Renormalization theory

ANNALS

OF PHYSICS:

S&496-560

(1969)

Renormalization

Theory*

P. IL Kuo Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14850 and Laboratory for Nuclear Science, and Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND

D. R.

YENNIE

Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14850

Standard renormalization theory is presented with the help of a new technique for handling the subtractions. Briefly, the approach used is to replace the subtraction of terms from the Feynman integrand by an equivalent set of differential and integral operations on the integrals. It is relatively straightforward to show that these operations lead to a unique finite part of every integral. Although we are concerned here primarily with the general analysis of renormalization theory to all orders of the perturbation expansion, the explicit calculation of individual terms in the expansion is straightforward, and certainly not more lengthy than in the usual approach. In this approach, it is a simple matter to demonstrate gauge invariance of the finite amplitudes without explicitly carrying out the integrations; this contrasts with the usual treatment.

I. INTRODUCTION

The modern phase of quantum field theory was initiated by Feynman, Schwinger, and Tomonaga nearly twenty years ago (1). One of the great intellectual achievements flowing from this development was Dyson’s program for a consistent treatment of renormalization to all orders of approximation (2). Although the main features of the program were clearly foreseen by Dyson, many intricate technical questions remained to be worked out by other authors. The most perplexing of these questions was known as the “overlapping divergence problem.” Very soon after Dyson’s work, Salam (3) and Ward (4) presented alternative approaches for overcoming this problem. We do not choose to review completely * Work supported in part by the Office of Naval Research at Cornell and by the Atomic Energy Commission under Contract AT(30-1)2098 at MIT.

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all the subsequent work, which has been vo1uminous.l However, two particular developments should be emphasized. One is the proof by Nakanishi and Weinberg (5) of the “power counting theorem”, which states, in effect, that one can determine the convergence properties of any Feynman integral by simply counting powers of the integration momenta. The other is the continuing effort to give the subject a more rigorous formulation (6). Reference to any of the original papers or to any text treating renormalization in detail (7) will reveal that the subject is a very intricate one with many subtle difficulties. Partly because of this intricacy, but mainly because of the development of nonperturbative approaches, there has not been very widespread interest in renormalization theory in recent years. In the belief that a mastery of renormalization theory may be of some value in the further development of field theory, we have arrived at an approach2 which we feel clarifies the formal problems of the theory and hopefully will simplify some of the mathematical investigations of the subject. The present work has three complementary aims. The first, and perhaps principal one, is primarily pedagogic. We wish to eliminate, or at least simplify, some of the intricacies which appear in the conventional discussion of renormalization theory. The problem is essentially this: One must examine a complicated multidimensional integral with various divergences and subtract certain terms from the integrand to make the integral converge. The subtractions must be shown to correspond to consistent renormalizations to all orders of the interaction. When the general situation is considered, the number of subtraction terms required may be large and an elaborate argument is necessary to show that the resulting integral is finite. Roughly speaking, our approach is to replace these subtractions by a set of differential and integral operations on the Feynman integrals. It will be easy to see, on the one hand, that these operations extract a unique finite part from each integral; and, on the other hand, that they do correspond formally to the correct subtractions required to give a renormalization. The second aim is to develop techniques which may simplify the actual calculation of higher-order contributions involving renormalization. It is generally appreciated that the work required to calculate a new order of perturbation theory is many times that of the previous order. However, we can anticipate that such calculations will be necessary in quantum electrodynamics in the near future if maximum information 1 The variety of approaches to the subject is indicated by the following list, which is intended to be representative, but not complete: Wu (8), Gaillard and Visconti (9), Bialynicki-Birula (IO), Mills and Yang (II) and Caianiello, Guerra, and Marinaro (12). Further references can be traced through these papers. 2 A more pedagogical (but less detailed) discussion of this work has been presented by one of the authors (DRY) in the 1967 Cargese Summer School, whose notes have been publishedi n book form: “Cargese Lectures in Physics,” Vol. 2, Edited by M. L.&y, Gordon and Breach.

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is to be extracted from various experiments in progress. The third aim is to provide a new framework which may permit a simpler discussion of some of the questions of mathematical rigor, or perhaps to raise some interesting questions such as: Is the present approach, which is formally equivalent to the usual one, also mathematically equivalent ? Our approach is concerned entirely with perturbation theory; we make no attempt to use the integral equation approach. The present approach may be more cumbersome, but for some questions, we feel it is more transparent. Equations involving infinite sums are then understood to be true in the sense that they represent an equality for each separate power of the expansion parameter; no claim is made about the convergence of the series or the validity of various formal rearrangements of the series (for example, regrouping the series so as to express it in terms of the observable rather than the bare charge). The general arrangement of the paper is the following. The remainder of this Section will be devoted to a discussion of some of the ambiguities of handling divergent integrals as they occur in field theory. Section II will deal with the terminology and formal algebra of renormalization in quantum electrodynamics. This discussion is a little more general than the usual one in that we permit the multiplicative renormalization constants to be defined in an arbitrary manner for each graph. The conventional mass-shell renormalization is only one of many possible schemes; it is of course the most important one because it gives the conventional definition of charge. However, the new generality permits us to carry out the renormalization in two steps. The first step is intermediate renormalizution,3 based on subtractions at the origin of momentum space. The second step is a finite transition to the conventional renormalized functions. The generalization also corresponds to the freedom of choice of renormalization point usually associated with the so-called renormalization group (23). In Section III we define a certain unique finite part of any Feynman amplitude called an intermediate amplitude. This involves a set of operations called Q-operations which remove all divergences and combinations of divergences. The trick which is presented there was discovered in a roundabout way. Initially we tried to make systematic use of identities like (3.1) for the remainder of a power series after the first several terms have been subtracted. This appears to be very complicated in general for several reasons: (i) In detail, it depends on the choice of the variables of integration (or routing of momentum), i.e., the set of “differentiated” graphs is not unique. (ii) Renormalization analysis is more complicated unless the routing in related graphs of different order is properly chosen (8, II). (iii) In quantum electrodynamics, gauge invariance is not self-evident from consideration of the s This two-step renormalization procedure was introduced by Bjorken and Drell (7). The details of this analysis will be carried out more explicitly in the present paper.

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integrand alone. We then tried to convert (3.1) into a mass differentiation identity by scaling the internal momenta (k = &‘) in (3.1); this turned out to be awkward. Finally, we noticed that what we really wanted was a sequence of differential and integral operations which projected the subtraction terms separately to zero and which at the same time did not change the whole integral. To project the subtraction terms to zero, we noted that, although they were divergent integrals, they were formally powers of external momenta times a homogeneous function of the masses. Differentiation with respect to the mass made the integral converge, and it was simple to construct a final operator which eliminated the resulting terms of certain predetermined powers of the masses. It was also possible to construct the final operator in such a way that it did not change the value of the complete integral with subtractions. The extension to the general situation with several divergences was not difficult. Section IV contains a demonstration that the finite amplitudes of Section III do in fact give a renormalization as defined in Section II. After the preliminary work, this is relatively direct and straightforward. Section V contains some concluding remarks and there are several appendices discussing some mathematical points and giving examples. Now let us turn briefly to an oversimplified discussion of some of the problems which arise in dealing with the type of divergent integrals which arise in field theory. As a model, consider the integral (with a Euclidean metric) I(q, Pv m) = IS_“, (k _ 4)‘Ad;kp2 + m2 ’

(1.1)

If this integral were convergent, we could immediately draw two conclusions: (i) Z is independent of q. (ii) Z depends only on the ratio (p/m). The first statement is proved by shifting the origin. The second is proved by noting that, if p and m are increased by the same scale factor, the integral can be restored to its original value by scaling k in the same manner. But, on the other hand, the integral is a function of the combination p2 + m2 and by scaling we can make this anything. Hence the integral should be a pure constant independent of q, p, and m. If we attempt to compute the constant, we find it is infinite; and hence the analysis seems meaningless. In field theory, we are undaunted by such a situation. It is recognized that the main (i.e., infinite) part of such integrals really represents a change (or renormalization) of some physical parameter such as mass or charge. We are really interested in extracting the residual varying part which depends on p and q. The contribution to the renormalization is said to be given by the value of the integral at some reference value of the external momenta, say q. and p,, . We treat the value of the integral at this reference momentum as a formal constant, Z(qo pp. , ml = A

(1.2)

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AND

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and define the varying part of Z by subtracting the original one:

the associated integrand

Z(q, P, m) = A + /I d2k l(k _ 4)2 : p2 + m2 -

from

Or - qo12 k po2 + m2 1

= A + Z’(q, p, m).

(1.3)

From now on A is treated as a constant and the integral occurring in (1.2) is always replaced by this same constant. Since we are dealing with ambiguous quantities, it is important to emphasize that whenever it occurs, the integral (1.1) should always be treated in the same manner using the dejinition (1.3). An alternative, probably more reliable but sometimes less convenient, approach is to introduce a regulator or cutoff in the integral. For example, we may insert a factor (12 (1.4)

k2 + A2 '

where II is very large (/I2 > q2, p2, m2) and consider the behavior as /l + co. Then we have the integrals ZCz, P, m, A) = /I

d2k

1

A2 k2 + A2 Q - q)2 + p2 + m2

(1.5)

and 44

= Go, po, m, 4.

(l-6)

In the present case, it is clear lim

A2-m lim V(q, z4 m, 4 ‘42-m

A(A)

= 03,

- Go , p. , m, 41 = I’(q, p, m).

(1.7)

The regulator then makes all integrals well defined and permits one to extract the desired finite part of the integral in a unique manner. Both approaches to determine the finite part suffer from the computational disadvantage that the number of different denominators in the integrand is greater than in the original integral. This complication gets worse in more complicated graphs. We want to indicate another method which will be used in the current work For convenience, we take the subtraction point at the origin (q. = p. = 0);

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501

this introduces no loss in generality.4 Then a simple scaling argument shows that I’ (now called I,,) is a function of two dimensionlessratios.

We note that F(O,O) = 0.

(1.9)

If we differentiate Z, with respect to m2, the two terms in (1.3) will converge separately. - aF = $ am2

G ($,

g)

- $

(1.10)

G(O, O),

where

G($ ’$1= -m2 j j d2k [(k_ ; + 4)2

p2

m212

(1.11)

= G($o). We should like to find a simple operation which turns G into F. One way would be simply to integrate (1.10) from m2 to infinity (noting that F --f 0 as m2+ 01). If we do so, it is necessaryto keep the G(0, 0) term in order to make the integral converge at infinity. It is just this type of subtraction which we wish to avoid. Instead, we may use the obvious behavior of F as m goes to infinity (namely, m2F increasesat most as a power of In m2) to write F(m2) = j:, =

drn12In ($)$

rnf2w

ml2 a m dm'2 In 7 ___ s ma i i am’2

2 G

(

-it2

(1.12)

2 3 - 212

1*

We now have then arrived at the following rules for obtaining the finite part of the original integral. (i) Differentiate the integrand with respect to mz. (ii) Carry out the momentum integration. (iii) Perform the successionof differential and integral operations specifiedby (1.12). The final operation may seemas unpleasant as some of the things we have tried to avoid; but in caseswhich we have considered, it can be bypassed rather easily. Examples of this are given in the 4 We may later recover I’ from Z’(P, q, 4

= UP,

am)

-

44~~~

40,m).

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KU0 AND YENNIE

appendix. The advantage of our method is that we need never explicitly carry out the subtraction at the origin of momentum space. F, as given by (1.12), automatically satisfies (1.9). This point may not be appreciated in our oversimplified example, but it seems to be a real advantage in the general case where p and q may themseIves be variables of integration. II. GENERAL TERMINOLOGY AND FORMAL ALGEBRA OF RENORMALIZATION

THE

A, TERMINOLOGY There exists a large battery of nomenclature necessary in the usual treatment of renormalization; its entirety is not needed in the treatment of this paper. For the benefit of readers who are not familiar with the conventional treatment and for the purpose of quick reference we compile here a glossary of terms that are used in this paper. Our usage of them does not differ from that of Dyson (2); a reader who is conversant with Dyson’s renormalization program will find nothing new in this subsection. A general familiarity with Feynman rules is assumed. We begin with the definition of the degreeof divergence, denoted by d, associated with a graph. It is just the power count of the integral corresponding to the graph, i.e., it is the number of internal momentum powers in the numerator, including a 4 for each d4ki , minus the number appearing the denominator. We shall often speak of a graph as being superficially divergent if its degree of divergence is non-negative (d = 0 for logarithmic divergence and d = 1 for linear divergence, etc.). By this power-counting definition, divergence of a graph implies the superficial divergence of the graph itself or any of its subgraphs. The crucial condition for a theory to be renormalizable is that the degree of divergence of any type of graph has an upper bound independent of the number of internal vertices. In quantum electrodynamics (QED), we have for any given graph d=4r-2b--f,

(2-l)

where f = number of internal electron lines, b = number of internal photon lines, and r = number of internal momentum integrations. As pointed out by Dyson, the dependence on the numbers of internal lines can be eliminated through the connections between the number of vertices, the numbers of external and internal lines, and r, so that d depends only on the numbers of external lines, d=4--F-B,

(2.2)

where F = number of external electron lines, and B = number of external photon lines. Thus an upper bound clearly exists. Barring graphs having no external

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(e)

503

(f)

FIG. 1. Examples of electron self-energy (ESE) graphs; the x in (e) and (f) represents a (- 6m) vertex.

lines, which have no physical consequence,and graphs with F = 0 and B = odd, which violate charge conjugation according to Furry’s theorem, there are only four types of superficially divergent graphs to be considered: (i) F = 2, B = 0, d = 1. Graphs of this type are called electron selfenergy (ESE) graphs. Some examples are shown in Fig. 1. The principal complication associated with electron self-energy integrals is the need for massand charge renormalization. The method for treating massrenormalization will be discussed here. Suppose the bare mass is m, ; then each bare electron propagator is (pi - m&l. Since it is desirable to expressall quantities in terms of the physical massm (= m, + am), we rewrite the bare propagator 1

1

p=p-m+13m P - m, 1 =-+ P-m

+

L(--6m) g-m

& &

(-Sm) &

+ *.a.

