RENORMALIZATION
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Tcxtbooks A N D HYMANBASS Editors: S A M U E I . EILENBERG
A list of recent titles in this series ;ippe;irs at the end of this volume.
RENORMALIZATION Edward B . Manoukian ROYALMILITARY COLLEGEOF CANADA KINGSTON,ONTARIO,CANADA
1983
W
ACADEMIC PRESS A Subsidiary of Hurcourt Bruce Jovanovich, Publishers
New York London Paris San Diego San Francisco Siio Paulo Sydney Tokyo Toronto
COPYRIGHTQ 1983 BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMIITED IN ANY FORM O R BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING. OR ANY INFORMANTION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
1 1 I Fifth Avenue, New York. New York 10003
United Kin dom Edition ublished b ACADEMfC PRESS. &C. ILONbONl LTD. 24/28 Oval Road, London NWI
iDX
Library of Congress Cataloging in Publication Data Manoukian, Edward B. Renormalization. (Pure and applied mathematics) Includes index. 1. Renormalization (Physics) 2. Feynman integrals. 3. Quantum field theory. I. Title. 11. Series: Pure and applied mathematics (Academic Press) QA3.P8 [QC20.7.R43] 510s [530.1’5] 82-8772 ISBN 0-12-469450-0 AACR2
PRINTED IN THE UNITED STATES O F AMERICA
83848586
9 8 1 6 5 4 3 2 1
To Tanya
This Page Intentionally Left Blank
CONTENTS
ix xi
i
Chapter 1 Basic Analysis 1. I Sets and the HeineBorel Theorem 1.2 Measure, Integration, and Fubini’s Theorem 1.3 Geometry in Rk Notes
5 12
26 28
Chapter 2 Class B, Functions and Feynman Integrals 2.1 Definition of Class B. Functions 2.2 Structure of Feynman Integrals Notes
30 30 42
Chapter 3 The Power-Counting Theorem and More 3.1 Statement of the Theorem 3.2 Proof of the Theorem in One Dimension 3.3 Proof of the Theorem in an Arbitrary Finite Number of Dimensions Notes
vii
43 46 54 62
Contents
viii Chapter 4
+O Limit of Feynman Integrals and Lorentz Covariance
E
4. I Class Y ( W )Functions (Test Functions) 4.2 Basic Estimates, E --* +O Limit, and Lorentz Covariance Notes
63 66 76
Chapter 5 The Subtraction Formalism 5.1 5.2 5.3 5.4
Basic Definitions and Examples The Subtraction Scheme Convergence of the Subtraction Scheme The Unifying Theorem of Renormalization and Basic Identities Notes
77 102 107
128 132
Chapter 6 Asymptotic Behavior in Quantum Field Theory 6.1 Preliminary Asymptotic Estimates 6.2 General Dimensional Analysis of Subtracted-Out Feynman 6.3 6.4 6.5 6.6 6.7 6.8
Integrands High-Energy Behavior General Asymptotic Behavior I Zero-Mass Behavior Low-Energy Behavior General Asymptotic Behavior I1 Generalized Decoupling Theorem Notes
136 139 151 162 164 171 175 178 181
Appendix Subtractions versus Counterterms A. I The Formal Unrenormalized Theory A.2 Equivalence Notes
183 187
192
References
193
List of Symbols
20 1 203
Index
PREFACE
Renormalization theory is still with us and very much alive since its birth over three decades ago. It has reached such a high level of sophistication that any book on the subject has to be mathematically rigorous to do any justice to it. Since the early classic works on quantum electrodynamics, it has been studied systematically and has become the method for computations in relativistic quantum field theory. The success of renormalizaztion has been recorded at least twice in the history of physics through Nobel prizes, once in 1965 for the work on quantum electrodynamics, and again in 1979 for the work on the unification of the electromagnetic and weak interactions. Quantum electrodynamics is in excellent agreement with experiments, and the unified field theories seem to be quite promising candidates for a more complete theory. In spite of the importance of renormalization theory in physics, very few field theorists seem to know the intricate details of this subject. This is essentially due to the complex nature of the subject. Accordingly a book on renormalization theory would be quite justifiable at this stage, if not a matter of urgency. We define two major lines of work on the subject. The first is to develop the subtraction formalism, provide its convergence proof, and extract general and valuable information from the subtractions, all done in a model-independent way, and is mathematical in nature. The other deals with model building, “modified” Feynman rules, symmetry principles ix
X
Preface
