We now exploit one of the most powerful consequences of renormalization: the fact that the final results for renormalized scattering amplitudes cannot depend on the choice of the renormalization scale ¿¿2. If it is really true that the choice of fj,2 does not change the final results, then there is a symmetry associated with this invariance, i.e., a group of transformations involving changes in ¡j.2 which leave all physical results unchanged. These transformations are referred to as the renormalization group. Following our discussion of the renormalization group, we will discuss asymptotic freedom and show that QCD is asymptotically free.

>Ve will use the notation of Chapter 16, where g0 denotes the unrenormalized coupling constant and gR the renormalized coupling constant. These two constants are related through renormalization, where Zg is a renormalization constant. In d dimensions, both of these coupling constants have the dimensions of e = 4-d, and the renormalized coupling constant gR is related to the dimensionless coupling constant g as in Eq. (16.17), where p, is the (arbitrary) mass scale used in the dimensional regularization scheme discussed in Chapter 16. Now, the unrenormalized coupling constant go is the only one of these constants which is independent of the renormalization scale p, (for the simple reason that the scale does not enter into its definition), and hence the dimensionless coupling constant g depends on the renormalization scale p, through a combination of Eqs. (17.1) and (16.17),

The implication of this dependence of g on renormalization scale will be studied in this chapter.

Now, as a specific example of how these ideas can be extended to scattering amplitudes, consider the unrenormalized scattering matrix for the interaction of n gluons (the n-point function). This amplitude depends on go and e, but not on the mass scale /i. Hence

where {p,} are the external momenta upon which M depends and the dependence of M on e is shown explicitly. The validity of (17.3) is regarded as self-evident; M cannot depend on fi since none of its arguments do. It is a mathematical statement of the physical fact that the physics must be independent of the renormalization point (i.

Now, the renormalized n-point function is obtained from the unrenormalized n-point function by multiplying by a factor of Z\j2 for each external gluon, so that

where we have displayed the fact that Mr depends on p, and will ignore any dependence of M on quark masses. The renormalized amplitude does depend on

Till p through Z:i , an essentially "trivial" dependence which is associated with the dimension of the amplitude (see below).

An equation describing the dependence of Mr on p can be readily obtained from (17.3). We have

2 dp

If we now express the total derivative on the left-hand side in terms of partial derivatives, we obtain the Callan-Symanzik equation [Ca 70, Sy 70]

where j3 and 7 are taken to be functions of g and dg

Equation (17.6) is an example of a renormalization group equation. It describes an arbitrariness in the behavior of M r which arises from the choice of the renormalization mass n. If we choose a different ¡1, Mr will still be finite, but it will n

Fig. 17.1 One-loop graph for the n-point function.

have a different value, and the behavior of Mr as /x is varied is constrained by Eq. (17.6).

At first glance, it may seem that the dependence of Mr on the renormaliza-tion mass scale /x is an amusing fact of no physical importance. However, as we will now see, this dependence gives a powerful new way to study the behavior of scattering at large momenta.

We can use Eq. (17.6), together with dimensional analysis, to deduce the behavior of M^ as the momenta {pj —> oo. We will continue to consider the simplest case in which there is no quark mass dependence. Then, from an analysis of the simplest Feynman graph which contributes to the one-loop graph with n

3g couplings shown in Fig. 17.1, we can determine that the dimension of M^ is

and that therefore Af(n) can be rescaled as follows:

where we have also replaced {p<} by {Ap*}, where A is a dimensionless parameter which is convenient to introduce, and fj, is the mass scale which entered through the renormalization of M. Note that Eq. (17.9) tells us that the variation of with respect to the momentum scale (as measured by the parameter A) is related to the variation of with respect to the renormalization scale /x, and hence n

Fig. 17.1 One-loop graph for the n-point function.

the renormalization group equations can be used to study the dependence of the scattering amplitudes on momentum scale.