(2.3)

We can view a factor (-am) in (2.3) as a new type of self-energy insertion (called a (-am)-vertex). Examples of such insertions are given in Fig. le,f. It simplifies matters to consider only the proper self-energy graphs. By a proper graph, we mean a graph which cannot be divided into two disjoint parts by cutting a single line, e.g. the self-energy graph, Fig. Id, is not proper. Let DG)( #, m) be any such proper ESE integral; its graph may contain (-Sm)-vertices internally, but the complete graph is not itself a (-Sm)-vertex. The sum of all such DG) is called

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KU0 AND YENNIE

,Z*( $J, m). Then the complete electron propagator and (- 6m) insertions5: si=

1 ~-m ___ 1 + p---m

-I- &g*

- a@&

(z* - 6m) +&z*

1 = p - m - Z*(p,

contains any number of Z*

-

sm+-+

0..

(2.4)

m) + 8m *

The final form of (2.4) may be regarded as simple a shorthand for the expanded form. The two expressions agree to any finite order of perturbation theory, and any analysis we make is true only in the sense of perturbation expansion. The significance of the physical mass is that the electron propagator has a simple pole at p = m. This requires zl*(p, m)lti-

= am.

(2.5)

We now want to redefine the meaning of graphs so that graphs with (-am)vertices will not be considered separately but will instead be incorporated automatically in the graphs without (-am)-vertices. For example, we wish to split Fig. le into a number of terms, one of which is taken with Fig. la, another with Fig. lb, etc. The combinations are represented only by Fig. la, Fig. lb, etc. Similarly, an appropriate part of Fig. If is to be combined with Fig. lc and represented by the latter graph alone. To carry out this program, let us define the mass renormalized self-energy for proper ESE graphs which do not contain ESE subgraphs, ,DG’(p,

m) = L!YG’(@, m) - SmcG’,

(2.6a)

where 8m(G’ = ZcO(m,

m).

(2.6b)

For graphs which contain ESE subgraphs we still use the above definition for J?“)(n m) but the definition of DG’( p, m) is modified as follows. Suppose a graph G contains senior ESE subgroups S, ,..., S, ; a senior ESE subgraph is one which is not a subgraph of another ESE subgraph. Then in DC), we substitute jpl’ ,*--, tf(sm) into the original amplitude in place of PI),..., Dsn), respectively. Since the subgraphs Si will have been treated in previous orders, this gives us an 5 The same result (2.4) can be reached by an alternate treatment. One may introduce a term into the free Lagrangian such that the mass of the electron appears as m. In the meantime a counter term (-Sm) is introduced in the interaction Lagrangian. The latter than generates all graphs which can be summed up as in (2.4).

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inductive definition of L?(G)for all G. It is clear that graphs containing (-am)vertices are now incorporated automatically into graphs containing the corresponding ESE insertions.6 If we now sum (2.6) over the set Q$ ,’ we obtain precisely the same set of integrals that we had with the graphs as originally defined with (-&)-vertices, (2.7a) (2.7b) and

Z*(p,m)= c .PG’(p, m) GECP,

w3)

= Z*(p, m) - 6m

yielding S&f, m) = [p - m - Z*(p, m)]-‘.

(2.9)

From now on, we need consider only graphs without (--6m)-vertices; unless explicitly noted, the contribution corresponding to a given G is z(GJ. Since the subtraction of 8m(G)takes away the leading divergence, the degree of divergence of z(c) is reduced to d = 0. (ii) F = 0, B = 2, d = 2. These graphs are called photon self-energy (PSE) graphs, and some simple examples are shown in Fig. 2. Any such graph G will give a photon self-energy contribution Ii’;‘(k).

Similar to ESE graphs, we also define Lr,$ to be the sum of 17$ over all proper PSE graphs.’ Being a second-rank tensor, 17,*, cannot be put in the denominator as L?* was in (2.4). However, LrzV has the formal property of gauge invariance given by (2.10) 17,*,(k) = k,X - k&v) 17(k2). 6 Practically speaking, a given 2%) now represents a sum of integrals. Those which would have (-&I)-vertices in the abandoned definition of graphs will generally be less divergent than the term with no (-&)-vertices. However, we need not pay special attention to this, and we stick to the convention of taking the degree of divergence as that of the most divergent term represented by a graph. ’ We shall represent the set of proper electron self-energy graphs by ~2’~. In the same way ol, and ol, will represent the sets of proper photon self-energy graphs and proper vertex graphs respectively. Only graphs without (-Sm) vertices are included.

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b)

h)

Id) --o-u-- --cmIf)

(e)

(4 FIG. 2. Examples of photon self-energy (PSE) graphs.

Although we do not wish to discuss the point here, this condition is usually taken to mean that the degree of divergence is reduced from d = 2 for I7$ to d = 0 for I7. Ultimately because of gauge invariance, the k,k, terms will not contribute to any physical amplitude and we may drop them. We may take the photon propagator to be (2.11) &Jc-2 (gauge terms may be added later). Then the complete propagator is effectively

DiJbv= + = 9

+ &$ (gA“k2fl) -!k?$.+ .-. (2.12) [l - II(kI.

(iii) F = 2, B = 1, d = 0. We call these vertex (V) graphs; simple examples are found in Fig. 3. The general form is

some

WC% P’). Notice that A, is defined without one power of e, , the bare charge. The complete vertex (which is also defined without one power of eO) has the simple form ~u=Yu+4b, where A&,

p’) is the sum of At”) over all proper V graphs.7

RENORMALIZATION THEORY

A(cl (e) FIG.

(iv) of photon apparent; by gauge divergence

(fl

3. Examples of vertex (V) graphs.

F = 0, B = 4, d = 0. Graphs of this type describe the scattering by photon. Here, as in case of type (ii), the leading divergence is only after suitable combinations of graphs are taken together as required invariance, the observable parts of the integrals have a degree of of -4. So these types of graphs are really convergent.

Finally, we define the notion of irreducible graph as one which does not contain a subgraph of types (i), (ii), or (iii). Although there are infinitely many irreducible V graphs (examples Fig. 3a,c,d), only the lowest order self-energy graphs (Fig. la, 2a) are irreducible. All higher order self-energy graphs are reducible. B. FORMAL THEORYOF CHARGERENORMALIZATION

From the foregoing discussion of superficially divergent graphs, it has been seen that the leading divergence for any graph is at most logarithmic after mass renormalization. A suitable further subtraction would render all graphs convergent; such subtractions will be defined and discussed in subsequent sections. An

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important feature of these subtractions is that they correspond to a multiplicative renormalization of the charge. It will prove convenient to decouple the discussion of charge renormalization from that of finiteness and treat it at the present point. We always have in mind perturbation theory in which the simpler graphs are analyzed first before proceeding to more and more complicated graphs. Thus when we encounter a superficially divergent graph (G) containing a superficially divergent subgraph (S,), the latter will have been analyzed completely at some previous stage. Further, if the subgraph is shrunk,* we will obtain a graph (called G/S) which will have been analyzed at some previous stage. These “shrinkages” arise in a natural way if we recognize that every vertex or self-energy can be split into a constant part plus a varying part. For example, we may set LFG’(p, m) = B$QY - m) + vpG’(p, n(Gyp) @‘
= c(C) a + yqpG’(p) a

p’) = Py,

+ “y-,&)($4

m),

(2.13a) (2.13b)

9

(2.13~)

p’).

Here we have simply chosen for each graph G a constant (B, C, or L); “v;, simply stands for the operation of subtracting the constant term from the original function. For the present discussion the constants can be defined in any manner; each graph has a unique constant. Later on, we shall define the constants by selecting some subtraction point a (E (a1 , a, , a&) and setting9 (a, - m) BCG) a = 2zG)(p* m)ld=al 9

dG) = IFG)(k2)l& YuLLG) = 4% Let us now introduce

the uniform

notation

9a )

P’L~‘.d=o, W for 2,17,

(2.14b) *

(2.14~)

or (1, and K, for B, ,

cl 3 L *

Now we can see how the shrinkages work. If & is internal to G, we may split the part Si into two terms, the V-part or the V-part. If we select the %-part, the resulting contribution to G is K,(‘*) times the contribution of the graph (G/Q. 8 To shrink a self-energy graph means to remove it and join the two lines that lead to it into a single line. To shrink a vertex part means to replace it by a simple vertex (r,). If there are several occurrences of a given subpart S, , our notation appears not to be sufficiently general. We can get around this by labeling the different repetitions &*, Si2, etc. Where necessary, we imagine this to be done, but do not call attention to it explicitly again. 0 These relations are understood as follows. In (2.14a), pais written pp. In (2.14~3, nrr is constructed with factors 7, , p, p’ in various orders (e.g., there is no factor pu , which is instead expre=ed as hw, + Y,P)).

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509

We seethat a general graph G can be decomposedin this way into a large number of terms of various types. These are (a) its %-part; (b) The constants KhS1’... KLse) from the g-parts of certain subgraphs S, , S, ... S, times a contribution corresponding to a graph G/&S, ... S, where S,S, ... S, have been shrunk.8 We want to define things so that this reduced graph is not itself a V-part and contains no g-parts; (c) A contribution corresponding to the complete graph G and not containing any %-parts. The latter quantity is obtained by removing all E-parts in all possible ways. Let us introduce the uniform notation WAG)for this contribution with no V-parts.

lb)

+El

FIG. 4. Examples of the decomposition of various self-energy graphs. A shaded area represents the constant part (Q-part) of the graph or a subgraph. When this shaded area is shrunkS, a graph of lower order is obtained. A box around a graph represents the residual contribution obtained when all its Q-parts have been removed. Thus a boxed graph with shaded subgraphs stands for the product of the constants of the shaded subgraphs and the residual contribution of the graph obtained by shrinking the shaded areas. 595/51/3-9

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The procedure we have just indicated is still somewhat vague because of a complication known as the “overlapping divergence problem.” This refers to the fact that it is possible for two subgraphs to be neither disjoint nor nested. In quantum electrodynamics this occurs only in the case of self energy diagrams, as is illustrated in Fig. 4. If we did not have this situation, we would simply take each internal Wcsi) and divide it into a V-part and a Y-part.lO When there are overlaps, we must proceed by induction. Identify all possible combinations of g-parts in G. These consist of sets S, .. . S, which are disjoint. If these subgraphs are shrunk, we shall obtain a graph G/S, . . . S, which was treated in a previous order, so that WAG’S-..SJ is known. Multiply this by KA’J .. . KkSe). Subtract all these terms from “y;l WCC).We then (by definition) have eliminated all %-parts from WtG). Hence $‘-w(G)

=

W,‘,“’ +

C

&h’

. . . &&)w~&...~,)

(a.. ...e}

(2.15a)

where the summation extends over all (nonempty) disjoint (i.e., nonoverlapping and nonnested) sets of subgraphs. This gives us a recursive definition for the Wa’s. We illustrate these definitions by a few examples. For simple graphs like (lb), (2b)ll we have Plb’(p,

m) - (p - m) &lb) = 2LP)Z2b’38)(p, = 2L~)pq

m) + ,xyJ’(p, m)

(2.16a)

p, m) + 2y”‘( p, m),

We adopt the notation that the constant part of a (sub)graph, i.e., (p - m) B~G’(L~G’, CA”‘) f or an ESE (V, PSE) (sub)graph be represented by putting the (sub)graph in shade. The a-amplitude is represented by enclosing the (reduced) graph in a box. The K,‘s which appear in (2.15a) then correspond to different disjoint shaded areas in the original graph. The graphical counterparts of (2.16a,b) are given by Fig. 4a,b. lo In the case of nested divergent graphs, if we take the Q-part of a graph, we need not consider its internal parts further; the definition of % includes all the internal properties of the graph. On the other hand, if we take the V-part of a graph, we must divide its subparts into Q- and V-parts. I1 The key to labeling the graphs may not be self-evident. We have chosen to use the figure number of the graph as its label.

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THEORY

For a slightly more complicated

graph like (2g), we have

n(2g)(k2)

+

_

,pg)

=

L~3dpm) +

Lp)~y/3C)(k2)

~hf)fl(%/d(k2) a a

+

+

p)p/3e)(j4

L(3it)L(3C)~~g/38SC)(k2) a a

(2. MC)

= L~=)na(2d)(k2) + L3j7”&‘) +

Graphically as follows:

+ ~2”

(P

+ Lp)Lp))

Ii’p’(k2)

it is Fig. 4c. In the above the identification (2b/3a) (lb/3a) (Wa) WW (2g/3e)

= = = = =

+ L7~g’(k2).

of reduced graphs are

(2a); (la); (2% WI; (2g/3f) = (2g/3a 3c) = (2a).

We may also write (2.15a) in the form VaWcG) = c K,(G --f G’) WF’)

+ W:’

(2.15b)

G’

where G’ is summed over all graphs reducible from G. The factor K,(G -+ G’) is obtained by summing over all “shrinkages” which give the same G’ from an initial G, e.g., K,(2b + 2a) = 2Lp’ &(2g -+ 2a) = LL*’ + Lp” + Lkm’Ly). Now let us consider each type of divergent structure in turn, starting with the vertex part. The bare vertex is y,, and the complete proper vertex is given by Ah.5 P’, m, ed =

C

J?‘(P,

P’, m, 4.