and spontaneous symmetry breaking, renormalization group equations, and operator product expansions, to mention just a few problems. This second line of work is a rapidly developing branch of research, and there is still much to be done. The first line of work is well established, and this book is devoted exclusively to it and deals with the basic facts of renormalization theory. Perhaps another suitable title for the book would have been “The Core of Renormalization.” The subtraction scheme we use has a very simple structure; we were inspired by the ingenious and classic work of Salam in its formulation. It is carried out in momentum space and applied directly to the Feynman integrand without ultraviolet cutoffs. A unifying theorem of renormalization is given that brings us into contact with other standard approaches of subtractions. In particular, the latter establishes the long-standing problem of essentially the equivalence of the paths taken in the ingenious approaches of Salam and Bogoliubov. The book deals with the basics of renormalization theory presented in a mathematically rigorous manner. It presents the subject in a unified manner and model-independent way. It is primarily designed for graduate students and researchers in quantum field theory, mathematical physics, and elementary particles. It may be used as a text in renormalization theory, or its subject matter may be incorporated in the second half of a serious course in quantum field theory. It may be also used as a reference book and for individual study. Although the textbook is mathematical in nature, one does not need mathematics beyond what one learns in a conventional undergraduate honors physics program to read it. Some familiarity with Feynman rules (at least with the structure of a Feynman integral and a Feynman diagram) is, however, essential. Excellent books and monographs exist in the literature on the latter, and they may be read simultaneously with this book. Most of the material given in the text has already appeared in the published literature. The first chapter deals with basics of mathematical analysis and, in particular, with the theory of multiple integrals and Fubini’s theorem that are so essential in this subject. The property of Feynman integrands as belonging to a very special class of functions, called the class B, functions, is given in Chapter 2. In Chapter 3 we deal with the classic power-counting theorem that provides a convergence criterion for Feynman integrals and also provides a method of studying their asymptotic behavior. The E + +O limit of Feynman integrals is dealt with in Chapter 4, where we establish their Lorentz covariance in this limit. In Chapter 5 we discuss the subtraction formalism and its convergence. Chapter 6 is devoted to the study of the asymptotic behavior of subtracted-out Feynman amplitudes in the Euclidean region. In an appendix the equivalence of the subtraction and the counterterm formalisms is given.
ACKNOWLEDGMENTS
Most of the book was written at the Royal Military College, and I could have found no more congenial atmosphere for completing this pleasant task than the one existing here. I should like to thank Professor S . Weinberg for the interest he has shown in my work on the subtraction formalism at its earlier stages. Professor W. Zimmermann’s analysis of his Bogoliubov scheme has been extremely useful in formulating our analysis of subtractions. Thanks are due to many of my colleagues who have repeatedly encouraged me to write a book on the subject. My thanks to C. Chamberlain, L. Craven, E. Engelhardt, B. Ison, I. Kennedy, and J. Morin for all participating in typing the difficult manuscript. The book would never have been completed without the patience and understanding of my wife. I am deeply grateful to her.
xi
This Page Intentionally Left Blank
INTRODUCTION
The years 1965 and 1979 marked, through the Nobel prizes,’ the unique role renormalization theory had played in physics. Renormalization theory is undoubtedly one of the most important developments ever in elementary particle physics. It has become the method for computations in relativistic quantum field theory and thus been able to confront theory with experiments, which, after all, is the ultimate goal of physics. The success of quantum electrodynamics is undisputably recognized, and the numerical predictions of this theory are in excellent agreement with experiments.* The unified field theories of the weak and electromagnetic interaction^,^ with Yang-MillsShaw4 fields, and in the same spirit the currently unified field theories of the strong, weak, and electromagnetic interaction^,^ seem to be promising candidates for a more complete theory. There have been even many interesting attempts to unite gravity6with the other interactions and develop a formalism
’
Schwinger (1972). Feynman (1972).Tomonaga (1972), and Salam (1980). Weinberg(l980), Glashow (1980). Cf. Brodsky and Drell (1970). See also Lautrup ef a/. (1972); Hinsch et a/. (1979). Salam (1968, 1980). Weinberg (1967. 1970. 1974, 1980), Glashow (1961, 1980), and some reviews by Abers and Lee (19731, Bernstein (1974), Big and Sirlin (1974), Taylor (1976). See also Schwinger (1957), Glashow (1959). Yang and Mills (1954). Shaw (1955). See also Utiyama (1956). Cf. Bjorken ( 1 972). Pati and Salarn (1973). Georgi and Glashow (l974), Georgi et a/. (l974), Buras et a/. (1978). Cf. Zumino (1975). Arnowitt et a/. (1975). Akulov et a/. (1975). Freedman et a/. (1976). Deser and Zumino (1976), Wess and Zurnino (1977). Brink et a/. (1978). See also Salam and Strathdee (1978).
