Treatment of Overlapping Divergences in the Photon Self-Energy Function*) Introduction

Transcription

1 Supplement of the Progress of Theoretical Physics, Nos. 37 & 38, Treatment of Overlapping Divergences in the Photon Self-Energy Function*) R. L. MILLSt) and C. N. YANG State University of New York at Stony Brook Stony Brook, New York, U. S. A. (Received August 29, 1966) An analysis is made of the generalization of Ward's identity to the problem of renorrnalizing the photon self-energy function in quantum electrodynamics. It is shown that it is nontrivial to select a path of differentiation in a consistent way through the different self-energy graphs, but that a suitable set of simple rules can be given which completely resolves the overlapping divergence difficulty in the photon self-energy function. Introduction In the renormalization program for quantum electrodynamics, the straigtforward procedure of renormalizing divergent subgraphs in a systematic way is possible only if no two divergent proper subgraphs**) overlap, that is, if for any two such divergent parts, either one is entirely within the other, in which case the former is to be renormalized first, or else it is entirely outside, in which case they can be renormalized independently. The problem of handling overlapping divergences, which in quantum e lectrodynamics occur only within proper self-energy graphs, has long been a source of confusion in efforts to present the subject of renormalization in the clear and elegant manner it deserves. While the problem itself has been treated quite generally/)-~) an elegant treatment 6 ) exists only in the case of the electron self-energy; here Ward's identity7) provides a simple relation between the derivative of the electron self-energy function and the vertex function, and the latter can be renormalized straightforwardly without overlap problems. A similar relation 5 ),o) does exist for the photon selfenergy function, which renders the renormalization procedure equally straightforward; nevertheless, the manner of differentiation is not immediately so clear as in the electron case because of the difficulty in specifying a path of differentiation in a consistent way through every possible proper self-energy graph. It is the purpose of this note to clarify *l Supported by the New York State Foundation of Science and Technology. tl Permanent address: The Ohio State University, Columbus, Ohio, U. S. A. **l A 'subgraph' will refer to a portion of a graph (a set of vertices and those lines which interconnect them) which does not include the whole graph. A 'proper' subgraph is one which cannot be disconnected by the removal of just one line.

2 508 R. L. Mills and C. N. Yang this procedure. In the interest of clarity we ignore here the matter of infrared divergence. Our concern is solely with the overlap problem for ultraviolet divergences. We shall also ignore here the overlap problem which arises in four-photon graphs. Let us first consider briefly the case*) of the electron self-energy function S'(p), whose terms, in the Feynman-Dyson perturbation expansion, contain overlapping divergent subintegrals. This difficulty can easily be handled 6 ),7) by solving for the derivatives (fj/fjprx.) s- 1 (p) and deducing S' (p) from these, subject to the proper boundary conditions on the renormalized self-energy function. It was shown by Ward 6 ). 7 ) that the vertex function r is related to this derivative by Now the graphs for r do not contain overlaps. So we can write r a (p, p) = ra + ;s contributions from all skeleton graphs with insertions r, S', D' for all vertices and propagators. (2) Difficulty with :~ It appears, at first sight, that by defining the derivative of the proper photon self-energy function n as a new type of "vertex function" J, one could follow exactly the same procedure as above. Upon closer examination, however, we found that one has to use caution in this procedure, since the differentiation of any particular n graph and the subsequent reduction to skeleton graphs depends on the path chosen for the external momentum through the graph. An explicit example of the danger associated with a wrong choice of path was exhibited in reference 5). We found (cf. reference 5), p. 1439) that to resolve the difficulty one way is to introduce a separation of each n graph into a sequence of core parts. If the core part contribution C is introduced explicitly, it turns out that the overlaps can be explicitly disentangled and the choice of path problem handled systematically by writing down equations similar to (2), which when iterated give correctly all the terms of n. This procedure will be described now. Core part C A n graph in which all interior self-energy parts are shrunk out and given the value S' or D' is called partially reduced. A partially reduced n *) For simplicity, we write D for DF and S for SF in this paper.