GM"

Using our definition, 4h

P’, m, 4

this can be rewritten = yrL, +

+ 4,(~, c

KdG -

P’, m, 4 (3 &“(P,

P’, m, 4

G.G’ =

Y,PL i- c [ 1 + c K,(G G’

G

G’)] &“(p,

p’, m, e,,). (2.18)

512

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AND

YENNIE

The summation over G refers to all graphs G which can be reduced to G’ by shrinking combinations of S-subgraphs, or equivalently, graphs which can be built up from G’ by inserting combinations of S-subgraphs. The coefficient of AA:‘) is easily determined because all the insertions are independent, and its coefficient becomes a product with one factor for each vertex, each electron propagator, and each photon propagator. The constant factor for each vertex is (1 + L,); the constant factors for the electron and photon propagators are geometric series which may be summed (i.e. interpreted) to be (1 - B&l and (1 - C&l respectively. If the graph G’ has n photon lines, the net factor will then be (2.19) Now Ah:‘) has precisely 2n powers the factor (2.19) into ep and write 4(P,

of e, ; thus it is possible to absorb most of

P’, m, eo> = yLLL + (1 + L,) A,,(p,

p’, m, e,)

(2.20)

#here

The complete vertex is then r,(p,

P’, m, eo) = yu + L(p,

P’, m, eo)

= (1 + La)[yu + L,(P, = (1 + L) r,,(~,p’,

P’, m,

e,>l

4 e,).

(2.22)

This simple result is quite significant. It tells us that we can easily express the complete vertex function in terms of the functions Ai:) defined by (2.15), thus confirming the cogency of that choice. Had we used some other subtraction, this simple result would not have been obtained. The subtraction (2.15) is thus said to correspond to a multiplicative charge renormalization. It may be noted that (2.15) is homogeneous in powers of e, . In calculating (2.20), we then simply replace e, by e, , which is equivalent to multiplying e, by a constant, as in (2.21). The other functions work out in a similar manner. The electron self-energy is expressed in terms of the u-amplitudes by

z*(#/, m, e,) = (P - m) 4 + (1 - 4) zfa(P, m eJ

(2.23)

RENORMALIZATION

513

THEORY

which gives for the electron propagator

Sk@, m, e,) = (1 - &J’

[P - m - %YP, w e,)l-l

= (1 - &Y The photon self-energy

&dP,

(2.24)

m, 4.

takes the form

17*(k2, f72,c,) = C, + (1 - C,) K&k2, m, 4 so that the photon propagator

(2.25)

becomes

Dk(k2, m, e,) = (1 - C,)-l& = (1 -

C&l

[l - 17,(k2,

m, eX1

(2.26)

DF,(k2, m, e,).

When the results (2.22), (2.24), and (2.26) are inserted into any skeleton graph, each vertex is to be multiplied by e, ; the factors (1 + L,), (1 - B&l, (1 - C,)-l then convert each of these eO’s to e, (when due account is taken of the special features of external line renormalization). The quantity e, has now disappeared completely from every physical amplitude, and all functions are expressed in terms of the a-functions and the charge e, . An interesting property of the formal algebra of renormalization is that it is transitive-this is related to what is usually referred to in the literature as the renormalization group (13). It has to do with the freedom of choosing the subtraction point. Suppose we were to choose b = (b, , b, , b3) instead of a in Eqs. (2.13), (2.14), and (2.15). We then would have reached a set of amplitudes Wp Wb(G’ = VpW’G’ - c K,(G -+ G’) Wb’G”,

(2.27)

G’

where KIG) is defined in complete analogy to KAG). The reader may convince himself that relation between the a-amplitudes and b-amplitudes is as follows:

Wb(G)= v,wF) - ; K&G -+ G’) Wr”,

(2.28)

where symbolically Kif’ = WCC)/ a b? i.e.,

(b, - m) Bg’ = z”‘(P,

m)~d-~~ , (2.29)

Cz’ = L?(G)(k2)l,2=, a e2 , y ” Ly’a = dG’(p au 9p’)l 2,=a’.%-b, .

514

AND YENNIE

KU0

The constants are related as follows:

1 + Lb = (1 + &Al + Lb) 1 - Bb = (1 - B,)(l - Baa) 1 - c, = (1 - C,)(l - c&J.

(2.30)

Other relations are r&p,

PI, m, 4

&&A

= (1 + Lb> rb,(p,

m, e,) = (1 - &X1

&,(A

PI, m, 6) m, eJ

(2.31)

D&dk2, m, e,) = (1 - Cab)-'D~o(k2, m, eb), with (2.32) It follows that if any set of u-amplitudes finite b are also finite.

are finite, then all b-amplitudes

with

C. THE COUNTER TERM FORMULATION OF RENORMALIZATION In the preceding discussion we have given a prescription for decomposing a Feynman integral into a sum of terms, each of which corresponds to a particular combination of subdivergences of the integral. When all Feynman integrals of a given type are summed, the contributions miraculously combine to produce a charge renormalization of the finite expressions. A somewhat more appealing approach is the counter term formalism (14) in which the various terms corresponding to shaded graphs are subtracted away as they arise. We shall now describe this approach and its connection to the one we have employed. The basic idea is to renormalize the field operators at the start so that the final propagators and vertices will have their correct limiting values near the mass shell. These limiting values are just the same as the lowest-order perturbation values (2.33a) &l(P)

1 z+p_,,

1 D,(k) --.k=-+O k2

(2.33b) (2.33~)

THEORY

515

#tN = L.1+ &W2 #Ad, 40) = 11+ GW2 k&4, [l - &)I

(2.34a) (2.34b)

RENORMALIZATION

Let us write this renormalization

in the form

e” = [l + B(e)][l

+ C(e)]‘/” e’

(2.34~)

where 8, C, and i are expressed in terms of the observable charge. They will ultimately be related to the functions B(e,), C(e,), and L(e,) defined in the previous subsection. This substitution is usually carried out in the original Lagrangian and the resulting terms proportional to J?, C, and t are the counter terms to which we have referred. They are regarded as part of the perturbation. The usual canonical scheme has some awkward features which we wish to avoid so we instead carry out this renormalization in the interaction picture. The interaction is = e$r4.#T

+ (-L~&4&

-

6mf.l + ~l$,$l>.

(2.35)

We must take into account that the factor pairings now have an additional constant. For example, the pairing of tir and $T gives, in momentum space, the factor 1 --= 1 1 + B 1/ - m

1 (8 - m) + &2/ - m) 1 =-LL--B@ f-m+

p-m

1 +fl--m

[-&a

- m)]

m)l jj-&

&

I--&P - m>l-j--& + .--. (2.36)

Thus we have a choice: either associate a factor (1 + B)-l with each electron line or introduce a new type of vertex -8( p - m) which is to be inserted in electron lines in all possible ways. The second procedure is followed. Clearly this new vertex can be inserted at all places where a self-energy could occur. It will provide the mechanism for canceling out the terms linear in ( p - m) so that both the bare and the complete propagator take the from (2.33b). The same procedure is applied to the photon propagator 1 --= 1 1 + C k2

1. + L [-@I k2 k2

$

+ $ [-f?k2] ;

[-ck2]

$ + e.0.

(2.37)

516

KU0

AND

YENNIE

Again, the second procedure is followed and the [- &] factors are represented by vertices inserted in the photon lines. Thus, in addition to the usual vertices, we have four types of counter vertices which are proportional to 2, C?,J?, and %z = 6m[l + 81. These are employed in a manner very similar to the way discussed for the -&n-vertices in Section II-A(i). Each of them is expressed as a sum of terms, with each term representing a single Feynman graph without counter terms.

Each term &fG) then provides a counter vertex corresponding to the graph G. The 2% discussion is entirely unchanged so we consider only the charge renormalization counter terms. Let W cG)be the contribution from a Feynman graph (G) without counter terms and W,fG) be the net contribution when all associated graphs with counter terms are added. Each of these associated graphs is obtained by replacing some of the vertex and self energy subgraphs in G by the corresponding counter vertex. The result is clearly WY)

=

w(G)

+

_

R(G)

c (-@%‘) (U...fZ}

. . . (-&‘Ss’)

w(G&...&)~

(2.39)

The constants @f) will have been determined in a previous order, and the constant &cG) is fixed by the condition that WiG) vanish in a suitable manner near the mass shell. The complete functions are

where

RENORMALIZATION

517

THEORY

We want to confirm now that the two prescriptions for defining the finite parts of self energies and vertex integrals agree. We outline how this may be done and leave the details to the reader. By considering (2.39) near the mass shell, one finds &Y?(e) = R(G)(e) -

C (-R(sJ(e)) {a...e}

. . . (-j+)(,))

K$I~-JJ(~).

(2.41)

Whence, using (2.13), (2.39) may be written WY) = y-,jAG) + c (-$%‘)

. . . (-@&‘)

~m,+WJ,)~

(2.42)

Now substitute from (2.15a) and note that all the terms corresponding to shrunken graphs cancel leaving @)

= w(G) m

(2.43)

as was to be shown. If one sums (2.41) over all graphs and uses the same analysis as led to (2.19), one finds [l - L(e)] L(e,) = t(e), or

11- &)I[1 + L(e31= 1 and

11+ &4lP - B(e31= 1, [l + C(e)]]1 - C(e,)] = 1. The equivalence of the two approaches has thus been demonstrated.

III.

INTERMEDIATE

AMPLITUDES

The first step in the renormalization program is to obtain a unique finite part of any Feynman integral. In the process, divergent contributions having the structure of renormalization terms will be subtracted away. In Section IV, it will be shown that these subtractions do correspond to a renormalization of mass and charge, as discussed in Section II. Following Bjorken and Drell (7), it seems most convenient to make these subtractions at the origin of momentum space rather than at the physical particle mass shell. Using the transitive property of renormalization, the subsequent transition to the mass shell is straightforward and introduces no new divergences (aside from infrared divergences, which may be considered to be well understood). The amplitudes obtained in the first step will be called intermediate renormalized amplitudes, or I-amplitudes for short.

518

KU0 AND YENNIE

In the following treatment., the only criterion for the convergence of a certain Feynman integral is the power-counting theorem (5) which states “A Feynman integral converges if the degree of divergence of the integral as well as the degree of divergence associated with each possible subintegral is negative.” A. IRREDUCIBLE GRAPHS Following the above theorem, the I-amplitude irreducible graph G is clearly given by W,‘“‘(p) = s dk [ Y”‘(p,

k, m) - Y”‘(0, k, m) 0.. - !f. avy(c) (0, k, m)] v! apv

-pG(O,k,m)=

s

dk

’ de v s0

associated with any divergent

V.

($)‘”

V”‘(,$p, k, m),

(3.1)

where VG)( p, k, m) is the integrand constructed according to Feynman rules; p, k stand for the momentum variables external and internal, respectively, to G; and v is the degree of divergence of G. The structure of the integrand of (3.1) clearly depends on the way the external momenta are routed through G, but it will be seen that the integral itself is independent of the routing. Now let us carry out the &differentiations in (3.1). For an electron line carrying total external momentum P and total internal momentum K, we find

This has the effect of introducing an additional vertex with factor (-P) in the electron line; we call this a P-vertex. For a photon, we find in a similar way

a at

i 1 (P + Q2 = (@’ + Q2 (--2P . K - 2P2) (cp : KJ2

(3.3)

giving a P-vertex with factor (-2P +K - 2fP2). In cases where there are more than one c-differentiation, we may obtain a double-P vertex from differentiating the numerator of (3.3); the resulting factor is (-2P3. The integrand of (3.1) now has (v + 1) P-vertices (counting double-P vertices as two vertices). Corresponding to these (v + 1) (or more) numerator factors of P, the degree of the internal momenta is reduced by at least (v + 1). For fixed 5, the internal integration accordingly converges. We assume that we may interchange the f and the k

RENORMALIZATION

519

THEORY

integrations. To show that WI is finite, we then have to show that the momentum integration does not lead to a function of 5 which is singular. Inspection shows that such a singularity could occur only for 5 ---f 0 as a sort of infrared singularity from neglecting the external momentum in the denominators. A glance at (3.2) shows that no such trouble arises from setting t equal to zero inside electron lines since the electron mass prevents an infrared divergence. Apparently, trouble might arise from too many differentiations (three or more) on one photon propagator. However, we note that a n-i an 1 ~a (@ 1 (EP + K)z = p” aKu i at 1 W : KY

(3.4)

and integration by parts permits the derivative to act on some other propagator. Since n is at most three, it is possible to avoid having too many differentiations on one line, and the integral converges. To see that the result is independent of routing, we introduce an equivalent way of defining W:“) by differentiating with respect to the mass WIG’(p)

= R:’

j dk (+j”

V”‘(p,

k, mx),

where n is any integer greater than v and the operator R!$ is defined by its properties: (4 R c)x-* = 0 if m is an integer satisfying 0 < m < n. (b) Rp)(a/axp F(x) = F(1) if F(X) is regular in 1 < x < co and F(x) = 0(x-l In” x) as x -+ co and m is any finite integer. An explicit integral representation of this operator is given in Appendix A and the class of functions C, for which R’,“’ is defined is described there also. Basically, the Property (a) is defined to project out the subtraction terms and Property (b) to make the remainder agree with (3.1). To establish this equivalence between (3.1) and (3.5), we write the integrand as

JAG)= Jrp + VP) + .. . + Y@) ” + vP’, where VG) 0 stands for the ith term in the Taylor’s expansion of the integrand and VA” is the remainder after v + 1 terms. We prove first that the subtraction terms are projected to zero by the combination of operations introduced in (3.5): RF’ s dk (&j”

V<(k, mx) = 0.