’ ’
’
I
2
Introduction
such that gravity may be treated by currently available methods of renormalized quantum field theory.’ In short, renormalization theory is very much alive and will undoubtedly stay with us for a very long time. Almost immediately after theinitial stagesin the development* ofquantum field theory, the so-called ultraviolet divergence problem was encountered in various computation^.^ As early as 1947, Bethe successfullycarried out a mass renormalization” in his classic computation of the Lamb shift,” and this year may be said to have marked the beginning for the need of a careful study of renormalization in field theory.” The first covariant formulation of quantum field theory, in a form quite suitable for practical computations, was due to Schwinger, Feynman, and T ~ m o n a g a , and ’ ~ the concept of renormalization was particularly touched upon in the classic papers of Schwinger (1948a, 1949a,b). Dyson (1949a,b) unified the works of Schwinger, Feynman, and Tomonaga and developed the initial stages of renormalization in quantum electrodynamics to arbitrary orders in the fine-structure constant. The first systematic study of renormalization historically was carried out by Salam in 1951 in a classic paper (Salam, 1951b; see also 1951a) where the subtraction scheme of renormalization, in a general form, was formally sketched. Surprisingly, this classic paper was not carefully reexamined until much later. In 1960 Weinberg established and proved one of the most important theorems in field theory. This theorem, popularly known as the “ power-counting theorem,” embodied a power-counting criterion to establish the absolute convergence of Feynman integrals. Salam’s work was first reexamined and brought to a mathematically consistent form in Manoukian (1976). In this latter paper, inspired by the classic paper of Salam, a subtraction scheme was developed and spelled out in momentum space with the subtractions, carried over subdiagrams, applied directly to the Feynman integrand with no ultraviolet cutoffs and taking into account all divergences. The absolute convergence of the corresponding renormalized Feynman amplitudes was then proved by the author (Manoukian, 1982a; see also 1977) by explicitly verifying in the process that the power-counting criterion of
’ Cf. Weinberg (1979). Born et a/. (1926). Dirac (1927). See also Heisenberg (1938) for later work. Oppenheimer(1930), Waller(1930). Seealso Heisenbergand Pauli (1929,1930). Heisenberg (1938). Weisskopf and Wigner (1930a,b), Weisskopf (1934a,b, 1936, 1939). l o Apparently H. A. Kramer (cf. Schweber, 1961 ;Weinberg, 1980) first emphasized the idea of mass renormalization. I Lamb and Retherford (1947). For a fairly detailed historical review of the early days of quantum field theory see, e.g., Schwinger (1958). Peierls (1973). Wentzel(l973). l 3 See Schwinger (1948a,b, 1949a,b, 1951a,b), Feynman (1949a,b, 1950), Tomonaga (1946), and also Koba et a/. (1947), Kanazawa and Tomonaga (l948), Koba and Tomonaga (1948). See also Schwinger (1958).
