ASYMPTOTIC BEHAVIOR in .NET

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The additional term is a four-divergence
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~ <p + x . oX = 20" [X" <p2(X)] ~)<p X2 X2
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Formally (i.e., barring possible singularities) the action, and therefore the equations of motion, are conformally invariant.
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<p4 theory in five-dimensional space with dynamical variables defined on a unit hyperboloid (minkowskian case) or unit sphere (euclidean case). Write the corresponding lagrangian and equations of motion. Expand the solutions of the classical free-field equations in terms of generalized spherical harmonics. (b) Show that the variation of the action of a massive theory under a dilatation [Eq. (13-41)] may be written as the integral of the four-divergence of a dilatation current. The latter is related to a modified energy momentum tensor (such that its trace vanishes in the massless case) as follows:
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Exercises (a) Express the massless
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(13-46)
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For a discussion see the work of Callan, Coleman, and Jackiw. (c) Investigate scale and conformal invariance in the presence of Fermi fields.
13-2-2 Modified Ward Identities
We perform a Wick rotation and study the euclidean theory. For our problem of short-distance behavior this implies no restriction. We write the normalized generating functional as
eG(j) =
9&(<p) exp
{-1 + f
4 d Xj<P}
(13-47)
The euclidean action 1 may be split into
(13-48)
A scale transformation <p(x) -+ A<p(X) may be considered here as a simple change of integration variable, under which
The change in the measure is absorbed into the normalization and we recover
QUANTUM FIELD THEORY
the naive Ward identity in the form
X <5J(x) ~) <5J(x) f d4 {m2 [~]2 + j(x) (1 + x . ax ~} eG(j) - (j = 0) = 0
(13-49)
The term in (<5j<5N could be replaced by a derivative with respect to a source for the <p2 operator. Of course, we expect Eq. (13-49) to be modified because of renormalization effects. In most of the cases encountered previously, such as gauge invariance or global symmetries, we were able to display a regularization and renormalization scheme preserving the symmetry and hence the Ward identities. But chiral anomalies have been the warning that modifications may appear in the quantum case. Before displaying the aforementioned modifications let us disentangle the algebraic structure of (13-49) rewritten as
The Legendre transformation to irreducible functions reads in euclidean variables, Eq. (6-97):
G(j) - q<p) =
d4 xj(x)<p(x)
<5G(j)
<5j(x)
(13-51)
.( ) __ <5q<p) }x <5<p(x)
with
qx, y; <p) =
d4Z <5j(z) <5<p(y) <5<p(x) <5j(z)
= <54(x
implying that the kernels -<5 2G(j)j<5j(x)<5j(y) and <5 2q<p)j<5<p(x)<5<p(y) are the inverse of each other. To simplify the notations we denote them
<5 2q<p) <5<p(x)b<p(y)
<5 2G(j) (x, y; <p) = - <5j(x)<5j(y)
(13-52)
in such a way that Eq. (13-50) takes the form
4 d x
{[(I + x
<P(X)]
<5;~)
<p2(X)
+ m2[ -
+ r- 1 (x, x; <p) - r- 1 (x, x; 0)] }
(13-53)
ASYMPTOTIC BEHAVIOR
Expand (13-53) in powers of irreducible function.
to obtain the corresponding identity pertaining to the n-point
To zeroth order in h, (13-53) reduces to the trivial cases encountered in the classical discussion (up to the Wick rotation): (13-54)
f 4x[(I
+ x :x)<p(X)J:~:; =
f 4x[(1
d d4 m
+ x a~)<p(X)J ( -
O<p + m2<p + g ~:)
(13-55)
f x 2<p2(x)
Equation (15-53) is indeed satisfied since [r- l(X, x; <p) - r- l(X, x; 0)] is of order h, as is readily seen by restoring the factors h (r -+ r/h, %<p -+ h Ojo<p). To order one this quantity may be written
and, formally, (13-56) Since
it follows that
(13-57) and Eq. (13-53) would be verified, if it were not for the necessary ultraviolet subtractions.
The functional
r~(cp)=i f d4xm 2[ _cp2(X) + r-i(x,x;cp)- r-i(x,x;O)]
(13-58)
is interpreted as generating the irreducible Green functions with an insertion of the operator -(m 2j2) Sd4x cp2(X). This was clear at the level of Eq. (13-49), where the term m2 Sd4x [bjbj(x)] 2 arose from the path integral over m2 Sd4x cp2(X). We check it at order one in Eq. (13-57). The corresponding diagrammatic expansion appears in Fig. 13-4. We call it a mass Insertion. Power counting indicates that this insertion introduces logarithmic singularities in the two-point function at order hi. In operator language the relevant Green function is Sd4x <01 Tcp2(X)cp(y)cp(z) 10). Let us specify our normalization conditions. Until we are faced with difficulties, it will be convenient to use inter-
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