Fig. 16.1 Two graphs for the self-energy in <j>3 theory. The shaded boxes surround divergent subgraphs, as discussed in the text.

and is simply the number of powers of momenta in the numerator (counting each ddki as d) minus the number of powers of momenta in the denominator. Clearly, if D > 0, the diagram will diverge; if D = 0, it is superficially logarithmically divergent, and it would seem to converge if D < 0. However, since each loop momentum is an independent variable, a divergence can also occur if D{ > 0 for any loop (or combination of loops) in the diagram (which will be referred to as a subdiagram and denoted by i), even if the overall divergence D < 0. For the diagram to be finite, both D and all Di associated with any subdiagram must be less than zero. This theorem is sometimes referred to as Weinberg's theorem [We 60],

To illustrate these ideas, consider the two loop diagrams for the self-energy in 03 theory, shown in Fig. 16.1. Both of these diagrams have a superficial ' ivergence D = 2d - 10, which is less than zero for d = 4 dimensions, but the self-energy insertion in Fig. 16.1 A (contained in the shaded rectangle) has a superficial divergence Dt = d - 4, showing that it diverges logarithmically in d = 4 dimensions. Diagram 16.IB has two overlapping vertex insertions, each with a superficial divergence Dx = d - 6, which converges in four dimensions. We will return to a more detailed discussion of these diagrams in Sec. 16.2 below.

To study the removal of infinities from a theory, we must express the superficial divergences D and {Dt} of each Feynman diagram directly in terms of the properties of the interaction Lagrangian £/. In order to keep the discussion general, consider an interaction of the form where F is the number of Fermi fields and B the number of boson fields which interact at each point in space-time. If Nb and Np are the total number of bosons and fermions external to a Feynman diagram (the sum of those in both the initial and final states), and if there are n vertices (for an nth order Feynman diagram), then, since each internal line couples to two vertices,

Another constraint comes from momentum conservation. The number of loops I is the same as the number of momenta unfixed by momentum conservation, which is the total number of internal momenta (lines) minus the number of vertices (each of which contributes one constraint) plus one (for the overall energy-momentum constraint which does not limit internal momenta). Hence e = nB + nF -n+ 1 . (16.4)

These three constraints enable us to re-express the superficial divergence in terms of the number of external lines (which depends only on the physical process under consideration), B and F (which depend only on the theory), and n, the order of the diagram. The result can be written


is the index of divergence of the interaction Lagrangian £/.

Note that the index of divergence depends only on the theory, and not on any particular physical process, while the remaining quantities in (16.5) depend on the number and kind of the external particles (which are fixed for any physical process under consideration) and on the number of vertices n in the diagram. In four dimensions, (16.5) reduces to

We will now distinguish three different classes of theories.

The first class has index / > 0. In this case, D will eventually become greater than zero as n increases, regardless of the physical process under consideration (i.e., for any N& and Np). Furthermore, as n increases, D becomes larger and larger. We conclude that if I > 0, there are always (higher order) Feynman diagrams which diverge regardless of the physical process under consideration. Such theories are called non-renormalizable.

Next, if / = 0, D is independent of n and in d = 4 dimensions is less than zero if D0 = 4 - NB - §7VF < 0. For such theories, all but a finite number of elementary processes are superficially convergent. For example, in a theory with a ipdTp<t> interaction, / = 0, and processes involving five external bosons, such as boson production in boson-boson scattering (Bj + B2 —>• B3 + B4 + S5). or two external fermions and two bosons, such as boson-fermion elastic scattering {B + F -> B + F) or fermion annihilation (F + F B + B), and all other more complex processes have no overall divergence; divergences which contribute to these cases come from simpler processes which occur as insertions inside of the Feynman diagrams. For QED, which has an index equal to zero, the only quantities which diverge are the electron self-energy, with D0 = — 1, the vacuum polarization, with D0 = -2, and the ■yee vertex, with D0 - 0. [The photonphoton scattering amplitude (7+7 7+7) also has D0 - 0 but does not diverge because of gauge in variance.] We will refer to such theories as superficially renormalizable and save the word "renormalizable" (which is often used) for a different meaning given below.