’*
3
Introduction
Weinberg was satisfied, thus completing the Dyson-Salam (DS) program. Shortly after the appearance of Salam’s work, Bogoliubov, together with Parasiuk (1957),14 in a classic paper developed a subtraction scheme and outlined a proof of its convergence. In 1966 Hepp gave a convergence proof of the Bogoliubov-Parasiuk scheme by using in the intermediate stages ultraviolet cutoffs, and in 1969 Zimmermann formulated the Bogoliubov scheme in momentum space with no ultraviolet cutoffs and gave a convergence proof of this subtraction scheme. Thus these two latter authors completed the Bogoliubov-Parasiuk (BP) program. Finally, the equivalence of the Bogoliubov scheme, in the Zimmermann form, and our scheme was then proved, after some systematiccancellationsin the subtractions, by the author (Manoukian, 1976) in a theorem that we have called the “unifying theorem of renormalization.” Since the Zimmermann form grew out of Bogoliubov’s work and our form grew out of Salam’s work, this theorem establishes the longstanding problem of essentially the equivalence of the paths taken in the ingenious approaches of Salam and Bogoliubov (in momentum space). The situation may be summarized by the following diagram: DS: Dyson (program)
Salam + Weinberg +
Author (completion)
I Author (equivalence) 7
BP: Bogoliubov-Parasiuk + Hepp-Zimmermann (completion) (program) The book deals with a mathematically rigorous formulation of renormalization presented in a unified manner and a model independent way. The subtraction scheme is introduced and its structure is studied, its convergence proof is provided, and finally valuable information from the so-called renormalized Feynman amplitudes is extracted. There are, however, many other interesting and important problems directly or indirectly related to renormalization that the reader may wish to read about. Some of these problems are gauge invariance,’ “modified ” Feynman ru1es,I6 anomalies’
’
See also Bogoliubov and Shirkov (1959) and Parasiuk (1960). Cf. Schwinger (1951a), Zumino (1960). Bialynicki-Birula (1962), Lukierski (1963). Taylor (1971). Slavnov (1972), Lee (1973). Fradkin and Tyutin (1974). I’ Cf. Matthews (1949). Lee and Yang (1962), Feynman (1963), deWitt (1964, 1967), Weinberg (1964a,b, 1969), Lam (1965), Fadeev and Popov (1967), Mandelstam (1968a,b), Fradkin and Tyutin (1969, 1970). Salam and Strathdee (1972), Bernard and Duncan (1975). Schwinger (1951a), Adler (1969), Bell and Jackiw (1969). I4
Is
Introduction
4
versus renormalizability,’ symmetry breaking’ versus renormalizability,” operator product expansions,’ renormalization group equations,” and the important concept of asymptotic freed~rn,’~ and there is still much to be done. As a rule of thumb, a theory is renormalizable if no coupling parameters appear in the Lagrangian that have the dimensions of some negative powers of mass. This in turn is related to the important fact that in a renormalizable theory the numerical values of only a finite number of parameters (such as masses and coupling constants) are “adjusted” or “fixed” e~perimentally.’~ On the other hand, in a nonrenormalizable theory, generally speaking, an “infinite” number of parameters are to be “adjusted ”or “fixed ”experimentally and such theories lose their predictive power^.'^ The mathematical structure of the subtractions we have developed, however, is general to apply to nonrenormalizable theories as well. All the subtractions of renormalization are carried out at the origin of momentum space.25 The following (partial) list of standard books on quantum field theory will be useful for the reader to consult when reading the present one so that he or she may benefit as much as possible from it: Bjorken and Drell(1965), Schweber (1961), Jauch and Rohrlich (1976), and Nishijima (1969).
Cf. Gross and Jackiw (1972). Bouchiat et a/. (1972). Cf. Higgs(1964,1966),Kibble(1967),Englert andBrout (1964),Guranliketa/.(1964).See also Goldstone (1961). Goldstone et a/. (1962), Anderson (1963). ” ’ t Hooft (1971a,b), ’t Hooft and Veltman (1972a,b), Lee (1972), Lee and Zinn-Justin (1972a,b,c), Becchi et a/. (1975, 1976). See also Fradkin and Tyutin (1974). Vainshtein and Khriplovich (1974), Costa and Tonin (1975), Slavnov (1975). ’ I Cf. Wilson (1969). Frishman (1970). Brandt and Preparata (l97l), Zimmerman (1973a,b). 2 2 Stuckelberg and Petermann (1953), Cell-Mann and Low (1954). Bogoliubov and Shirkov (1959). Callan (1970), Symanzik (1970). ‘t Hooft (1973), Weinberg (1973). 23 Politzer (1973). Gross and Wilczek (1973). See also Politzer (1974). Buras (1980). Marciano and Pagels (1978). 2 4 Cf. Matthews and Salam (1954), Salam (1952), Bogoliubov and Shirkov (1959) for such details. ” A renormalization with subtractions performed at the origin ofmomentum space iscalled an intermediate renormalization; cf. Bjorken and Drell(l965). l9