(3.6)

520

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YENNIE

We note first that the x-differentiations make all the k-integrations convergent. In (3.6) it is then permissible to rescale the integration variables by k + kx; the x-dependence then factors out as a pure power satisfying the condition in (a) and the application of R, gives zero. It remains to be shown that Rk’ j dk (&)”

V$%?, k, mx) = j dkVf’(p,

k, m).

given

(3.7)

To see this, we express VhG)in terms of the 5 integral, as in (3.1). Now carry out the t-differentiations. Then since the k-integration converges, we assume that the x-differentiations may be taken after the k-integration. At this stage we have Rt’

(2,”

j dk j; df 9

= Rk’ (&)”

($r+’

x” j dk j’ d[ 9

YcG)(&, k, mx) (-$)‘+’

V(‘) (-$I,

k, m).

0

Taking into account the x-dependence of the (V + 1) P-vertices, and using the same arguments as in proving the finiteness of (3.1), we see that as x + co the integral vanishes sufficiently rapidly so that (3.7) is valid. Clearly (3.5) has some advantages over (3.1). These are (i) The form (3.5) is clearly independent of the particular routing of the external momenta through G. (ii) The number n used in (3.2) may be any number greater than the degree of divergence of the graph. Therefore, for a given renormalizable theory, one can choose a universal n for all graphs (for QED, one may choose n = 3). (iii) It is clear from the proof that the procedure (3.5) does not alter the amplitude of a convergent graph. This property will be convenient later in verifying the gauge invariance of the Z-amplitudes. As a convenient shorthand, we introduce the operation Q W’“‘(p)

= R:’

j dk ($-)”

Q on an integral

V”‘(p, k, mx).

e consists of three steps: (i) Multiply all masses in the integrand by x; (ii) Differentiate n times with respect to x; (iii) Apply Rkn) to the whole integral.

RENORMALIZATION

THEORY

521

In this notation we have, if G is irreducible and divergent, @)

= p&G)

and if G is irreducible and convergent we also define @)

= Q&G’ = w(G)+

B. REDUCIBLE GRAPHS When there are divergent subgraphs the situation is much more complicated and the operation Q is not sufficient to produce a finite result. The power-counting theorem suggeststhat one should apply the same subtraction procedure to every subintegral which is intrinsically divergent. Following Dyson, we note that such integrals are associated with superficially divergent proper subgraphs (or S-subgraphs, for short) and combinations of them. The complication of renormalization theory has always been that of defining the subtraction procedure when subgraphs overlap each other. However, it will soon be clear that our Q-operation has the desired property that it yields an unambiguous result even when such overlapping divergences are present. To see what this means, we first spell out the prescription for obtaining the Z-amplitude associatedwith any given reducible graph G. Let S, , S, ,..., S, be all the S-subgraphs of G, ordered so that if Si C Si then i > j; also define G to be S,, . The ordering is only partially determined by these conditions since some pairs of subgraphs may be disjoint or may overlap. For such pairs it is necessary to show that the ordering does not affect the final result. It is necessary to go into a little detail on the treatment of ESE subgraphs. We recall that we have defined graphs in such a way that (-&P)-vertices are not shown explicitly, but are understood to accompany each ESE subgraph. The recursive definition of ,J?(G)of Section II can be symbolically summarized as

The inputs to this are the z(G)’ s, which are given by a set of Feynman integrals, whose divergences we want to remove. Suppose for a moment we have achieved this and call the finite result JYiG).We then use,?Yi:“)asinputs to (3.9a) to obtain ,??jG)

It is clear only finite quantities are involved here. But inserting 2;‘) into ,Yi’:“) also implies that the role of 8rncG)is now played by a finite constant 8mjc’ = 2p)(m, m, Zj”).

(3.9c)

522

KU0

AND

YENNIE

This means that when we remove the divergences of Ds) we should also replace 8mcs) by am,tsJ. This short discussion preambles the first step of the prescription which we now state: (i) Write out the sum of Feynman integrals for any graph, with one term for each combination of (--6m)-vertices. Consider each term separately, and replace each ( -8mtS)) by the finite constant (-6miS)). Note that different terms in the sum will now have different sets & since we have collapsed some self-energy graphs and replaced them by --6m, (‘) . We no longer are concerned with the internal structure of these collapsed graphs and those &‘s and any of their subgraphs are omitted from the set {&}. (ii) Multiply all masses and &ml’s in S, by xi for all i = 0, 1,2,..., s such that & is in the set for the particular integral. Let m, be a collective symbol to represent this dependence on the xi’s. (iii) Apply

to the integrand. (iv> APPLY

to the whole integral. (v) Sum the separate contributions. One of course has the option of choosing ni = maximum degree of divergence plus one or as large as is necessary to remove the divergence in Si . Moreover, in listing all the S-subgraphs one is free to include finite subgraphs as well; one may in fact include all subgraphs. As a shorthand, we shall occasionally represent this sequence of operations by the notation

@) =

Qs

.a. QIQow’“’

= fi R’,:” 1’ dk fi

(-&-,“I

V’G’(~, k 4.

(3.10)

i=O

The order of the Qi’s is that of the Rz)‘s since the differential operators (a/ax,)“9 obviously commute. The proof that the result is always finite is saved for the next subsection; however, it should be clear that the differentiations make the k-integrations hnite.

RENORMALIZATION

523

THEORY

Let us illustrate this prescription, and the reason for the order of the R operations, by applying it to the photon self energy contribution corresponding to Fig. 2c. With an appropriate assignment of momenta, the resulting intermediate amplitude is

1

1 x Tr Ye p - mx

1

x Tr IL

1

p! - mx ”

P+k-mx

1 p

_

mx

(-smP”‘x)

p

2

mx

yy

p

+

&l-

mx

1

i

(3.11)

1

where (dp) is shorthand for d4p/(2a)4. A particularly convenient routing of the external momentum has been selected. We may obtain the expansion in powers of k by expanding the electron propagator 1 I-k-mx=

1 p-mx+

1

1

p-mxkp-mx

When this is substituted back into (3.11) and the differentiations are carried out, the first three terms are seen to vanish by scaling (p, q --f px, qx) and using Property (a) of R,(%). In the final term we may take the x-differentiation outside the integral. If we now carry out the y-differentiation and scale the integration variables ( p, q + xp, xq), a factor x-l may be extracted from both integrals, and the only x-dependence inside the integral is in the factor 1 p - (k/x) - m . As x + co, the integral goes to a well-defined limit of R, to write

and we may use Property (b)

524

KU0

AND

YENNIE

Now we may treat the internal electron self-energy in the same manner, 1 Pfd-my

1

+ &$-~)~+(&~jz/+;-my-

=- d-w

The first two terms give zero by scaling (4 + qy) and using Property (a). In the last one, the y-differentiation may be taken outside the whole integral. To investigate the y-dependence as y -+ co, we again scale (p, q + yp, yq) and extract a factor y-l. The resulting integral is

j @!)W$ Tr[Yup_ :mly)Ye(A ’ d - k/Y)

”

(P - im/y)

PjZp + : _ mf (3.13)

l 1. P - (k + m)ly

ky

As y -+ co, we perceive the danger of an infrared-type divergence at the lower limit of the p-integration. Counting powers, we see that the divergence is logarithmic and hence the integral will contain only a term In y. Applying R>‘(a/ay)nl and using Property (b), we find finally

1 x

p

+

d _

(13 m

YO -

am1

l

(j&h)” 1jj& YtJ

P-h-ml

(3.14) where L$“‘(p,

m) = - je, j (&I 3/o (A

P)’ p + : _ m y” -$ - 8mp’

is just the lowest-order electron self-energy contribution of Fig. la. It is clear that (3.14) is equivalent to the I-amplitude corresponding to the lowest order photon self-energy graph (Fig. 2a)

with the following insertion 1

[p-ml

-

&

Sp)(p,

m) --L-

3-m

.

RENORMALIZATION

525

THEORY

Now we have to study the effect of interchanging the R operations. The trouble arises if all the x- and y-differentiations are applied to the electron self-energy subgraph. Consider the situation for k = 0. After the differentiations, we scale as usual (p, 4 + xp, xq) and extract a factor x2-% times a finite integral independent of x0. This integral is just what would be obtained by setting x = 1 after the differentiations. The operation Rr) would immediately give zero, but we try to apply RF) first. We study the behavior of the integral for large y by scaling (p, q + yp, yq) and extracting a factor y2-?. The remaining integral contains a y-dependence solely through the factors

Elsewhere the result is equivalent to setting y = 1 after differentiation. to expand the integral in inverse powers of y by using

Now we attempt

repeatedly. The first term y” leads to no trouble, but the next term (y-l) does not converge because of an infrared divergence as p ---, 0. If instead we keep a finite remainder after the first term, the integration will be cut-off logarithmically at p - (m/y). The net result is that the combined function upon which R$) and Rf) must act has terms of the form ML0 z-n1 x Y 9x 2--n@ Yl--n1 In y In the first one, either order of operation gives zero. (However, it is necessary that n, > 3, whereas usually for an electron self-energy contribution n, = 2 would be sufficient.) In the second one, R, cannot act first, but if R, acts first, the result is zero. In the present example, the inability to interchange R, and R, is due to our particular method of projecting out the undesired contributions. If one follows the conventional procedure of inserting the finite electron self-energy into the photon self-energy and subtracting, the result will be just (3.14). Thus our procedure appears to put an unnecessary restriction on the order of operation. However, in the case of overlapping divergences, one cannot simply insert finite vertex parts and then make an overall subtraction. In general, the result continues to diverge. But if the overall subtraction is made first, the remaining internal divergences can be handled in the usual way. The order of subtraction thus seems more intrinsic when there are overlapping divergences and our procedure is well-adapted to it. The price we pay in having a universal prescription is that it forces us to work in a specific order even when that would not otherwise be necessary. Perhaps it would be possible to devise a more flexible method, but that will not be attempted here. It should be emphasized that if we make the overall and internal subtractions first, then we may take the R’s in either order. The subtractions together with scaling give for the first term of (3.10)

(3.16) 595/5rl3-10

526

KU0 AND YENNIE The preceding discussion shows that if we carry out the operations in the order indicated, the result is equivalent to setting x = y = 1 in the integral. Suppose instead we carry out the y-operations first. Then for fixed x, the discussion after (3.13) applies and we may drop the y-operations and simply replace y by 1. The resulting function of x has the proper behavior at infinity and we can drop the x-operations and replace x by one. This is of course equivalent to the original result.

C. PROOFOF FINITENESS

The next step is to prove that the I-amplitude associated with any given Feynman graph G and defined in (3.10) is finite. The procedure is generally the same as illustrated by the example of the preceding subsection, but in detail it is more complicated because of the possibility of overlaps. By the power counting theorem, the momentum integral in (3.10) is amply convergent because of the combination of x-differentiations on the integrand. It remains to be shown that the R!J) operations are finite. Since the momentum integral is convergent after the x-differentiations, we may choose any convenient routing of the external momenta and any convenient set of independent internal momenta. The proof then proceeds by successive expansions of the integrands of various subgraphs, starting with the whole graph G = S, . l2 After each expansion, Property (b) of the R-operator is used to eliminate the x-operations. The necessary study of the asymptotic behavior as a function of x is supplied in Appendix B. In our analysis, it will be convenient to choose a canonical set of momentum variables. Such a set is defined by the property that each S-subgraph has its own internal momenta. Although Salam (2) has shown in general that such a set exists, the situation is so simple in quantum electrodynamics that we will give a brief demonstration here. Consider first the situation where there are no overlapping divergences. Each senior subgraph will have been considered in some lower order and we may assign internal momenta to it in the same manner as when it was treated as a whole graph. Having done this, we assign additional momenta in an arbitrary manner until a complete set of integration variables is obtained. We have finally to consider the situation with overlaps. This is illustrated for a general photon self energy in Fig. 5; the treatment of electron self energy graphs is parallel. In Fig. 5, the bubbles stand for proper e+ - e- scattering graphs, i.e. for ones which cannot be separated by cutting the lines of an e+ - e- pair. Self I* It is understood that the following discussion applies to each Feynman integral after the contribution associated with each graph has been broken down into the various contributions corresponding to different combinations of (-&n)-vertices. Although the integrals with (-Sm)vertices appear to be less divergent than ones without such vertices, the degree of divergence is always taken to be that of the original graph. This is necessary in order that the result correspond to the proper renormalization. Our technique for renormalization is clearly based on the scaling properties of the integrals.

