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SYMMETRIES
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a t5rS (4)) k(X)1k1 t5 I(X) = 0
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(11-159)
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Consequently all counterterms required by renormalization have to be symmetric, provided that the normalization conditions are. Indeed, assume the property true up to the L-Ioop order. Since !l's was the most general symmetric polynomial of degree four, the effect of counterterms could only renormalize the mass J1 and the coupling constant A, and affect multiplicatively (all components of) the field by the wave-function renormalization factor Z. Since !l's + f1!l'~L) enjoys the same invariance as !l's did, it follows that r computed with it up to order (L + 1) fulfils the same constraint (11-159). Its divergent part of order (L + 1) is the local symmetric polynomial generating the counterterms of order (L + 1). This inductive proof appears pedantic here. In more elaborate cases the parametrization of the transformation may be modified order by order by renormalization, but the framework described above remains useful. Consider now the complete lagrangian !l' including the linear breaking term e . 4>. A translation on 4>:
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4> = 4>' + v
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(11-160)
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is necessary in order to generate the correct perturbation theory. Omitting the prime, we have therefore
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G(j)
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S >>(4)) exp {i Sd4x [!l's(4) + v) + 4>. (j + e)]}
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S >>(4)) exp {if d4 x
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[!l's(4)
+ v) + 4>. eJ}
(11-161)
where the denominator insures that G(O) may undo the translation
eG(j) =
O. Since 4> is a dummy variable we
S >>(4)) exp {i Sd4 x [!l's(4)) + (4) - v) (j + e)]} S >>(4))exp {if d4 x [!l's(4)) + (4) - v) eJ}
G(j)
(11-162)
and express G(j) in terms of the symmetric functional as
Gs(j
+ e) -
Gs(e) - i
d4 x j . v
(11-163)
where v is determined by the condition that the derivative of GO) with respect to j vanishes at j = 0:
t5Gs (j) Vk = it5A(x)
Similarly,
ir(4))
j(x)=c
(11-164)
= G(j) - i d4x j. 4> = Gs(j + e) - Gs(e) - i d4 x j. (4) + v)
(11-165)
QUANTUM FIELD THEORY
with Thus
!5G(j) !5Gs O+ c) Q>k(X) = ~( = .~. () - Vk )
lUJk X lUJk X
(11-166)
r(q,) = rS(q,
+ v) -
rS(V)
d4x c q,(X)
(11-167)
!5rs(V) Ck= - - !5Vk
The identity (11-159) expressed in terms of r(q,) states that
f d4x [Q>k(X) + VkJIi{;~I~~~ a [!5r(q,)
(11-168)
Let us look in more detail at the content of (11-168). For instance, take one derivative with respect to q, and set q, = 0:
Iii !5Q>I(X) - Cl
It/J=O + Vm Tml
4 !5 r(q,) d Y !5Q>k(X)!5Q>I(Y) t/J=O - 0
(11-169)
Taking into account that !5r(q,)/!5Q>I(X) It/J=o = 0, which follows from the definition of v, and the anti symmetry of the matrices T, this means the following relation for the inverse propagator r:;J(p2) at zero momentum: (11-170) This shows that c and v are collinear, a fact already clear from Eq. (11-167). If we think of c as an external magnetic field then the magnetization v is along c. A mass parameter for the transverse states (with respect to c) is defined according to (11-171) and
c=mh
(11-172)
The quantity mT plays the role of a pion mass in the (J model and generalizes a result obtained in Sec. 11-4-1. From (11-167), it follows that once the symmetric theory is renormalized so is the broken one. Explicitly, if (11-173) the corresponding renormalized functional for the broken symmetry case is (11-174) A rescaling of the symmetry breaking parameter c has been necessary:
Co = Z-1/2 C
(11-175)
= Co
in such a way that c . q, remains invariant, c . q,
q,o, and insures that
SYMMETRIES
(11-176)
bT's,reg(vO)
bVok
Equation (11-167) implies that the amplitudes for the case with broken symmetry are obtained by resumming tadpole insertions in the symmetric theory. For instance, the p-point Green functions (p> 1) may be written as (Fig. 11-13)
rip) .. ,k, (x b k,,.
.. ,
x) p
;, L..., n.
~ fd 4y 1 "'d4y
r(n+p) ,k"I, .. ,1 (x 1, .. , x P' y b S,k" ...
.. , y) n
(11-177)
For compactness we have assumed v along the first axis in isotopic space. This discussion of renormalization may also be carried out when the breaking terms are more complex and involve higher-dimension operators. The lesson to be drawn from the work of Symanzik is that breaking terms of dimension w < 4 only require counterterms of dimension lower or equal to w. For instance, a mass-breaking term (w = 2) will not affect the counterterms of degree three or four which will remain symmetric. A soft breaking (w < 4) leaves a remnant of the initial symmetry. A hard breaking (w = 4) will a priori destroy completely the symmetry. These properties on the ultraviolet divergences have their counterpart on the asymptotic behavior at large momenta of renormalized Green functions, at least in the euclidean region. This is another aspect of Weinberg's theorem (Sec. 8-3-2). Thus a soft breaking will not affect the asymptotic regime which will remain symmetric. The reader may wonder what is the link between this discussion and Ward identities
= <TiJ j~(x)Al(xd" An(xn)
<T A 1(Xl)'" O(XO - xg) UMx), Ap(xp \]'" An(xn)
(11-178)
encountered in quantum electrodynamics or in the applications of current algebra. It is easy to convince oneself that the former are relations of the kind of (11-178) integrated over x. The fields Ap(xp) are identified with </>k (xp) and j is the Noether current j~(x) = [iJ.</>k(X)] 7k~[ </>I(X) + VI]. Upon integration the left-hand side vanishes as a total derivative without boundary contribution in the absence of massless states. On the right-hand side iJ"j~(x) is equal to CI1/;:' [</>m(X) + vm and the last ] term involves the variation of the field
o(XO - yO)UMx), </>k(Y)] = 04(X - y)7kH</>I(Y)
We therefore obtain the identity (11-168).