Chapter 1 / BASIC ANALYSIS
1.1
SETS A N D T H E HEINE-BOREL
THEOREM
Consider two points x = (xl,. . ., x k ) and y = (yl,. . . ,yk) in the kdimensional Euclidean space Rk.Define the distance between x and y by Ix - yl = ,(xi - yi)2)1'2. A neighborhood of a point x E Rk is a set N , ( x ) defined by N , ( x ) = {y: Ix - y I < E } , where E > 0 is called the radius of N,(x). A point x is a limit point of a set S c Rk if every neighborhood of x contains a point of S different from x.l This leads to a definition of a closed set as a set containing all of its limit points. A point x is an interior point of a set S c Rkif there is a neighborhood N,(x) c S. This leads to a definition of an open set as a set such that every point in it is an interior point. An elementary result in set theory relating open and closed sets is the following: A set is open if and only if its complement is closed. Let S' be the complement of S in Rk,i.e., S' = Rk - S , which denotes the set of elements in Rkbut not in S. Suppose s' is closed and let x E S (i.e., x 4 s').Then by definition of a closed set, x cannot be a limit point of s' as the latter contains all of its limit points. This means that there exists a neighborhood N,(x) of x such that S' n N,(x) = Qr and hence N,(x) c S. Thus x is an an interior point of S and S is open. Conversely, suppose that S is open and let x be a limit point of S'. Then for every E > 0, N,(x) contains a point of S' different from x [i.e., the N,(x), for all E > 0, are not subsets of S ] , and so x is not an interior point of S. Since S is open, it
(c:=
' The symbol c stands for containment. A t B means A is a subset of B ; it may also imply that A equals B. To exclude equality we write p. 5
1 Basic Analysis
6
follows that x E S', S' contains all of its limit points, and hence it is closed. Other useful results related to open and closed sets are the following: The union of an infinite number of open sets is open; the intersection of an infinite number of closed sets is closed; the union of a finite number of closed sets is closed; and the intersection of a finite number of open sets is open. We discuss the important concept of a cover of a set. We say that a family of sets { S , :p E V }covers a set S if S , 3 S , where Vis some set and need not be countable. For any set S c Rk,the family of neighborhoods { N e ( x ): x E S } obviously covers S . Again the set S need not be countable. Another, though trivial, example of a cover of a set S c Rkis the family { R k } . The cover { N , ( x ): x E S } of S provides an example of an open cover as every set N,(x) in this family is open. A basic property of an open cover is the following: Every open cover of a set S c Rk has a countable subcover. The demonstration of this result is as follows. Let IS, : p E V } be an open cover of a set S . For any x E S , there exists at least one p E Vsuch that x E S,. Consider a neighborhood N,(x) of x such that N,(x) c S,. Let y = (yl, . . . ,yk) E Rksuch that the yi are rational and ly - x I < r/2. Since between any two real numbers we may find a rational number, we may let t denote a rational number such that ( y - X I < t c r/2. The latter implies that x E N,(y), N , Q ) c N,(x) c S,. The family of all neighborhoods with rational radii about points y = (yl, . . . ,yk) E Rkwith the yi varying over the set of rational numbers is obviously a countable set since the set of rational numbers is countable. We may then introduce a family of neighborhoods { N ' , N', . . .}, where for some n, N" = N , ( y ) with X E N" and N" c S,. Accordingly we may introduce indices p , such that x E N" c S,,. Now for any X E S ,we may find a neighborhood N" such that X E N" c S,,; hence the family {S,,, S p 2 , .. .} provides a countable subcover of S by open sets. We introduce the concept of a box in Rk. Let x = ( x l , . . . , xk), rn = (ml, . . . ,mk) and let I denote the set of integers and N the set of natural numbers. Consider the set
UpEY
B , ( r n ) = { x : m j / 2 " I x j < ( m j + 1)/2",j= 1 ,..., k}
(1.1.1)
for n~ N and m j E I . Let 9, = {B,(m) :all mi E I } .
