4x ljJ( -
O)ljJ -+
f d4x ljJO(1 + a ~ + b ~: + .. -)ljJ
leading to a propagator with a behavior smooth enough at large momentum to insure the convergence of all Feynman integrals. This regularization will be understood in the sequel without being explicitly written out. Consider now the generating functional for connected Green functions in the symmetric theory
eGs(j) =
f .@(ljJ)exP{i f d4x C5es(ljJ) +j.ljJJ}
Gs(j) = Gs[j + bw(Tj)J
(11-156)
As a consequence of the in variance of the (regularized) lagrangian under the transformations (11-155), Gs(j) satisfies
(11-157a)
or, equivalently,
. () X
4/ bit{x)
bGs(j)
(11-157b)
In order to exhibit the structure of the divergences we need a similar identity for irreducible Green functions obtained after the Legendre transformation
ir s(ljJ) + i
d4x j.ljJ
Gs(j) (11-158) bGs(j)
c/>k(X) = ~( ) lVJk X
Since conversely
. Jk(X) brs(ljJ)
= - bc/>k(X)
we derive that rs(ljJ) enjoys the same in variance properties under the transformations (11-155):
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