3 Treatment of Overlapping Divergences 509 graph can be uniquely separated into a sequence of core graphs in the manner shown in Fig. 1. Each core graph has one incoming electron line and one outgoing electron line on the left and similarly on the right. It must also be such that by breaking any two internal electron lines, one cannot separate it into two disjoint parts, one on the left, core core Fig. 1. Separation of a partially reduced 1t: graph into a sequence of core graphs. In this example the 1t: graph is separated into three core graphs as marked. the other on the right. We shall refer to the electron line that enters a core part from the left or goes out of a core part towards the right as the upper electron line. If we denote by C(kt, k2, ks) the contribution from all core graphs with the same external momenta k 1, k 2, ks and - k 1 - k 2 - ks we have, as illustrated in Fig. 2, n(k) =rs' S'r+rS' S'CS' S'r+rS' S'CS' S'CS'S'r+. (3) The spin indices are suppressed in this formula in an obvious way. This formula is not suitable for renormalization; the presence of r, which prevents proper cancellation of the renormalization factors Z1 and Zs, is related to the occurrence of overlapping divergences. To resolve this difficulty we differentiate (3) after assigning in Fig. 2 the external momentum k to the upper electron lines S'. Using c c c Fig. 2. 1t: as a sum of terms involving C. All electron lines are S', not S. an - as' S' + as' S'CS' S' ak -r-ak r r-fik r it 1s obvious that ( 4) becomes + S' S' ac S' S' + S' S'C as'_ S' r ~ r r ~ r +. (4) r=r+rs' S'C+rS' S'CS' S' + (5) }!!_=r as S'r+rS'S'-ac S'S'r ak ak ak ' (6)

4 510 R. L. Mills and C. N. Yang a formula illustrated m Fig. 3. This removes the overlapping divergences completely since C can be reduced to skeletons without overlaps. A skeleton C graph is one in which all vertex parts are shrunk out and given the value r. Therefore, ac 8k Fig. 3. Graph corresponding to Equation (6). A triangle denotes derivative with respect to k. All electron lines are S', not S. C= 2: contribution from all skeleton C graphs with insertions r, S', D' for all vertices and propagators. as illustrated in Fig. 4. (7) c I X ll Fig. 4. C as a sum over skeletons. S'. The blobs are r. All internal lines are D' and Renorrnaliza tion Equations (1) (2), (6) and (7) are equations involving S', D', r and C. Iteration of these equations gives the correct Feynman diagrams for S', D', rand C. Furthermore, these equations are of the correct form for renormalization, since the overlap problem, as exhibited by the appearance of factors r in (3) is eliminated through the differentiation process which leads to the replacement of (3) by (6). To complete the renormalization program one would still have to show that subtractions from (6) and (2) actually render the integrals convergent. This is a problem that we do not propose to discuss here. But we do want to sketch a demonstration that this problem is not more involved in (6) than in the skeleton r diagrams. To do this we shall show that fhr:/fik can be written as a sum over skeleton graphs with all possible selfenergy and vertex insertions (including lower-order fin/fik subgraphs). It is not necessary that all possible skeleton fin/fik graphs appear, as indeed they do not, but it is necessary that all graphs with the same skeleton be included in order that the removal of renormalization factors should correspond to subtractions in the various subgraphs. To accomplish this, we use Eq. (6). In the first term on the right we use Ward's identity (Eq. (1)), and, for the second, we give rules for differentiating C. In each C graph the path of differentiation must enter

5 Treatment of Overlapping Divergences 511 and leave by the upper external lines, as noted above. For each skeleton C graph we arbitrarily specify a standard path; then in any C graph we follow the standard path through the skeleton, use (1) and (6) for selfenergy subgraphs, and an analogous prescription for r subgraphs. That is, for each skeleton r graph we choose an arbitrary standard path (one for each way of entering and leaving), and for reducible r graphs we use the standard path for the skeleton, and the above procedures at lower orders for self-energy and vertex subgraphs. It is convenient, though not essential, to follow electron lines wherever possible, and to follow the electron direction along each electron line. One must then prove that the graphs generated by the above procedure satisfy the requirement discussed above. This can be done by induction. (Note in particular that a skeleton 8C/8k graph, for example, may arise from the differentiation of a nonskeleton C graph.) Acknowledgement One of the authors (RLM) wishes to thank the members of the Physics Department and the Institute for Theoretical Physics at Stony Brook for the hospitality he enjoyed during his visit. References 1) A. Salam, Phys. Rev. 82 (1951), 217; 84 (1951), ) S. Weinberg, Phys. Rev. 118 (1960); ) J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw Hill Book Com pany, Inc., New York, 1965), Chap ) N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience Publishers, Inc., New York, 1959), Sec. 26. Klaus Hepp, Communications in Math. Phys. (in print). 5) T. T. Wu, Phys. Rev 125 (1962), ) ]. C. Ward, Proc. Phys. Soc. (London) A64 (1951), 54. 7) ]. C. Ward, Phys. Rev. 78 (1950), 182(L).