( 1.1.2)
The latter defines a family of boxes with each box having the volume 2-k". Obviously for any boxes B, E $9" and B,. E W,,,with n > n', we can either have B, c B,. or B, n B,. = 0.Let S be any nonempty open set in Rkand let s E S . For a fixed n E N , x is contained in one and only one of the boxes in 9,.Since S is open, we may find, for n sufficiently large, a B, E 9, such that x E B, and B, c S. Accordingly, we may write S as the union of those elements in B1 contained in S, those elements in W 2contained in S but not lying in the boxes
1.1 Sets and the Heine-Bore1 Theorem
7
in A?’, those elements in B3contained in S but lying neither in the boxes in A?, nor in the boxes in A?2,and so on. That is, S may be written as the countable union of disjoint boxes in u B2u &I3 u . + .. This construction is quite useful for defining the concept of a “volume” for the open set S as the sum of volumes of these disjoint boxes, just mentioned, in A?’ u B2u A?3 u ... . The volume of a box in A?,,, as we have seen above, is 2-k”. The following construction is of particular interest. Let n E N be fixed. The family B,, in (1.1.2) is countable; i.e., we may write A?,, = { B ’ , B2,. . .}, where B’, B 2 , .. . are pairwise disjoint boxes, each box of volume 2-k”, and Q
Rk = (JBi.
(1.1.3)
i= 1
Consider a set of real numbers E, i.e., E c R’.If for all a E E, there is a number b E R’ such that a < b, then the set E is said to be bounded above and b is called an upper bound of E . If b is an upper bound of E and if for every c b, c is not an upper bound of E, then b is called the least upper bound (1.u.b.) of the set E. Similarly, we define a set E of real numbers to be bounded below if for all a E E , there is a number 1 E R’ such that I I a, then I is called a lower bound of E. If 1 is a lower bound of E and if for every d > 1. d is not a lower bound of E, then I is called the greatest lower bound (g.1.b.) of the set E. Suppose E is a closed set and is bounded above. Let b = 1.u.b. E;’ then it is easy to see that b E E. To show this, suppose that the contrary is true, i.e., b 4 E. Then for every E > 0, there is an a E E such that b - E Ia Ib ; otherwise (i.e., a < b - E for all a E E ) b - E would have been an upper bound of E, contradicting the fact that b = 1.u.b. E. This means that every neighborhood of b contains a point of E different from b since we have assumed that b $ E. It then follows that b is a limit point of E and hence it must be in E , as the latter is closed, thus leading to a contradiction with the hypothesis that b E. Consider a set S c Rk.The set S is said to be bounded if there is a point y E R‘ and a positive constant 0 < C < co such that Ix - yl < C for all x E S. A sequence may be defined as a function whose domain 9 is a subset of (or coincides with) the set of natural numbers N : ( a , :n E 9 c N } .
-=
1.I .I
The Heine-Bore1 Theorem
We show that each open cover of a closed and bounded set S contains a finite open cover. We have already shown that every open cover of a set contains a countable open cover. Accordingly suppose that {On}= {On: n = 1,2,. . .} is a countable open cover of the set S : 0, =) S. Then we
un
* Here we are invoking the “completeness axiom” of real numbers, which states that the bounded set E has a 1.u.b.
I Basic Analysis
8
show that there exists a finite number of open sets S , , . . . ,Sy E (0,) that cover S : S c Si. An example of a closed and bounded set is provided by a closed and bounded k-cell Rk defined by
u;?"=,
W = ( x : a j < x j < b j , j = 1,..., k},
(1.1.4)
where x = ( x l , .. . , xk)E Rk,and (1.1.5) Then we note that for all x and y E W i n (1.1.4) we have Ix - yl Ir. We consider the proof of the above statement first for a closed and bounded k-cell as defined in (1.1.4). Suppose there exists an open cover (0,) of W that contains no finite subcover. Let d j = ( a j + bj)/2, and divide the closed interval [ a j , bj] into two (closed) intervals [ a j ,d j ] , [ d j , bj] for j = 1,2, . . . , k. For k = 2, for example, we then generate 4 closed 2-cells: W ' , W 2 ,W 3 ,W4 such that W' = W. In general we then generate 2k closed k-cells: W ' , W z , . . . , W Z ksuch that 0;: W' = W.By hypothesis at least one ofthese 2kclosed k-cells, say, W j = W,,cannot be covered by a finite number of open sets in {On}.We repeat the same construction, as done above for W, for the k-cell W, and thus generate 2k closed k-cells. Again by hypothesis at least one of these latter k-cells, say, a k-cell W, ,cannot be covered by a finite number of open sets in {On}.Continuing this process, this generates a sequence of closed and bounded k-cells W,, W,, . . . such that W 2 W, 3 W, 3 I . , where by hypothesis the cannot be covered by a finite number of open sets in {On}. We first show that there is at least one element common to all the K, i.e., is not empty. The 4 are of the form
0s
w
w ={ x : a i < x j I b : , j = 1,..., k}, for i = 1,2,.