RENORMALIZATION

527

THEORY

FIG. 5. Illustration of the procedure for defining a canonical set of momenta when there are overlapping divergences.

energy insertions in connecting lines are ignored since they can be treated in the same manner as senior subgraphs which were just discussed. The canonical set can be obtained by defining internal momenta for each bubble13 first and combining these sets with the set k, ,..., k, . Let c!,,be the number of independent momentum variables in S,, and Vso~(p, k, m,) be its integrand,12 where p stands for the momentum variables external to S, and k = (k, , k, ,..., kto) for a canonical set of internal momenta. The following expansion is then made. + V’“‘(P, k m,)= ; $ [@”+%P,k,m,,] e=o

#)(A

k, m3

(3.17)

WkO

where dois the degree of divergence of So and VR “0) is the remainder,which behaves like (k)-“Go-l as k + co.14 When (3.17) is substituted into (3.9) the x,-dependence of all the terms except the one involving the remainder can be factored out of the integral. The dependence is x>-+” 0 for the mth term in (3.17). Since no > do by choice and 0 < m < do, these terms satisfy the requirement of Property (a) and consequently are eliminated by Rzo. The remainder term converges without wax,)~o, which may therefore be moved outside the integral. One way to carry out the expansion is to represent the remainder as in the integrand of (3.1) and use the &differentiations given by (3.2) and (3.3). As a result the original graph So will be replaced by a set of graphs each of which has do + 1 P-vertices. The particular set of graphs obtained in this way will depend on the original choice of routing the external momenta, but the final result for the integral will not. I3 If the bubbles have internal divergences, it is always possible to find senior S-subgraphs and assign momenta to them in the same manner as when they were first treated in a lower order. It is impossible for two such subgraphs to overlap and give the complete bubble. r4 This sentence is a shorthand for the statement that if we replace ki by Akt for all i and let A + cc, the result will behave like A- Irdo+I). It should also be mentioned that this behavior refers only to graphs without (-8m)-vertices. Those with (--Sm)-vertices will decrease more rapidly.‘*

528

KU0 AND YENNIE

At this point the I-amplitude

has the form

The k-integral converges because the expansion makes the overall degree of divergence negative and any divergence of the subgraphs is eliminated by x-differentiations. The study of large x,, behavior is the same as that for the irreducible graphs; since the x-differentiations do not act on photon lines, the infrared problem is not aggravated. Hence we may use Property (b) to eliminate the x,-operations and replace x,, by unity in VA”o’. Now consider a graph corresponding to a given term in F’$ro). We call this SA and similarly label its subgraphs S; . .. S: in accordance with their correspondence to the original S-subgraphs. Because of the P-vertices, some of these subgraphs may have been rendered convergent. We are now going to proceed through these subgraphs in succession and treat them in a manner analogous to that for the whole graph. Examine the first subgraph S; . If it converges because of the P-vertices,12 take (a/ax,y, outside the integral and use Property (b) (as justified in Appendix B) to eliminate the x1 operations and replace x1 by unity in the integrand. If S; is not convergent, expand the subintegrand associated with S; in powers of the momenta external to S; ,I5 until a convergent remainder is obtained. The new vertices in this remainder are called P,-vertices. It is easy to see that all terms except the remainder vanish upon application of Property (a) of R2’.16 In the remainder (a/ax$ may be taken outside the integral and Property (b) may be used to eliminate the x,-operations. We proceed in this manner from one S-subgraph to the next in sequential order, until all S-subgraphs are exhausted and all x’s are set equal to unity in the remainder, which we may write WY’ = j- dkVA”(p, k, m)

(3.19)

where vkG’ stands for the sum of the remainders after all the expansions. It is also understood that (3.19) is really a sum over integrals corresponding to the different possible (-&z)-vertices. The above steps are plausible, but by no means obvious. We must consider the situation where subgraphs overlap. If two S-subgraphs S, and S, overlap, I5 The momenta external to S; include p and any k’s external to S; . I6 Any of these terms will have the same structure as a term when S, was considered as a whole graph (if momentum labeling is appropriately chosen). Hence when momenta internal to S’ are scaled by x1 the result will be x;~(O < m < nt) times a finite integral.

RENORMALIZATION

529

THEORY

then in renormalizable theories their union S,,, has a degree of divergence greater than the sum of twe two separate ones (&, > d, + d,). Let us consider the situation where SaB = S,, . After the initial P-vertex expansion, we know that di is negative and hence either di or dbor both is negative: The external momentum expansion has eliminated the overlapping divergence. Thus there will be no ambiguity about how to proceed through the steps of the preceding paragraph. This is illustrated in Fig. 6. The l-amplitude is given by

After the initial we may write Wjeb) = +-‘R$)

expansion there are four new graphs Fig. 6b-6e. At this point

j dk, dk, (-$-,,

(-&)“’

V?(kI,

k2, k, mz)lzO=l . (3.21)

For the term corresponding to Fig. 6b, S; is convergent and we may use Property (b) to eliminate the x,-operations and replace x, by unity. This is followed by an expansion of S; . One of the terms of this expansion is eliminated by Property (a)

k

(d)

FIG. 6. (a) Fig. (2b) with the overlapping p-vertex expansion of (2b).”

(e)

subgraphs SI and S, labeled; (b)-(e) terms in the

530

KU0 AND YENNIE

and the other is a remainder which is kept. In this remainder the x,-operations are eliminated and x2 is replaced by unity. For the terms corresponding to Fig. 6c and 6d, no further expansion is needed and the x, and x2 operations are eliminated. The final term is similar to the first one except that the expansion of 5’; is performed first. The final result for WI is finite. Although the set of graphs which is obtained in the reduction depends on the choice of momenta, the result is unique because the integral in (3.10) is finite and we may choose the integration variables as we please. This subsection has been concerned with showing that the final R, operations lead to no difficulties, and for this we found it convenient to use a canonical set of momenta. We have deferred most of the discussion of the asymptotic x-dependence to Appendix B. However, we would like to indicate the general idea here briefly by reference to the simple situation where there are no (-&)-vertices and where there is no “surplus” convergence due to an excess of any type of P-vertices. Suppose we are considering the ith subgraph at the stage where we have made various expansions and have taken the (C@x,)‘Q outside the integral. We have to examine the behavior of the integral as xi + co. Hopefully Property (b) will be satisfied. Scale all k’s by the substitution ki -+ kxj and take a factor x;l outside the integral. The remaining x,-dependence is in the denominators external to Si and there it takes the form 1 (3.22) E - bh) ’ or

(if we retain a photon mass). A power count of the L’s external to & shows that the integral would be logarithmically infrared divergent as xi -+ co. Hence we expect the integral to give at most a power of In xi, and the necessary conditions for Property (b) are confirmed.

IV.

DEMONSTRATION

THAT

THE

Q-OPERATIONS

GIVE

A RENORMALIZATION

A. GENERAL APPROACH In Section III we derived a prescription which yields a finite I-amplitude associated with any Feynman graph. This was accomplished with the help of Q-operations which eliminated terms which had an obvious renormalization structure. It is the purpose of the present section to show that the terms eliminated

RENORMALIZATION

531

THEORY

add up in just the right way to give a multiplicative renormalization as discussed in Section II. Since the integrals are divergent, the subtracted terms are really infinite and the renormalization is formal. In order to make the operations meaningful, we shall assume implicitly that all the Feynman integrals have been regularized in some consistent manner so that all the integrals are convergent and orders of operation can be freely interchanged. The regularizing masses are to be multiplied by X’S in the same manner as the physical masses; this is necessary to preserve the dimensionality properties on which the Q-operations are based. The use of regularizing masses is discussed in further detail in Appendix C. AI1 divergences of course reappear as the regularizing masses become infinite; the Z-amplitudes are of course unchanged by the regularizing procedure since they are given by integrals which are finite without regularization, once the proper order of the Q-operations has been chosen. To illustrate our ideas, we confine our attention to quantum electrodynamics; the transcription to any other renormalizable theory is expected to be straightforward. We wish to show that the subtractions of Section III correspond to a renormalization with the subtraction point chosen to be a, = a2 = a3 = 0. The method of proof will be inductive. We aim to show t""'(p,

m) = (p - m) BF' + Z;(G)@, m) + c K(G ---f G') Ep"(p,

m)

(4. la)

G’

I’I(~)(@ = Cp’ + ZIjG’(k2) + C K(G -+ G’) Z7~‘(k2)

(4.lb)

G’

~:‘
(4.lc)

G’

where ,FG'(p, m) = PG'(p, m) - 8rntG), am(') = 2YG)(m, m); .ZfQ,m)

= z;'G'(p,m)

- amjc',

GmjG' = @(m,m). The Z-functions have been defined in Section III. We assume that (4.1) has been proved up to some order in eo2, and then consider functions of the next higher order in eo2. These are expanded about the origin J+“GYP, m) = (8 - m) I??) + ,FG'(p , m) - BmrcG)>

(4.2a)

532

KU0

AND

YENNIE

where 2?(1/,

m) = z’G’(p, m) - 9G’(0, m) = QJ’“‘tn

ml >

6m' = Zlltc)(m, m), Lp(@

= cp

+ 17'(G)(p)

(4.2b)

and A’G’(p I.6 3p’) = .yllLF’ + A”G’(p II 2p’) *

(4.2~)

The “prime” functions correspond to the V-functions of Section II. The procedure will be to express the W’ functions, which are overall convergent, in terms of the lower order functions which have been previously worked out. Before giving the general argument, this will be illustrated with a number of examples, which should be studied carefully. B.

EXAMPLES

The examples are labeled by the figure numbers. Ex. (3b):

The initial subtraction gives (4.3)

where we have used the fact that the overall subtraction may conveniently be expressed as the Q,, operation acting on A,(3b). We are permitted to route the external momenta through the graph in any convenient manner. For example, if we route it along the electron line and carry out the expansion as in (3.17), we will obtain a result which is illustrated by the graphical equation of Fig. 7a. In this figure, the p-vertices are represented by cross lines labeled 0 (to represent a subtraction of momenta external to S,). The Q, and momentum expansion operations give equivalent results; and we may, if we like, let the Q, operations act on the separate terms in thep-expansion without changing them. Now consider the decomposition of the divergent subgraph S, . This is given by /1(3a) = y&L?’ 0

+ Q,&’

(4.4)

and is represented by Fig. 7b. In this example, there is no difficulty in inserting (4.4) into the integrand of the last term of (4.3). We merely have to be aware that all

RENORMALIZATION

b

(3b) L

W

ypLo

(30) =

YpLo

= YpL,

(30) +A+

CW

(3I.d b

a 1-1 . . .A/& . . . * . .)

+(.

WI +

533

THEORY

(34 + L,

Q0 $a)+Q

I Q 0 *:3b)

(c) FIG. 7. (a) The p-vertex expansion of (3b); (b) decomposition finite part; (c) complete decomposition of (3b).

of (3a) into its C-part and

masses appearing in (4.4) are to have a factor x,, from the QO operation. The important point is that the number Li3a) .IS independent of x,, since it depends only on the ratio of masses (including regulating masses). Thus the contribution from the constant part of (4.4) gives simply LbSa)Q,A~?, and we find /1’3b’ LA

= yuLfb’ + L~)Qo/l~) =

.yuLfb’

+

L$a)&a)

+ QIQ,,Afb’ +

in agreement with (4.1). This equation is illustrated

(4.5)

&b”’

graphically in Fig. 7c.

Ex. (lb): This example is more complicated because of the overlapping divergences. For this reason we cannot simply substitute the decomposition of the left-hand vertex, for example. To eliminate the overlap, we must now make explicit use of the momentum expansion. The subtraction equation J$lb’(fj, m) - &(lb)

= (p - m) B$b’ + QoZ’lb’(p, m) - 8m’(lb)

(4.6)

534

KU0 AND YENNIE

is represented graphically in Fig. 8a; we have chosen to route the external momentum along the electron line. As before, the Q, operation is permitted to act on each separate term in L’ ‘(lb) without altering it. We see that in every term at least one of the vertices is now finite. In place of the vertices which remain divergent, we again insert the decomposition (4.4). The constant part of (4.4) then yields contributions L~3”Q,Z’1~), while the varying part is added to the other finite terms. Now we want to see that this sum is simply Q,QIQ,J(lb). To do this we insert superfluous Q, and Q, operations into the vertices which were originally finite because of the p-expansion. Then each finite graph has the complete set of operations QzQ,Qo .17 But, following the discussion of Section III C, the Q, operation applied to the original integrand is equivalent to the p-expansion. Hence we obtain the result Qo2?1b’(p, m) = 2Lp)Q@‘)(P, = 2L$=)Zj9P By definition,

m) + Q~Q~Qo~(‘~)(P, 4 3 m) + Zjlb’(p,

m).

we have

Hence we find (4.7) which is illustrated

in Fig. 8b.

Ex. (1~): This is the first example with an internal has no overlap. As usual, we make the initial subtraction p’c’(p,

m) - &q(lC) = (p’ - m) It(“) + Q0 2+“‘(#, 0

self-energy, but it

m) -

&IZ’(~~).

(4.8)

I7 There is a subtlety in this argument which may worry the discerning reader. In the first graph of C’“b) we decomposed the S, subgraph first and then inserted the superfluous operation Q1. Thus in the finite part the operations appear in the order Q,Qz . We have asserted that because of the regularization, the orders of operation should not matter. Suppose we had tried to insert the superfluous Q1 operation before decomposing the S, subgraph. Then one of the electron lines in S, will have its mass multiplied by x1 and the other will not. The decomposition (4.4) is then no longer valid. However, there is clearly a more general one with a similar form. The more general one has an Lisa which depends on x1, so we might question whether the shrunken graph has the right value in this case. It does! The important point is that Lp*J should depend at most logarithmically on x1 and when Property (b) is used to eliminate the superfluous Q, operation on the shrunken graph, the result depends only on the value of Lp*) at x1 = 1. It is clearly more convenient to carry out the substitutions in the order first presented, but there is no ambiguity in the result.

RENORMALIZATION

. (lb) c = (p-m)B,

535

THEORY

(lb)

+ (2

,(lb)

(lb)

-6m’

)

(a)

(lb)

_(lb) z

= (g-m)Bo

(So) + 2 La

_(la) -(lb) X, + X,

(b) FIG.

8. (a) The p-vertex expansion of (lb);

(b) the complete decomposition

of (lb).

