Asymptotic Freedom and the Renormalization Group
2.1 Principal Chiral Fieldst
In this section we shall study the large p limit for the ^ = 2 principal chiral field described by the Lagrangian:
I/Trid^g ^ c^g)
(l/e^ = I and g e G)
(2.1)
As we discussed in Chapter 1, the infrared interaction of the massless particles described by (2.1) is logarithmically strong. It is our aim now to reveal the structure of this logarithmic interaction.
Let us study the effective Lagrangian which arises from (2.1) in the loop approximation. In order to find it we write the quantum field g{x) in the form:
g(x) = h(x)-g,^(x) (2.2)
where ^ci(-^) is some classical solution for the Lagrangian (2.1), namely
— yj
(2.3)
=0 f^^ = c^g’ g
Our programme is to integrate over the field h{x) so as to obtain an effective action depending on g^x{x). This approach is more or less standard in field theory but it requires some clarification. At the first sight, due to the invariance properties of the integration measure,
^h{x) = ^(h(x)g^x(x)X the result of such integration would seem not to be dependent on ^Iso it is not quite clear, what kind of
t A principal chiral field is one which defines the principal bundle over the base space.
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DOI: 10.1201/9780203755082-2
GAUGE FIELDS AND STRINGS
classical solutions must be included. Both of these questions can be answered if we consider a finite size system and fix the value of g(x) at the boundary F :
g ( x ) \ , = g i O i e V (2.4)
We are considering the functional
n g io ^ = S>g{x)Q-d/ft)S[ÿ(x)] (2.5)
g(x)\r = g(4)
(where S is the action). This functional is just the Euclidean analogue of the Shrodinger wave function. The classical limit ft -► 0 would corre
spond to taking the minimum of the action S with the Dirichlet boundary conditions (2.4). Integration over ft(x) in the decomposition (2.2) corresponds to the inclusion of quantum fluctuations. It is clear from the above discussion that we should fix the boundary condition for ft:
ft(x)|r = / (
2
.6
)To sum up, we shall compute the integral over ft and obtain an eflective action depending on It must be understood, however, that ^ci(^) is not an independent variable: in fact all classical solutions are parametrized by their boundary values g(0 is ^ — 1-dimensional).
So, we are effectively computing the T-functional (2.5), which as we already said is an analogue of the Shrodinger wave function and, on the other hand is analogous to the on-shell amplitudes of the Minkowskian theory (they also depend on ^ — 1-dimensional fields).
All this information about boundary conditions and T^-functionals can be kept subconsciously so far as we are interested in infinite systems. Actually, most often we need not bother to express ^ci(^) through g(0- Substituting (2.2) into (2.1) we get:
= = LI' + g;, '(h ' d^h)g^,
2el
+ \ Tr(R'>(;i - ' d^h)), R f = (d^gjg:, '
^0 (2.7)
If we want to examine only one loop corrections to the classical action, we have to consider only small fluctuations of the matrix h. We can write:
h = ^ \ (j) + (2.8)
(where 0 belongs to the Lie algebra of G.). Substitution of (2.8) into (2.7) gives:
¿€q ZCq
(2.9)
(we have omitted the term d^(j) because of the equation of motion d,Rl' = 0).
Now the term in the effective action which is quadratic in will be given by the Feynman diagram t
^ I j ' A C D j ' B C D
= -<?))«
• d^p (2p + q \ ( 2 p + q),
( 2n y %p\p + q f (2.10)
If we extract the ultraviolet divergent contribution from (2.10) the formula takes the form:
1 d^x{Rl\x)nRl\x)r X -CXG) ' PmPv
( 2 n f p*
+ finite part c,(G) \ 1
H- finite part (2.11)
(A is a momentum cut-off).
We see that if we introduce the renormalized coupling constant by the formula:
1 1 Q(G)
An log A (2.12)
t We have introduced the generators of the fundamental representation of the Lie algebra of the group G: [F*, i®] = Tr (j) =
are the structure constants of this algebra. Notice that
GAUGE FIELDS AND STRINGS
the effective action will be finite in the given order. The result of the above computation can be interpreted in two different ways. The most naive interpretation has already been given: namely we compute the effective action and see that the ultraviolet divergence in it can be absorbed into a redefinition of the coupling constant. Another interpre
tation is the following. Let us suppose that we have a theory with the cut-off A (which is of the order of the inverse lattice spacing). If we integrate out the fields h(x) with the wave vectors A < p < A we shall obtain the effective action which in the low energy limit has again the form (2.1) but with a renormalized value of el, Cq(A):
el(A) = el (A) + ^ C , ( G ) \ o g ^ = elA (2.13)
This formula follows directly from (2.11), provided that we restrict the p-integration by the condition A < |p| < A. As a result we conclude that the physical theory, formulated with the cut-off A and the bare charge el must be equivalent (for small momenta \p\ A) to the one with the cut-off A and the specially chosen new bare charge el. This statement is called renormalizability and the transformations from el to el and from A to A the renormalizability group. The formula (2.13), as is seen from its derivation, is correct provided that:
el log y ^ 1 (2.14)
Nothing prevents us from repeating the procedure and passing from A to A < A etc.
The most important consequence of renormalizability is that it controls the momentum dependence of different physical quantities. Let us consider as an example the behaviour of the effective charge e^(p) for the fluctuations with momentum p. This quantity can be defined in different ways. One of the possibilities is to consider the four point function with all momenta equal to p. There are many other options and we shall comment on this ambiguity later.
