ASYMPTOTIC BEHAVIOR in Visual Studio .NET

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It may happen that there exist relations among the operators ON as a consequence of a specific
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dynamical scheme. As an example this is the case for the equations of motion. The bare connected Green functions of the regularized <p4 theory satisfy in euclidean space the equations of motion
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(13-156)
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with ( - 0 + millA the inverse of the regularized propagator. Take a derivative with respect to j(y) and let x ..... y. We obtain an identity which translated on the proper functions reads
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'1'(-
O+mi)A<p(x).b
b ~<p4(x).b - <p(x) b<p(x) f'b
(13-157)
The renormalized version of this relation mixes operators of dimensions two and four. As a consequence there exist functions ai(g) and b(g) such that we have, identically,
a,(g)f'~b<p(q; Pal + a2(g)f'~nl(q; Pal + a3(g)f'~~~)2(q; Pal + b(g)f'~;<p2(q; Pal
f'(n)(p', ,Pa
+ q, ,Pn)
(13-158)
In the massless theory, giving the dominant behavior the contribution proportional to b will vanish. From the Callan-Symanzik equations relative to the two sides of (13-158) we conclude that the matrix yii pertaining to operators of dimension four must have a zero eigenvalue.
A similar phenomenon occurs when one of the operators ON (or a combination of the ON) is the generator of a continuous symmetry, conserved current, energy momentum tensor, etc. The corresponding anomalous dimension vanishes. We have already encountered this case in electrodynamics, the consequence of which was the appearance of a unique f3 coefficient in Eq. (13-28) for the invariant charge. A more general situation was encountered in the (J model when a symmetry is softly broken, i.e., by terms of dimension d less than four in the lagrangian. We recall the result of Symanzik showing that the counterterms of dimension higher than d could be kept symmetric. In particular, wave-function renormalization is symmetric. Under such circumstances let Jt) be the bare current (of dimension three) and Do its divergence. The latter is of dimension smaller than four by hypothesis [see Eq. (11-3)]:
8,JI;
t5(XO - yO) [J8(x), CPo(y)]
TCPo(X)t5 4 (x - y)
(13-159)
with CPo the bare field being a vector in the internal space and representative of the generator. The Ward identity
8~<01 TJ8(X)CPO(Yl) CPo(Yn)IO)
the matrix
= <0 ITDo(x)CPO(Yl) ... CPo (Yn) I0)
<0 ITCPO(Yl) TCPO(Ya) CPO(Yn) I0) t5 4 (x - Ya)
(13-160)
QUANTUM FIELD THEORY
becomes in renormalized form
Z} 1 zn/2a~<01
TJ,Jx)cp(yd'" CP(Yn)IO>
Z 1 zn/2 <0 I TD(x)CP(Yl)' .. CP(Yn) I0>
+ zn/2
L <0 I Tcp(Yl) ... rcp(Ya)' .. CP(yn) I0> t54(X -
(13-161)
if we assume no anomaly. In writing (13-161) we have taken into account the fact that wave-function renormalization is symmetric, i.e., independent of the component of cp. Since the renormalized functions are finite it follows that Z1 = ZD is finite and a proper normalization consistent with (13-159) yields (13-162) As a consequence, exact or softly broken symmetries correspond to currents for which Y1=YD=0 (13-163)
For the divergence D to have a dimension smaller than four it must contain an explicit dependence on the massive parameters of the theory. In a fermionic theory, for instance, where chiral in variance is broken by a mass term of dimension three, the axial current conservation is softly broken. In the absence of anomalies (13-164) With a mass-independent renormalization, m depends on the dilatation factor A as in Eq. (13-96). Let Ym be the anomalous dimension of the operator !f/ljJ. According to (13-163) we shall have a a { - q. aq - Pa apa
+ Ym(g)m am + P(g) ag + 4 + Ym(g)
~ [da + Ya(g)] } r:~y5ifJ(q; Pa) = 0
(13-165)
The explicit dependence on m has contributed an extra term Ym, to be interpreted as the anomalous dimension of i!f/y 5 ljJ, equal therefore to the one of !f/ljJ : Yi1iy5ifJ = Y1iifJ This result is not surprising since both anomalous dimensions may be computed in the chirallimit where they obviously coincide.
Let us apply these ideas to hadronic symmetries and to their relation with the effective lagrangian of weak nonleptonic interactions. Consider a model for strong, electromagnetic, and weak interactions based on a gauge theory for a product group Gs @ Gw along the lines sketched at the end of Chap. 12. The group Gs will typically be the color group [for instance, SU(3)] and Gw will be spontaneously broken down to a U(l)Q phase group for the electric charge.
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