Finally, if the index I < 0, then only a finite number of diagrams associated with a finite number of physical processes will diverge. An example is 03 theory in four dimensions, which has / = -1. In this theory only one diagram (or subdiagram) diverges: the lowest order meson self-energy, with n — 2 and D — 0. All other diagrams, except those containing this lowest order self-energy insertion, will converge (and after this self-energy is regularized, all other diagrams will converge). Such theories are said to be super-renormalizable.

Regularization Schemes

The term regularization is used quite generally to describe the process of removing infinities from any set of Feynman diagrams associated with any of the above classes of theory. These infinities are removed according to a definite procedure or prescription referred to as a renormalization scheme. Using this scheme, any Feynman diagram can be written as the sum of a finite and an infinite part, with the infinite part depending on a number of infinite constants, referred to as renormalization constants. These constants cannot be determined by the theory and are regarded as free parameters to be fixed by experiment. For non-renormalizable theories, the total number of required renormalization constants grows with the order n, and an infinite number are needed to remove all infinities to all orders. For theories which are superficially renormalizable, the number of renormalization constants is finite and can be absorbed into a finite number of parameters, such as the charges and masses of the particles in the theory. This process of redefining the theory by absorbing the renormalization constants into the parameters of the theory is referred to as renormalization, and the proof that this is possible is definitely non-trivial and is the subject of much of this chapter.

Using the ideas introduced here and in Chapter 11, we will now describe in more detail how Feynman diagrams are regularized and how infinities are systematically removed. To keep the discussion general, but not too abstract, consider 03 theory in d dimensions. Its index is

so that the theory is super-renormalizable in d = 4 dimensions, superficially renormalizable in d = 6 dimensions, and non-renormalizable in d = 8 dimensions.

Consider the lowest order (n = 3) vertex correction shown in Fig. 16.2A. This scalar vertex correction depends only on the square of the three external momenta, A(p, q) = F (q2,p2,p- q), and is divergent in d > 6 dimensions. The infinite part,

Fig. 16.2 The lowest order vertex correction in <p:i theory is shown in (A), and the Feynman diagram generated by the corresponding counterterm Ao is shown in (B).

which we will denote by Ao, can be removed by subtracting it from the diagram. Detailed examples of how these infinite parts are defined and subtracted will be presented in the next section. The finite part which remains after the subtraction will depend on precisely how the infinite part is defined and how the subtraction is carried out, and this dependence generally appears as a dependence on some momentum scale, which will be denoted by fi. Hence, the renormalized vertex correction AR is written where the dependence of AR on the momentum scale /i, which inevitably arises in the subtraction, is shown explicitly. The infinite subtraction constant, A0, can He treated in one of two equivalent ways. The first method, which we used in Chapter 11, is to absorb it into the lower order graphs (the coupling constant in this example), which are said to be renormalized by the subtraction. In this example, the three-point coupling to third order then becomes (recall that the coupling constant go must multiply the vertex correction to give the diagram in Fig. 16.2)

where T is the full vertex function, the renormalized coupling is gR = go + ff0Ao. and g0A„ = gR AR to third order in perturbation theory. Defining the renormalization constant Z\ by

(as we did in Chapter 11) gives

a logarithmic divergence in d = 8 dimensions. To remove this singularity, it is necessary to add a counterterm to the 4>3 Lagrangian of the form

where A0 is determined from the singular part of graph 16.3A (see Prob. 16.2). However, since a <t>4 term does not appear in the original Lagrangian, this counterterm cannot be absorbed into one of the original parameters of the theory, and its appearance changes the structure of the theory. This need not be a disaster; in this example A0 can be treated as a new parameter and determined from a measurement of 4><i) scattering at some fixed point, allowing us to predict the scattering at other points. But the appearance of a new counterterm certainly reduces the predictive power of the theory, and because the index I of this theory is positive (remember that d = 8), we can expect many new divergences to appear in higher order. This will introduce still more counterterms, further reducing the predictive power of the theory. In practice, non-renormalizable theories are useful only in cases where a good estimate can be obtained from the first few orders in perturbation theory. Chiral perturbation theory, based on the non-linear chiral models discussed in Chapter 13, is an example of a non-renormalizable theory which has enjoyed considerable success. For a discussion of effective Lagrangians, see Donoghue, Golowich, and Holstein (1992). We will not discuss non-renormalizable theories further.