To develop this connection, begin with the Callan-Symanzik equation (17.6) for the amplitude (17.9), which is

Because of Eq. (17.9), we know that, apart from a "trivial" factor derived from the overall dependence of M^' on p4~n, the action of the partial derivative pd/dp on Mcan be replaced by differentiation by -Xd/d\. Making this substitution gives

+ /3(g)§-g~n'r{g)jM<£]({Xpi},g^) = 0 . (17.11)

This equation now relates the variation of MR on A (the momentum scale) to its variation on g and will permit us, under certain conditions, to estimate the behavior of Mr for large A (large momenta).

We will now show that the solution to this equation can be written in the following form:

({A Pi},g,ri = S ({Pi} , gr( A, g), p) , ■/here S is an overall scaling factor

4-n exp exp

and gr(A, g) is the running coupling constant which depends on A and g and which is interpreted as the effective coupling constant for A > 1. The running coupling constant is defined by the equations

Before we show that (17.12) really does satisfy Eq. (17.11), note that it tells us that M at large momenta {ApJ with A —* oo can be obtained from M at some fixed momenta {Pi}, provided we replace the coupling constant g by the running coupling constant gr and multiply by a scaling factor dependent on 7. We see that knowledge of the running coupling constant is sufficient to determine much of the behavior of the scattering at high momenta.

To prove that (17.12) is a solution to Eq. (17.11), begin by noting that Eq. (17.14b) can be integrated if we regard g as a parameter independent of A. Then dgT d\

and using the condition (17.14a), we obtain the following integral equation for the running coupling constant:

This equation gives an implicit solution for the running coupling constant gr(A, g) which will be discussed shortly. But first note that this equation can be used to show that the running coupling constant satisfies the following equation:

This can be readily demonstrated by differentiating (17.15) with respect to g, which gives

1 dgT 1

From this we conclude that

0(g)^L=0(gr) , and comparing this with Eq. (17.14b) gives (17.16).

The proof that (17.12) solves Eq. (17.11) can now be completed. Differentiating (17.12) gives dgr d dX dgr

A aWu\ \ \ cid9r d [ dx d9r{x,i xM^fel.Sr,/x) , (17-17)

where S is the scaling factor of Eq. (17.13). Combining these equations and using (17.16) give dx dj dgr{x,g)

dgr dg

Now we may use (17.16) again to carry out the integral over x giving i \ at \ f* dx d"t d9r{x,9) 7(9t) - 0(g) — jr--p-

and hence Eq. (17.11) is obtained.

An understanding of scattering at large momenta is greatly facilitated by an understanding of how the running coupling constant behaves as A —> oo, and we will discuss this in the next section.

Before turning to this discussion, rewrite the solution (17.12) for the case when 7(00,0) = 7oc is finite. In this case it is convenient to subtract the integral over 7 as follows:

J — l[9r(x,g)} = J — |7[ffr(a:,s)]-7=o|+7oologA , (17.20)

and write the solution in the following form:

where

S+ ~ A4~n~"7o° exp n J^ ~~ [t ff)] — 7c»]| • (17.22)

Note that the integral over x now converges to a finite value as A —> 00, so that 'he dimension of Mr has changed from 4-nto4-n - wy^. The quantity 7oo is referred to as the anomalous dimension of the field, since it indicates how the scaling behavior of Mr departs from that predicted by simple dimensional analysis.

From the defining Eqs. (17.14), it is clear that the behavior of the running coupling constant gr depends on /3(gr), and from the implicit solution, Eq. (17.15), it is clear that the zeros of (3 play a special role. As the upper limit of the integral in (17.15) approaches a value of g at which 0(g) = 0, the integral will diverge, and hence A must either approach 00 or 0, the only two points at which log A also diverges. Clearly, the behavior of log A depends on whether or not the integral is positive or negative as the zero is approached. We are therefore led to distinguish the two possibilities shown in Fig. 17.2. In the first case, the zero will be denoted by g+, and such a zero is referred to as an ultraviolet fixed point. In this case

17.3 BEHAVIOR OF THE RUNNING COUPLING CONSTANT 579

Fig. 17.2 Behavior of the /3-function near (A) an ultraviolet fixed point and (B) an infrared fixed point.