Since our theory does not contain any dimensional parameter except for A, we must have:
e \ p ) = e^(log(A/p), el) (2.15)
Let us express e^(p) in terms of where p is some fixed value of the momentum. Inverting (2.15) we have:
«0 = eo(log(A /p), e \ p ) ) (2.16)
Substitution of (2.16) into (2.15) gives:
e \ p ) = e\\og{p/pX e \ p \ log(A//i)) (2.17)
The first thing renormalizability tells us is that (2.17) actually does not depend on its last argument, since we can change A and compensate for it by changing el without changing e^(p). Hence:
e \ p ) = e\\og(p/fiX (2.18)
We see that e^(pX being expressed in terms of e^(fiX does not contain any divergences and does not depend on the structure of the theory at distances of the order of lattice spacing. But this is not the end of the story. Actually we have a further constraint on (2.18) which follows from the fact that the point p was quite arbitrary. Hence, just as it was with A, it must be possible to compensate for a shift in p by changing e^{p). We have, therefore, a functional constraint on e^{p) which is quite easy to solve. Namely, it is clear that: dependence. It is not fulfilled in a fixed order of perturbation theory and
perm its us to ob tain nonperturbative expressions.
It follows from (2.19) that in a theory without dimensional para
meters a so-called dimensional transmutation takes place:
e \ p ) = /(log(p//l)) X = fie (2.21)
All quantities depend on a universal correlation length 2 “ ^ which should be kept fixed as the lattice spacing A"^ goes to zero. No other arbitrary parameters enter into the theory. (The last is not generally true: there are theories with several effective charges, like massless scalar QED, in which physical quantities depend on the ratios of these charges). In order to see how (2.19) improves perturbation theory, let us write it in differential form:
d e \ p )
= P(e\p))
d \og(p/p)
l^(x) = f' (g(x))
(2.22)
This is the Gell-Mann-Low equation. Iterations of its solution in
Solving the differential equation (2.22) we get:
e \ p ) e (p) ■■
1 + e \ p ) log|
©
(2.25)
It is easy to check that while (2.24) does not have the form (2.19), (2.25) does (with f { x ) = 47r/C„((^) x). Another useful expression for e^{p) is:
What is the range of applicability of (2.25) and (2.26)? It is defined by the fact that we have neglected all higher powers in e^(p) in the expression for the jS-function. Therefore the condition is:
e \ p ) < 1 (2.27)
The real meaning of this improvement to perturbation theory, invented by Gell-Mann and Low, is that it replaces the expansion in the bare charge e l which may not be small by the expansion in e^(p), which in many important cases is small.
For example if we rewrite (2.25) as: smallness is called asymptotic freedom. For p < / perturbation theory
is untrue and qualitatively different methods are needed. They will be discussed in later chapters.
Let us proceed to the computation of the correlation functions. Take as an example:
9 ( x - y ) = {Tr(g-\x)g(ym (2.29) The tactic is again the following. Let us integrate over rapid fluctua
tions with A < |/?| < A and use the renormalization group argument. If we write
g{x) = (1 + (l>ix) + j(j)\x))goix) (2.30) and integrate over the 0 with the wavelengths in our interval we get:f
C
@(p) = ®o(P)l * “ V X (2.31)
From (2.31) we deduce by the repetition of the arguments leading to the Gell-Mann-Low equation, that if we set
Slip) = d(p)/p^
then:
where
dlog dip)
dlog(p/p)= yie\p))
y(e^) = -h Oie"^) n
(2.32)
(2.33)
(2.34) Integrating (2.33) we get:
1 / C e ^ (p ) , ,
®(p) = p M + log(pV p^)'
,2n^\4C2/C^
for p > X (2.35) which is the desired answer. Of course, the range of applicability is again p P L
As we said before the definition of e^(p) is ambiguous. This ambiguity leads to an ambiguity of the j?-function and of the y-function. However
tHere we put = — C2/. For the fundamental representation of G = SU{N):
C2 = {N^ - 1)/2N; C,{SU{N)) = N.
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the next result for any correlation function is unique—the jS-, and y- ambiguities compensate each other. Moreover, the first two coefficients in the j9-function are more or less universal. This can be checked by a change in the definition of e^{p). Suppose we chose:
e \ p ) = e \ p ) + C , e \ p ) + C ^ e \ p ) + ... j?-function is indeed universal. In the case when several leading coeffi
cients of the ^-function vanish, this remark relates to the first nonvan
ishing one.f
So, our conclusion is that the Gell-Man-Low renormalization group is a useful tool in the region where the effective charge is small. In order to use it one has to compute the coefficients before the leading powers of log(p/p) in physical quantities. This is usually easy in the lowest order, but becomes increasingly tedious in the higher orders, since one has to separate a nonleading contribution log(p/p) from the leading ones ^ (log(p/p))". In the asymptotically free theories the renormaliza
tion group gives order by order a small distance expansion which goes in inverse powers of log(p/p). For example the two loop order solution of equation (2.2 2) gives:
eHp) = -
1
log l o g y ) ^ )?, \og(p/A) I Pi log(p/A)1
\ og\ p/ A) p > X (2.39)
Another useful thing to remember is that the inverse correlation length A is expressed in terms of the bare parameters A, Cq as
A = const • A(Co)^^^^‘c^^^‘^o (2.40)
t In the general case we have freedom to redefine coupling constants, leading to Riemannian geometry in the space of coupling ‘constants’.
which is the consequence of the relations:
dX dX del d \ d el
X = ^(t){el)
(2.41)
(the first is the renormalization condition and the second is just a naive dimensional statement). The equation (2.40) indicates a consistent limiting procedure necessary to reach the continuum limit of the theory.
For the asymptotically free case (j?i < 0) it is described by the condi
tions:
^0 ^ 0 A -► 00 2 -► const. (2.42)