Our discussion up to now has focused on how the infinities are removed in "lowest order." The central problem in the proof of renormalizability is to show that the addition of a finite number of counterterms is sufficient to remove all i-ifinities from the theory. For example, return to the diagrams shown in Fig. 16.3. In d — 6 dimensions, 16.3A is finite, but 16.3B is infinite because of the diverging vertex subdiagram. A counterterm added to the Lagrangian renders this vertex correction finite, as we have discussed, and the same counterterm inserted in the <t>4> scattering box, shown in Fig. 16.3C, will also insure that the two diagrams 16.3B and 16.3C are finite. To prove renormalizability, we must show that such a procedure works for all diagrams to all orders.

The demonstration of the renormalizability of QED will be a major goal of this chapter. Before we discuss these problems further, it is helpful to consider a few more examples and to develop a technique for evaluating multi-loop diagrams.

To clarify some of the issues which will arise in the construction of a general proof of renormalizability, we look at <p3 theory in six dimensions. As discussed above, this theory is superficially renormalizable (has an index equal to zero) and will provide a simple illustrative example.

First, consider the dimensions of the coupling constant in (j>:i theory. The action is dimensionless, so in d dimensions, the Lagrangian density must have the


The mass shift and wave function renormalization come from A(q2) and B(q2), as in Sec. 11.3.

We are interested in the behavior of each of the functions near d = 6 dimensions. Note that each is singular in the small parameter e = 6- dase—>0, because

It is convenient to separate A and B into two parts: a singular part, which will be denoted by A0 and B0, and a finite part An and BR. The separation between these two parts depends on the renormalization scheme, because the singular part can include any finite terms which it is desirable to include, and will also depend on the mass scale n which enters through the substitution g2R = g2^ of Eq. (16.17). To make the results well-defined and unique, we must define, as part of the renormalization scheme, what finite terms are to be included in A0 and B0 and which are to remain in AR and BR. In this chapter we will adopt the somewhat unconventional scheme of including in A0 and B0 all finite terms which do not depend on momenta or on the scale parameter /¿2. With this choice, the finite t .rms emerge only from the expansion of the factor

so that we have uniquely

with corresponding finite terms

Note the explicit appearance of the scale parameter /j2 in the finite terms. We will say no more about these finite pieces now.

The infinite parts (16.24) can be expanded in a Laurent series in the small parameter e. Using the expansion of the T-function, r(l + f) =\-\l + 0(e2) , (16.26)

where 7 = 0.5772 • • • is Euler's constant, the terms which survive as e —► 0 are


In the scheme which we will use in this chapter, these constants will become the counterterms discussed in the previous section. In some treatments, only the 1/e part of (16.27) are included in the counterterms, leaving the finite part to be combined with AR and BR. This is the minimal subtraction scheme [Ho 73], and in this scheme the counterterms are


where the subscript MS refers to "minimal subtraction." Alternatively, the MS scheme [BB 78] includes the log(47r) and Euler's constant 7 in the counterterms, so that

Since this combination of e, 7, and log(47r) arising from the expansion of (47r)£/2r(e/2) occurs frequently, the MS scheme is quite popular in QCD. In this chapter we include all of the terms in (16.27). With this convention, our <p3 Lagrangian becomes

C = \ [d^d»cj> - m202] - fBo^d^ + \AQm2<t>2 (16.30)

where the A$ and Bq of Eq. (16.27) are precisely the correct factors required to cancel the divergence arising from diagram 16.4A. These counterterms will generate the Feynman diagram shown in Fig. 16.4B, as discussed in the previous section.

where V, defined by this equation, will be used below. Carrying out the integral over k then gives

In d = 6 dimensions, this is singular due to the pole in T(e/2), where e = 6 - d as before.

Following the renormalization scheme we aie using in this chapter, the coun-terterm implied by (16.48) is obtained by taking (jx/X)': —> 1, giving

and the finite term is

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