In the second case, the zero is denoted by g- and is referred to as an infrared fixed point. In this case

To understand the behavior of gr near these points, suppose that gr is less than g± but close enough to g± so that the zero at g± is the only zero under consideration. Then, as gr —► g± from below, the integral will diverge to + infinity at g+ and - infinity at g~, and hence A —► oo as g —> g+ at A —> 0 as gr —► The same conclusion is reached if gr > g±\ in this case the sign of the integral is changed, but the sign of ¡3 is also changed, so the same argument holds. We conclude that ne two solutions gr(X,g) are

In such a case, g+ is the effective coupling constant at infinite momenta, and g~ is the effective constant at low momenta.

The function (3 will usually approach zero as some power of g when g —* 0. The two ways it can approach zero are illustrated in Fig. 17.3. In the first case, (3(g) is negative for small g; in the second case it is positive. From the previous discussion, we see that, in the first case.

This case is referred to as asymptotic freedom. The name comes from the fact that the effective coupling constant of the theory approaches zero as the momenta approach infinity. This means that the scattering amplitudes at large momenta can be calculated using perturbation theory, and all of the wonderful perturbative methods which we have described can be applied to high energy calculations,

g+ = gr(oo, g) ultraviolet fixed point g- = gr(0, g) infrared fixed point .

Fig. 17.3 Behavior of the /3-function for small coupling. In (A) the theory is asymptotically free, but for a theory with a /3-function given by the dashed curve, the asymptotically free region cannot be physically realized if g >

even if the effective coupling constant at moderate energies is large. As we will show in the next section, this remarkable property is a feature of non-Abelian gauge theories and of QCD in particular.

In order to prove that a theory is asympotically free, it is sufficient to demonstrate that (3(g) < 0 as g —► 0 in lowest order perturbation theory [GW 73, Po 73], This is because such a demonstration insures that gr is small at large momenta, and hence also justifies using perturbation theory to estimate p{g). However, such a demonstration does not guarantee that the theory as physically realized by nature is asymptotically free. For example, if the theory has a /3-function with two other zeros at finite g (as shown by the dashed line in Fig. 17.3A), and if the physical value of g > <72. then the coupling will evolve to <72 as A —► 00, and we will never ' ^ach the asymptotically free region. Alternatively, even if there are no other zeros, the coupling constant may run so slowly that it does not become small until the momenta are so large that they are inaccessible experimentally. For QCD, neither of these situations seems to hold, and it appears that the asymptotically free region of QCD is physical accessible.

Anticipating the results we will obtain in the next section, we assume that in lowest order perturbation theory where 0O is a positive number. In this case we have an asymptotically free theory where g+ = 0, and the effective coupling constant at large A is obtained by solving Eq. (17.15). Substituting (17.25) into (17.15) gives

from which we obtain

constant does indeed "run" according to Eq. (17.28). A recent determination of how the QCD coupling constant "runs" is shown in Fig. 17.4 [BP 92]. From these results a value of A ~ 150 - 250 MeV is obtained.

Note that the running coupling constant (17.28) approaches zero very slowly as Q —> oo, and hence the force between two quarks at very short distance does not really go to zero, as is sometimes assumed.

17.4 DEMONSTRATION THAT QCD IS ASYMPTOTICALLY FREE

We will now show that QCD is an asymptotically free theory. Eqs. (17.2) and (17.7), the demonstration requires that we compute

Combining

where Zg is a renormalization constant defined in Eq. (16.121), which in lowest order perturbation theory has the form

where k is a constant to be determined below. Hence Eq. (17.29) has the following solution to third order in g:

2 g2n Zg f

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