,v=-zzT +%aBJia) +tiI(“) Ca)t (#-m) (a) P

+ W-mlBo

(Id

+Bo

(IaLb) 2

+ 5,

(Ic)

(b) FIG.

9.

(a) The complete decomposition

of (la); (b) the complete decomposition

of (14.

536

KU0 AND YENNIE

The internal self energy Z (la) has the decomposition P"'(p',

mxJ - xo6m(1a) = (p'-

mx,)B~)

illustrated

+ Q12$"'(p

in Fig. 9a,

, mxJ - xoBm'(ia). (4.9)

Again, the constant B A’“’ depends only on mass ratios and is hence independent of X,. The varying part of (4.9) corresponds exactly to the prescription of Section III B, and the complete result is z(lc) = (p _ m) BP’ This is illustrated Ex. (lOa):

P

+ Bo(la)Ztr(la)+ z$d.

(4.10)

in Fig. 9b. Suppose we generalize the preceding example by inserting a

= (@l-m)

(S,)..,IS,)

Ba

+x1

+ xK(S,S’

S’)x,

6’ 1

(b)

,m-‘) c

(lOa) =(a -m)Ba

~ (IW +x1

u (lOa’) + IZ~~lOa+lOa’P1

(cl FIG. 10. (a) Lowest-order ESE graph with a general ESE (S,) inserted in the electron line resulting in a graph calIed (lOa); (b) the complete decomposition of S, ; (c) the complete decomposition of (lOa).

RENORMALIZATION

537

THEORY

more general self-energy part with graph S, , as illustrated of (4.9), we now have .Fsqpt,

mx,) = (p’ - mx(J lp

+ ,IF(

in Fig. 10a. In place p’, mx,)

+ c K(& + s? .P’)(pt,

mxo),

(4.11)

S’

which is illustrated very schematically in Fig. lob. As usual, the 2: are given by complete sets of Qi operations as defined in Section III B. Let us denote the graph obtained by replacing S, by s’ in (lOa) by (loa’). By our induction process, this graph will have occurred in a previous order. We note that K(S, -+ s’) = K(lOa -+ 10a’) and hence we find ,FlN) = (p - m) BP)

+ ,$m) + @l),p)

+

c

qloa

+ lOa’) Zcjloa’)

lo&'+l&

(4.12) = (p - ITZ)BP)

+ 2p)

+ C K(lOa -+ loa’) ,2p’). 10a'

The result is illustrated

in Fig. 10~.

C. GENERAL ARGUMENT After these examples, the general argument is rather straightforward. We always start out by making the overall subtractions to separate the divergent expression into a constant part plus a varying part. In the analysis of the varying part, the simplest situation occurs when the original graph contains no overlapping divergences of senior subgraphs. Then one simply inserts the known decomposition of all the senior subgraphs. By inspection, the result is then the desired decomposition (4.1). We must then turn to a discussion of overlapping divergences. A convenient first step is to identify all self-energy subgraphs and substitute their known decomposition. This is unique since such graphs cannot overlap each other (they may of course have internal overlaps or self energies, but these were understood at a previous stage). When any self-energy subgraph is shrunk or has a shrunken subgraph, the whole graph will have been considered in some previous order and will give a well defined set of contributions to (4.1). Thus we need consider further only the ,J?j’i) or flj”j) part of each self-energy subgraph in the further analysis. As in Ex. lb, the momentum expansion is essential to eliminate the overlaps. We choose any convenient routing for the momentum and expand in terms of

538

KU0 AND YENNIE

graphs with P-vertices. This will immediately separate the divergences, and we may insert the known decomposition of the senior subgraphs in each of these new graphs. Since the self energies have already been treated, these decompositions involve only terms in which vertex graphs are shrunk. Now consider any particular shrinkage of vertex graphs (including the case of no shrinkage at all), and group together all the graphs with different sets of P-vertices but with the given shrinkage. From the power counting theorem, no P-vertex appears inside a shrunken vertex. We assert that the total contribution from these terms is just one of the terms in the decomposition (4.1). The noshrinkage contribution is obvious. For any other combination of shrinkages, we note that the original choice of momentum routing gives a possible routing in the shrunken graph (called the induced routing), but not necessarily the one used when the graph was first considered. This is of no consequence because, as has frequently been stated, the final result is independent of the routing of momentum. A little reflection shows that the fixed shrinkage has precisely the same set of contributions as would have been obtained originally if the induced routing had been used. The contribution from a given shrinkage is then the product of K’s for the shrunken vertices times the finite part of the shrunken graph. It is clear that all possible disjoint shrinkages will occur somewhere in the sum, and hence (4.1) is confirmed. D. RENORMALIZATION

ON THE MASS SHELL

Up to this point we have succeeded in prescribing for each graph a finite I-amplitude which is related to the unrenormalized amplitude according to multiplicative renormalization. What remains to be done is to shift the subtract point to the physical mass shell by a finite renormalization. Since no infinities (or alternatively, regularizing masses) are involved this can be done directly with the functional relations (2.31) with the specialization a = (0, 0, 0), b = (m, 0, m). In Dyson’s notation, they read rdp,

P’, m, 4 = (1 + ~2 rl,(p,

P’, m, ed,

&dP, m, a) = (1 - Qbl Gdpc, m, 4,

&(k’,

m, g) = (1 - CJ’

(4.13)

D;,l(k2, m, e&,

where

Wp, m, 4 = [P - m - ZYP, m, 41-‘, Ddk2, m, 4 = b [l - 171(k2, m, 41-l,

(4.14)

RENORMALIZATION

THEORY

539

and

(4.15)

+ LlY el2 -_ (1 (1 - Z&)2 er2* The Dyson functions r, , Sk, , DF1 can be calculated directly from (4.13) once the Z-functions are known to any order without going back to individual graphs. It is to be noted that the Z-amplitudes satisfy the Ward identity (Appendix D), whence it follows that

L, + Bl = 0 to any order; consequently el = eI,

(4.17)

i.e., the intermediate charge is in fact the physical charge. This makes the relations (4.13) completely trivial.

V. DISCUSSION

We have given a new presentation of renormalization theory which differs from previous ones more in technique than in substance. At the possible risk of being pedantic, we have presented the formal details of renormalization in a rather complete and explicit manner. Our set of formal operations corresponds most naturally to a subtraction method in which one works “from the outside in”, i.e., in which the overall divergences are subtracted before the sub-divergences. In this order of subtraction, overlapping divergences do not occur in quantum electrodynamics or in the other usual renormalizable theories. In contrast, the previous formulations work “from the inside out” using counter-terms which make the sub-divergences finite. We have indicated in Section II C how the two methods lead formally to the same result for the renormalized functions. We regard our operator scheme as an intermediate framework from which we can show, on the one hand, that the intermediate amplitudes are finite and, on the other hand, that they correspond to the correct formal subtractions. The easy step in the proof of finiteness is to see that the x-differentiations make the momentum integrals converge according to the power counting theorem. The

540

KU0

AND

YENNIE

hard step is to show that the resulting function of the xi’s is in the class of functions on which the R, operations may act in the proper order. The manner of showing this is slightly indirect. We subtract terms from the original integral which are zero because of Property (a) of the R, operations and then try to show that the new function satisfies Property (b). The subtraction terms are chosen to correspond to the initial terms in an external momentum expansion of the graph or subgraph. The new integrals are convergent without the x-differentiations. We should emphasize that we have not given a rigorous proof that these integrals do actually satisfy Property (b). Rather we have tried to make this very plausible by examining the salient features of these integrals in Appendix B. It is conceivable, but unlikely, that we have overlooked some pathology which could cause difficulty. The demonstration of multiplicative renormalization in Section IV was purely formal and consisted in showing that all terms were properly accounted for with the correct weight. To make the steps meaningful, we had to imagine that some relativistic regularization was used. In Appendix C, we give a brief argument that, when the regularizing masses tend to infinity, the intermediate amplitudes are given by our prescription. One particular advantage of our approach over other ones is the ease with which we can demonstrate gauge invariance of the intermediate amplitudes. There are two situations to consider, depending on whether the photon does not or does participate in the divergent integral. In the first situation, it turns out that by introducing superfluous x-operations one can combine the contributions from various graphs and prove gauge invariance without actually completing the integration. This contrasts to the situation where external momentum subtractions are used; in that approach, there is no easy way to see that the sum of integrals is gauge invariant without actually carrying them out. The case where the photon participates in the divergence is more complicated and we have considered it only for covariant gauge photons. The result is that the new intermediate amplitudes differ from the mass-renormalized ones by finite renormalizations. The gauge invariance question is discussed in more detail in Appendix D. A point which has not been emphasized until now is the possible advantage of using the x-operator technique in computing the renormalized intermediate amplitude for a physical process. In place of the large number of terms which would be generated by subtractions (as in Section III), we have a single expression for each original Feynman integral. Of course, it must be admitted that if the x-differentiations are carried out explicitly, many terms will result in our formalism also; besides, we have additional parametric integrations to do at the end. It is expected that in practice not all the x-operations need be carried out explicitly. This is suggested by the actual calculation of the lowest-order expressions which is given in Appendix E. There it is shown that the differentiations and integrations contained in the Q-operations need not be carried out. In these cases, we can

RENORMALIZATION

541

THEORY

evaluate the amplitudes with the use of the defining properties only. More generally, consider any divergent graph. The integrand can be parameterized inside the Q-operation using the usual Feynman parameter techniques (15). The number of parameters involved is the same as for the unrenormalized amplitude, but smaller than that for a subtracted integrand, which has additional denominators. In most cases, one can carry out the momentum integration before doing the Q-operations; the apparent divergence is transferred to the lower end of the parametric integration. Of course, when the Q-operations are carried out in proper sequence, they render the parametric integral convergent. Rather than carry out the Q-operations, one can make the external momentum subtraction at this stage. At this point the external momentum expansion is route-independent and hence simpler than in the original Feynman integrand. The guideline is that when a correct subtraction is made, the corresponding Q-operation collapses. Hence we suggest that the Q-operations should not be viewed as a divergence-removing device but rather as a prescription which dictates what subtractions to make. One has the freedom to subtract at any stage of the parametric integration. This contrasts with the usual method of having to perform the subtractions at the very beginning and facing the explosion of the number of terms. The problem of carrying out the renormalization in terms of the Feynman parameter representations has been considered by Bogoliubov and Parasiuk and Hepp (16). A treatment closely related to the present one has been given by Appelquist (27). He formulates the finite amplitudes ab initio without reference to the ones defined here; of course, the two approaches lead to equivalent results. These approaches are more rigorous than the present one, but hopefully our approach is more accessible to those who do not have an expert working knowledge of Feynman parameter techniques. Also, we do not know of any transparent way of discussing gauge invariance in the Feynman parameter language. A brief mention should be made of the fact that renormalization theory can provide some information on the analytic form of the terms of a perturbation expansion. This property of the theory is usually developed by a study of the renormalization group (13). It is also possible to do this by a direct study of the implications of the subtractions which are necessary to produce the finite functions. We shall illustrate this with a very simple example. Consider the lowest order photon self-energy L7(2a) with cutoff. It is a function of k2, m2, and /I2 (cutoff mass). Because it is dimensionless and finite, it can be expressed as a function of two dimensionless ratios, say (kz/A2) and (m”/A”). The subtraction scheme tells us lim {lI(2a)(k2/A2, m2/A2) - Ilcza’(O, m”/A’)} n+m

= ZI,(2&)(k2/m2).

To proceed further, it is necessary to make a highly plausible conjecture: 595/51/3-11

(5.1) For

542

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AND

YENNIE

k2 > m2, we can neglect m2/A2 in the first term of the left side of (5.1). Next we introduce some new variables u = In k2/m2,

v = In m2/A2.

W-9

Then, using the conjecture, (5.1) takes the form

.‘j% [G(u + 4 - ffW1 = &)

(5.3)

when u > 1. This is a functional equation; it is easy to see that the solution must take the form F=ffcu, G = g + 4~ + 4, (5.4) H = (g -f) + cu. This may of course be confirmed by direct calculation and the constants f and c determined (g depends on the details of the cutoff). This example indicates the general idea of the approach. It has been investigated in much greater detail by Rae (18). ACKNOWLEDGMENTS

This work benefitted from discussions with many people. We would like to thank particularly Professors J. D. Bjorken, T. Kinoshita and T. T. Wu and Drs. T. Appelquist and A. Rae. We are grateful to Prof. B. Zumino for asking the appropriate question about gauge invariance which stimulated the discussion in Appendix D. We also thank Professor K. Gottfried for a critical reading of the manuscript.

APPENDIX

A

Our aim is to construct an operator Rp’ which has the Properties (a) and (b), which we reproduce here for convenience. (a) Rt’x” (b)

= 0

RF) ($-,”

if

m is an integer satisfying 0 < m < n.

F(x) = F(1)

if F(x) is regular in 1 < x < co

and F(x) = 0(x-l In” X) as x -+ 03 and m is any finite integer. It is easy to achieve Property (a) by having RF) start with the operation (a/axy xn. To satisfy (b), it is clear that we need some sort of integral operation.

RENORMALIZATION

543

THEORY

Let us try Rp’ = I’

where h,(x) is to be determined

m

dxh,(x)

($r

x”:

so that Property (b) is satisfied. This means

J1 dxh,(x) (2)” cc

x” (g)”

(A.11

F(x) = F(1)

for suitable F(x) satisfying the conditions of (b). To find the conditions which determine h,(x), we integrate by parts repeatedly. The terms coming from the limits must be made to vanish by imposing the conditions

dmh, dx”

rel =

m = 0, l,..., n -

0,

1

(A4

and m = 0, l,..., n -

1,

or

pi x- (n-m)d”h dx” = 0, from the conditions

m = 0, l,..., n -

on F(x). With these conditions (- 1)” ,‘, dxx” $$

(&)”

1

(A-3)

on h, , Eq. (A.l) becomes

F(x) = F( 1).

(A-4)

Calling g, = (- 1)” xn(dnhn/dxn), the necessary conditions on g, , analogous to (A.2) and (A.3), are d% dx”

z=1 = 0; d”-‘g, dxn-l

m = 0, l,..., n - 2,

(A.3 =

(-l)"-1

and Jilim x- (n-m) *

= 0.

dx”

’

m = 0, I,..., n -- 2.

The solution of (AS) is &l(x) =

(1 - ,>,-I (n -

I)!

(‘4.6)

544

KU0

AND

YENNIE

which also satisfies (A.6). From the definition (-1)”

d”hn dx”

(n -

subject to the boundary conditions

of g, , we find (1 - x)“-1

I)!

(A-7)

x”

(A.2). The result may be expressed

zdx,(x- X’y-1 (1- x’)n-1 h,(x)= Kn(-- 1)” 1)!12s X’n

1

(A-8)

It is easy to confirm that (A.3) is satisfied. The first few functions h, are readily computed: h,(x) = --In x,

(A.9)

h,(x) = -(x + 1) In x + 2x - 2, ha(x) = --$(x2 + 4x + 1) In x + 2(x2 -

(A. 10) (A. 11)

1).

With the specific form of h, , we may now rewrite the condition sufficient condition lim xi In x & ’ ( 1

.X-m

F(X)

= 0;

i = O,..., (2n -

on F(x) as a

1).

(A.12)

This defines the class of functions on whose nth derivative Rc) is defined. For the type of functions which occur in field theory, if (A.12) is true for i = 1, it is true for all higher i. Hence one may always use a larger n than is strictly necessary in a given situation. If a function satisfies neither the condition for Property (a) nor the condition for Property (b), it may be possible to split it into the sum of two functions which do. An important example which occurs in the discussion of the lowest-order divergences of quantum electrodynamics is

Noting the behavior of the function as x + co, we subtract a function (l/x) from the operand; by Property (a), this changes nothing. But this gives us I&?’ (&r-l

[-$&-

- $1 = Rp’ (&,”

= $ In(1 + a) by Property (b).

[+ ln(x2 + a) - In x]

(A.13)

RENORMALIZATION

545

THEORY

Another example which is needed is R,1”) g-,“-”

(1 + &

x) -&

= RC2) z 1+ 2 t

x) x x2 + a

= Rp

(1 + &x)(-&

gy-’

= R’“’3c (+-I” = 4 ln(1 + a). APPENDIX

- +j

($ ln(x2 + a) - x In x) (A. 14)

B

In this Appendix we want to make plausible the assertions of Section III C regarding the behavior of the integrals for large xi . We recall that we have worked from the outside in, first expanding in powers of the external momentum of S,, , then doing the same for each subgraph Si in turn. At any given stage, if the number of powers of external momentum in a term is less than the degree of divergence of the particular Si (i = O,..., s), that term is eliminated by Property (a) of Rci'. We can envisage two ways of proceeding. The first is to make the p-vertex expansion of each Si in turn, take the derivatives with respect to xi outside the integral, examine the behavior of the resulting integral as xi ---f co (with the xi (j > i) derivatives still acting on the integrand) to verify that Property (b) is satisfied, and then set xi = 1 using Property (b). The second approach is the same except that one makes all the p-vertex expansions first and then shows that Property (b) is satisfied for each individual xi for all the other xg’s held fixed. At this stage we see that the R-operations can be applied in any order and the result is to set all xi’s equal to one. The two methods clearly give the same final result. This observation could be of practical importance since it shows how we can always compute the finite part of an expression without actually using the R-operations: Choose any momentum routing, make all necessary momentum expansions working from the outside in, and throw away the renormalization terms. The following discussion applies to either approach just outlined. We fix our attention on the xi variable and suppress the dependence on the remaining xj’s. We use a canonical set of momenta and let s E {sl .. . So} be those internal to Si and G = {e, . . . &} be those external to S, , but internal to S,, , As usual, p will represent momenta external to S, . We have various situations depending on how the p-vertices are distributed in the graph for a given term. There may also be

546

KU0

AND

YENNIE

m-vertices, which arise either from (--Sm)-vertices or from xi tions; we need not distinguish their origin in the following important characterization of any given term is the distribution inside and outside of & . Let these be given by the following

(j > 1) differentiaanalysis. The most of various vertices numbers (all > 0):

a: degree of p internal to Si ; ~2~+ b: degree of [internal to & ; c: number of m-vertices in Si ; do + e: degree of p external to Si ; f: degree of [external to Si ; s number of m-vertices external to Si .

The use we have made of Property (a) tells us (a + b) > 0,

(a + e) > 0.

(B. 1)

We also know that if we had no p or m-vertices the total degree of the (unrationalized) denominators in S, would be 4~ - di , while that of the whole expression would be 4or + 4/3 - d,, . Hence we can write our expression somewhat symbolically as

(de) (Pri(e); , cm) , m Ne

(s, 4 P, mxdNt

’

03.2)

with Nc=4b+di+e+f+g, Ni = 4a + a + b + c.

The meaning of the numerator powers is obvious. The expression (s, /,p, m#“’ represents the total denominator inside St ; the symbol indicates that it is a homogeneous function of degree Ni in the indicated variables. Similarly (8, p, m)Ne represents the denominators external to S< . The numbers Ni and N, have been chosen to agree with the number of expansions which have been made to produce the p- and m-vertices. It is clear that when xi is very large, only the large values of s will be emphasized. This suggests that we should rescale the variables internal to Si : 4%+ xi&z ;

(a = l,..., CX).

The result of the scaling is (de) (p)“‘”

(of trn)”

(4 P, mlNe

(B-3)

RENORMALIZATION

547

THEORY

If the integral converges as xi + co, (B.2) tells us that Property satisfied. We have two cases to consider.

(b) will be

Case I: N,>4P+f+d,+b, or e+g>b. In this case, both the s and G integrals converge by the power counting theorem without the &dependence inside the S-factor. Hence Property (b) is satisfied. Case II: e + g < b. Now the e-integration cannot converge as a whole without the &dependence inside the Si factor. This indicates that large values of L (cc xi) are emphasized in the integral. Naively, we want a transformation like

e - x,e.

(B-4)

This leads to an overall negative power of xi times an integral which converges as a whole as xi -+ co. However, we must be more cautious than this. It is possible that some subintegral (characterized by variables el) converges separately. If we make the transformation (B.4) for those variables, we will be emphasizing the wrong region of integration. Hence we want to make the transformation (B.4) only for the variables (called F’) which are complementary to E’. Suppose first that the set d’ is empty. Then we can make the transformation (B.4). The result is 1 (P>“‘” w (4 xa+e+g J-n (de) (e, p/x, , m/xp D

m

(ds) (PI” (odita (4”

Ni ’

(B.5)

(s, 4 5 z , m 1

If b > e + g, the integral will not diverge as xi - co; if b = e + g, it will diverge logarithmically. In either case, since a + e + g > 0, Property (b) is satisfied. Now suppose the set 6” is nonempty. Consider the subgraph s” containing all the lines through which momenta e” and s circulate. The momenta p and 4’ are external to this subgraph. In place of (B.2), we have an integral (omitting the G’) of the form (ds)(p)a (e”)di+a-a’ ce’)“’ (mxi)c (s, L”, d’, p, mxJNi

(dt”) (‘)

(B 6) ’

with N”=4/3”+d,+e”+f”+h’+g”.

As before, all exponents in the numerator

are nonnegative;

however, we do not

548

KU0

AND

YENNIE

exclude the case where d” < 0. At this point we scale the variables J” and s by xi with the result 1 Xa+e”+h’+g’+b’ i

(J3.7)

In general (independent

of d”), we have a + e” + h’ + b’ > 0;

consequently the overall power of xi is negative. Inside the integral, we can let xi -+ cc without creating a divergence. (It is easy to show that b > e” + g” + h’ + b’ from the condition that the I’ integration converges separately.) Hence Property (b) is satisfied.

APPENDIX

C

In order to justify the formal renormalization, it is necessary to use some method of regularizing the Feynman integrals so that all the operations become well defined and the subtraction constants become finite. It is necessary to show that the Z-amplitudes are independent of the regularizing masses. This seems fairly obvious since the Z-amplitudes are finite and hence should be insensitive to the changes in the integrand at high virtual momenta. We keep in mind the Pauli-Villars method of regularization (29), which we do not describe in detail here. The important point is that in some denominators or combination of denominators the physical masses are to be replaced by other masses which will tend to infinity. We aim to show that the contribution of these subtraction terms to the Z-amplitude will be zero. Let the scale of all these regulating masses be II. We may try initially to scale all internal momenta by:

From the type of argument given in Appendix B, the result will be an overall factor of II of negative degree times a new integral in which all physical masses will be divided by (I. If the integral doesn’t diverge, we are finished. If it does diverge, we must have an infrared type divergence associated with a subset of momenta. Then, as in Appendix B, we scale only those k’s which do not have

RENORMALIZATION

549

THEORY

an infrared divergence. Because the k’s being scaled have a convergent integral in the ultraviolet, we still obtain a negative overall power of A. We conclude that the I-amplitudes become independent of A as A + co.

APPENDIX

D

In this appendix we are concerned with the question of gauge invariance in quantum electrodynamics. We want to show that the intermediate amplitudes do have all the usual gauge-invariance properties. One way to do this would be to compute the Feynman functions in different gauges and verify directly that they have the expected properties. This would obviously not be practical in any but the simplest cases. Instead we hope to use the Feynman identity CD.l) in the usual way to obtain cancellations between various diagrams. We want to bring out the fact that for the purpose of discussing gauge invariance our method of defining the intermediate amplitudes seems superior to the P-vertex expansion, which must in the end yield the same result. This can be understood by considering (D. 1) in which k is considered to be an external momenta. In order to make a P-vertex expansion, the k in the denominator (but not the numerator) of (D.l) would have to be multiplied by a factor 5 (0 < .$ < 1) according to (3.1). But this destroys the identity .18 This is not so surprising since mutilations of the momentum dependence of an expression quite generally complicate gauge invariance considerations. On the other hand, if all masses in (D.l) are multiplied by the same product of xi)s, the identity is still valid. Thus we may hope that the gauge invariance proof will be more straightforward. If the factor k refers to an external photon, both masses on the left-hand side of (D.l) will in fact be multiplied by the same product of xi’s and we may use (D.l) as it stands. However, if k refers to an internal photon, the masses may be multiplied by different products of x~‘s. We then have a more general identity 1 K-k-mx%-mY

1

I8 If we try to obtain a correct identity by dividing the right side by I, the t-differentiations longer yield a convergent result.

no

550

KU0 AND YENNIE

The last term on the right-hand side of (D.2) looks (and is) troublesome. However, we note that it has one less power of internal momentum and hence may give a less divergent integral. If, in a given situation, it turns out that Property (b) is satisfied for the parameters which differ in X and Y, then those parameters may be set equal to one and the last term of (D.2) will vanish. However, as will be seen later, there exist situations where Property (b) is not satisfied and the last term of (D.2) does not vanish. This will turn out to be related to the finite difference between the renormalization constants in the intermediate and mass-shell renormalizations. GAUGE-INVARIANCE

WITH RESPECT TO EXTERNAL PHOTONS

Let us consider as an example the lowest-order vacuum polarization.

The identity (D. 1) gives k’I&‘(k)

= ieo2R~’

CDe4)

Although the definition of (D.3) requires only IZ > 3, it is convenient to take n > 4. Then the two terms in the integrand of (D.4) will separately give convergent integrals. If we shift the origin of kl in the second integrand by -k, the two terms cancel and we see (D-5) We note that this was accomplished by consideration of the properties of the integral and that it was not necessary actually to evaluate (D.3). In fact, since (D.5) justifies (2.10) for this single graph, it reduces the actual labor in calculating (D.3); this will be seen in Appendix E. The next example we consider is the fourth order vacuum polarization illustrated in Fig. (2b,c). We redraw these in Fig. 11 and include a convenient momentum labeling. The mass subtractions can be treated trivially as in the preceding example, and we ignore them in what follows. We note that if we use the identity (D.1) we will obtain six graphs that have the structure of Fig. llb. The place where a photon line could be inserted to recover each of the original graphs is indicated on the figure. Now, in order to facilitate the subsequent analysis, it is convenient to assign the x-operations according to the divergences which appear in Fig. llb and retain them when the additional photon line is inserted. This will give superfluous operations in l7 (Zb,2c’*aC”),but will permit us to treat each

RENORMALIZATION

551

THEORY

k,+ k

k,tk

k-g2--

2

(2a) k,tk

k,tk

Yea-

-

b

(2c’)

(a)

(2c“)

L a I t (kp) (2c”)

I

-t201

(2c’)

(b)

FIG. 11. (a) The three contributions to the fourth-order vacuum polarization which are related by gauge invariance; (b) the common graph which is obtained if an external photon line is removed from the indicated place in each of the preceding three graphs. x-variables are assigned to each of the three graphs so that they will have the same x-variables in this new graph.

term of (D.l) separately since there will be sufficiently good convergence of all integrals. With these rules, we obtain

=

e

4Rh)R(%)R(%) 02

Y

5

1 1 1 1 x Tr yu kl - mxz yo It2 - mxy ’ k, + k - mxy ‘a kl + k - mxz 1 1 1 1

1

+ Yu kl - mxz ’ Jtl + h - mxz ya k2 + Jt - mxy y” kl + k - mxz

1

1

1

1

+ YIAJ$ - mxz ‘CJIt2 - mxy y” ItI - mxz ’ /(1 + it - mxz I = 0.

03.6)

552

KU0 AND YENNIE

The result is obtained by noting that four of the terms cancel pairwise in the integrand and the remaining two cancel when the origin of integration for one of them is shifted. All these steps are justified if we take n, > 4, n2 > 2, n3 > 1. As mentioned earlier, we can make these assignments even though some of the operations may be superfluous in the original expressions. The generalization of the foregoing analysis to any related set of vacuum polarization graphs or light scattering graphs is quite obvious. Similarly the generalized Ward-Takahashi identity (20) for the intermediate amplitudes is easy to derive: (P’ - PL 4w9

P> = z;(P) - z;(P’)-

(D.7)

Details are left to the reader. GAUGE-INVARIANCE

WITH RESPECT TO INTERNAL PHOTONS

A general gauge transformation

is given by the substitution

NW g uy -+ g”” - kllfY(k) - k”‘(k). Under this transformation, physical matrix elements are unchanged, but the various off-shell quantities are transformed. Although such general transformations are needed (for example, in establishing the connection between the Coulomb and covariant gauges), we shall restrict our attention to the simple case where the additional term in (D.8) has the covariant form

k,k,/k2.

(D-9)

With this choice, power counting considerations are not altered. Let us start by studying the simplest case, the lowest order electron self energy of Fig. la. With a “gauge” photon, it becomes

X

- mx) + (p - mx) p + ; _ mx (a - m4l.

148

We want to compare this result with what would have been obtained regulated the photon propagator and not subtracted at the origin

cj+;’ = s -

X

ieo2

-(P

1

(dk)

-

p

4

+

_

(p'

had we

(12

1 k2

(D-10)

A2

-

(12

4

_

p

k2

+

;

_

m

(#J

-

ml/.

(D.ll)

RENORMALIZATION

553

THEORY

(Note the absence of any mass shift: Sm is gauge invariant.) The first term in (D.11) gives a contribution to B, which is not gauge invariant. The corresponding term in (D.lO) is eliminated by the x-operations. Let us rearrange the second term of (D.lO)

(dk)

4

k2

:

X2x2

X I(P - m) p+~_mx(~-m)+m(l-x)p+~_mx(P-mm) 1 + (P - 4 p+k-mmx

m(1 - x) + m2(1 - x)2

1 p+R--mx’

t (D.12)

The first term in the bracket of (D.12) can be seen to agree with the finite part of (D.11) by the use of Property (b). In the remaining terms we cannot set x = 1 (and hence eliminate them) because they do not satisfy Property (b). We may, however, expand these terms in powers of a/ until we reach terms where Property (b) is satisfied; such terms will vanish because of the factors of (1 - x). The result is that the bracket of (D.12) is equivalent to I (P - ml p+j-mx(P-m)+m

k 2 mx (P - 4 + (P - m> #-mx”’

1 + m2 k - mx - m2U - 2x1 k _’ mx d

(D.13)

where we have dropped terms which vanish upon application of the x-operations. At this point we may set x = 1 in (D.13) using Property (b). What is the meaning of the additional terms in (D.13) that do not correspond to terms in (D.11) ? They are clearly mass and charge renormalization terms which arise from having subtracted at the origin rather than at p = m. Thus when we renormalize on the mass shell, (D.11) and (D.12) will give identical results. Accordingly, if we simply drop these terms as they arise, we automatically obtain the function properly renormalized on the mass shell. Now let us consider a somewhat more complicated example, the vacuum polarization contribution from Fig. 11, with an internal gauge photon. It is a straightforward, but slightly lengthy, calculation to show that only the last term of (D.2) survives cancellation in every case.

554

KU0

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YENNIE

= eo4R(n2)R(%)Rh) z Y 0 x (kl _ k2)21- ~2X2y2z2 m2x2(y- 4”

x Tr I yu kl 1

1

1

1

- mxz h2 - mxy yy

1

k2+#--f?W

1

&+k--mxz

1 + Yu hl - mxz yy Al + h - mxz k2 + P - mxy kl -I- /I - mxz 1 1 1 1

+ Yu kl - mxz /cz - mxy kl - mxz ”

1

kl + /c - mxz I

1 1 x Tr yLLkl - mx ” Al -I R - mx xsm$@’ Al + A’ - mx 1

I

(D.14)

where

Our aim is to show that this complicated expression reduces to zero; we present an outline of the proof. Consider the first integral and take they and z differentiations outside the integral sign. Next use the properties of the x-operations to expand up powers of k and discard the first three terms (cc k”, kl, k2). Any resulting terms which go to zero at least as fast as (l/y) and (l/z) can be discarded because of the factor (y - z)“. For example, in the expansion of the first term, all powers of k must be in one vertex or the other if a term in the expression is to be retained. There are two such terms. Consider one of them, say the one where all powers of k occur in the vertex associated with z. Using Property (b), we may set x = z = 1. Because of Property (b) the vertex containing the y is then independent of k and kl and reduces to a multiple of ye . The multiplicative constant is clearly the same as the one which results in the calculation of the second order vertex with a gauge photon. Similarly, in treating the second term, we may immediately set z = 1. Then, since the k,-integral is finite, it must give precisely the same result as a function

RENORMALIZATION

555

THEORY

of k, + k that the last term of (D.12) gives of p, namely a result of the form: axm + b(kl + k).

It is the role of the mass subtraction in the second integral of (D.14) to convert this to b[&

+ # -

xm].

But by Ward’s identity the constant appearing in the vertex in the preceding paragraph is (--b) and all these contributions cancel each other. These two examples should indicate the general nature of the problem of handling gauge photons in this formalism. We shall not attempt a general discussion. We believe the general conclusion is the following: Any nonvanishing contributions arising from the last term of (D.2) correspond merely to the difference between intermediate and mass shell renormalization. The net effect on photon self-energies and light scattering will be zero, and the spurious charge renormalizations (B and L) will be altered in a compensating manner.

APPENDIX

E

The practical aspect of our renormalization technique is illustrated here with some examples of lowest order. It is shown that the method of mass differentiation provides a simple way to extract finite parts from divergent integrals. (i) Lowest order PSE graph. The Z-amplitude graph is given by

for the lowest order PSE

That this alone satisfies gauge invariance has been shown in Appendix

D, i.e.,

17;;;‘(k) = (g,,k’ - k,k,,) L$@(k2).

To obtain l’I:2a)(k2), which can be regarded as the coefficient of -k,k, in (E.l), we combine denominators using Feynman’s method and work out the trace, $ Tr( > = J1 da 2p,p, 0

Because the p-integral

+

PA

+ p&,

-

guv(p2

-

m2x2

+

P * k)

(p2 + 201~. k + ak2 - m2x2)2

is convergent

we are allowed

to change variables this step is can be interchanged with the shift. By

P-+-P’ = p + ak and to discard terms odd in the new variables;

possible because the x-differentiations

556

KU0

covariance considerations we write

YENNIE

we also replace phi

-241

i Tr{ > = 1: da

AND

by &gevprz. Dropping

the prime

- a) k,k, - guy[$p2 - m2x2 - a(1 - a) k2] [p2 + ar(1 - a) k2 - m2x212

Keeping only the term k,k, we see that #@(k2)

= 4ie,2&’

j (dp) (&)”

j; da Lp2 + ,(;”

a;kt)e

m2x2]2

* (E-2)

We present two ways to evaluate this integral: the regular way and the short-cut way. In the regular way the integrand is differentiated once with respect to x, and the remaining (L3/axp-l is moved outside the p-integral. Then, we find II?’

= 4ie:R:’ -$

(&,“-’

&(I - a) m2x ,: da j (dp) Lp2 + a(l _ a) k2 _ 01(1 - a) m2x a(1 _ a) k2 _ m2x2

-1

= - 2w2 j 0 doLR:’ (&)“-’ =- eo2 j’dcm(1 27r2 0

m2x2]3

- a) In [l - a(1 - CC)$1.

(E.3)

The last step has been explained in Appendix A. In the short-cut method, we regard the integrand of (E.2) as the limiting lhll

c++~

Note if E > 0 the p-integral

I (44

value

241 - a) a) k2 - m2x212+ ’

[p2 + a(1 -

can be performed without any differentiation

I [p” + ar(l - CZ)k2 - m2x212+ = ii??

1 1 E(E + 1) [a(1 - a) k2 - m2x2]2 ’ (E.5)

The result is then expanded in powers of E: 1 1 -- i - - In ) ol(l - a) k2 - m2x2 ) + G In2 1ol(l - CX)k2 - m2x2 I + -]. 16+~+ 1 [E The first term vanishes on differentiation with respect to x, the second term has a finite limit as E + 0. All subsequent terms vanish in this limit, and the result is n:28)(k2) = g

Jl dora(1 - LX)RF’ (&,”

ln(mV

- a(1 - CX)k2). (~6)

RENORMALIZATION

THEORY

557

The logarithm function does not satisfy the requirement of Property (b); however, we can subtract from it In m2x2 which vanishes under Rt’(a/ax)” [by Property (a)]. Hence

R’“’ 5 (&)”

h(m2x2 - a(1 - a) k2) = R!$‘)($-I” =ln

The last step is by Property find (E.3).

(

l-

In (1 -

at1 ix:)

k2 )

cy(1 - a) k2 m2 i.

057)

(b). When (E.7) is substituted into (E.6) we again

(ii) Lowest order ESE graph. The Z-amplitude graph is given by

for the lowest order ESE

Let us describe the short-cut method first.

=-

eo2

1

-(I - IX) pc + 2mx ’ E(E + 1) [ol(l - cy)p2 - cum2x2]’

8.rr2s o

= -

-eo2 l j daRt’ 8.rr2 o

(&)”

[-( 1 - a) p + 2mx] ln(olm2x2 - a( 1

- 4P”>-

To apply Property (b) we note that R’d 32

[-(1

- CI) p + 2mx] ln(arm2x2) = 0

if n > 2, so the above is equal to ,$@(p)

= f-$ ,: daR$” (&)” =-

[(l - a) p - 2mx] In [ oun2x2 -Gii2-

e02 I’ da[(l - a) p - 2m] In (1 8rr2 o

(’ ,f)“).

a) p2 ] (E-9)

558

KU0

To obtain zp)(p)

AND

YENNIE

it is only necessary to subtract Zj:“‘(m),

2yyp) = 2$@(p) - 2$@(m) = $,‘da

I[(1 - a) p - 2m] In [ 1 - (1 - a) $1 0

+ (1 + a)mIna.

(E.10)

I

For the regular method the two terms in the integrand in (E.8) are handled separately. The treatment of the first term exactly parallels that of J7:%‘(kz). The second term requires two differentiations Rp &,“-’

J (dk) (&)’

= RF' ($-)+'

x[fv + 1y(l - a)p” - anw]-2

(1 + $

x) 4wn2x / (&)[k2

= $&’ (&)+‘(I + &) = &

m2x2

+ ~$1 - LY)~” -

-‘;;m

a~n~x~]-~

a)p2

In (1 - (1 - CY)$),

(E.ll)

where the last step follows from (A.14). When (E.ll) is substituted back into (E.8) the same result (E.9) is obtained. From (E. 10) the on-the-mass-shell electron self-energy function can be computed

It is interesting to note that while L?jla)(p) is free from infrared divergence, 2?p’( p) is not; the divergence comes from azl*‘/@, which diverges at p = m. (iii) Lowest-order vertex function A$@(p’,

p) = -ietR$

(E.13) The quantity in square brackets is parametrized

I at,0

to give

Y~(#’+ Ic + mx) YAP + k + mx) Y’ @‘)

[k2 + 2(01,p + asp’) . k + fflpa + a,~‘~ -

(0~1 + 4

m8x213

RENORMALLZATION

559

THEORY

where (d/l) = 2da, da2da, 6(1 - a1 - 0~~- 01~).

The integration variables are now shifted by the amount ar,p + 01~p’, and only even terms are retained, with the result s CC+0

$+xy,k

+

Y"(P'

-

%P

-

a,@+

mx)yu(P

-

%P

-

a,P'+

4r"I,

(E.14)

where D E k2 + alp2 + a&2 - (alp + CX&)~ - (aI + a2) m2x2. In (E. 14) only the term containing

ky,$ needs special attention. Under integration by -$yuk2, then either the regular or the E-method can be

sign we replace kr,k used to show

-i

= ~

yLI ln [

%P2

+

“2P12

-

(w

+

c~~P’)~- (al + a2) m2 1

The integration of the remaining terms in (E.14) is straightforward the results are combined, we find

alp2

+

x 1yu ln ( + A 3&f - alp 2

1’

-(a1 + 01~)m2

and, when

(w + N~P’)~- (al + a21m2 1 -(a1 + a2) m2 a2P’ + 4 y,(t, - alp - a2pI’ + m) y”

o12Pf2 -

(a1 + 01~)m2 + (alp + a,~‘)~ - alp2 - a2p12

(E.15) The further reduction

of AiF’ proceeds in a conventional way. We note that of 01~= 01~= 0 so that the on-themass-shell vertex function

A:F)(m, m) diverges in the neighborhood

(lgy(p 3p’) = f@)(p , p’) - &y

contains an infrared divergence. RECEIVED: July 31, 1968

m, 4

560

KU0

AND

YBNNIE

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