( 1.1.6)
. . . Let cj = l.u.b.(afi:i
=
1, 2,. . .},
( 1.1.7)
and hence by definition a$ Icj for all i = 1,2, . . . .Because of the property of the k-cells W,3 W,3 , it is readily seen that the set (a;, a;, .. .}, with j fixed, is bounded above by bj'. It is also clear that cj Ibf, for all i = 1,2, . . . , otherwise we would arrive to the conclusion that c j is not a least upper bound as defined in (1.1.7). Accordingly uf 5 cj 5 bf for all i = 1,2,. . . . Let c = ( c 1 , c 2 , ..., ck). Then C E for ~ all i = 1,2, ..., i.e., c ~ n p D =4, , and hence the latter is not empty. Since {On} covers W,it then follows that we may find a member 0, such that c E 0,. Also since 0, is open, c is an interior point of 0,; i.e., we may
1.1 Sets and the Heine-Bore1 Theorem
9
find a neighborhood N,(c) c 0,.By construction, for any x and y in a set H!, our subdivisionsimply that ( x - y ( I 2-$,and in particular, Ix - cI I2-'r for all X E W,. Accordingly, by choosing i sufficiently large, we may make 2-'r c E, and hence we arrive to the conclusion that, for the corresponding i, c 0,.This leads to a contradiction to the hypothesis that H! cannot be as the open set 0, covers covered by a finite number of elements from (0,) This completes the demonstration that the closed and bounded k-cells can be covered by a finite number of open sets. We now generalize the above result to any closed and bounded set S c R k. Let W be a closed and bounded k-cell containing the set S: W 2 S. Consider the set (0,) u s',where the family (On}is an open cover of S, i.e., S c 0,. Obviously (0,) u s' is an open cover of Was the set S' is open and S u S' = Rk.We have already seen that a cover {On}u S' contains a finite cover of W. If this finite cover of W does not include the set S', then we have demonstrated the validity of our statement since W 3 S. On the other hand, if the finite cover does include the set S', then the finite open cover is of the form uy= Si u S' 3 W 3 S, where the Si E {On}.But S n S' = 0, hence S c UK Si and {Sl, . . ., S,} is a finite subcollection from the family (0,) and covers the closed and bounded set S. This completes the proof of our statement (the Heine-Bore1 theorem), which may be summarized as follows :
w.
Un
Every open cover of a set contains a countable open cover, and ifthe set is closed and bounded, then the latter cover contains afinite open cover.
Theorem 1.1.1 :
A useful result that follows by the application of the Heine-Bore1 theorem is the following:
Rkof a real function f is closed and bounded and f is continuous on 9.Then f is bounded, i.e., I f (x) I IM for all x E 9. Also there is a point xo E 9 such that 1 f (xo)l = l.u.b.( I f <x)l:x E 9}. Lemma 1.1.1: Suppose that the domain 9 c
Since f is continuous on 9,then for each x E 9,and any given E > 0, we may find a 6 , > 0 such that 1 f(y) - f (x) I c E, or I f (y) I II f (x) I E, for all y E N,,(x) c 9.In particular, we may choose E = 1. The set (N,,(x) :x E 9} provides an open cover of 9, and from Theorem 1.1.1 we may find a subcollection, say, {N,xl(xl), . . ., Ndx,(x,)}, that covers 9.f is bounded in each Nb,,(Xi)asIf(Y)l IIf(xi)l + l f o r a l l y ~ N , ~ , ( x i ) . L e t= M 1 + max{If(x1)l, . . . ,I f (x,)] }. Since for any y E 9 we may find an N,,,(x;), with i E [l, . . .,m], we then have for all y E 9,I f (y)l I M. Let b = l.u.b.{ I f (x) I : x E .9}.Suppose I f(y)I c b for all y E 9.From the continuity condition off, we may find for each x ~ and9 E = a(b - If ( x ) l ) > 0, where0 c a c 1, a px > 0 such that If CV) I (1 - a ) I f (x) I + ab for all Y E Np,(x)* Let {Npx,(x,), * * Np,,(xn)> be a cover of 9.Let M, = max( If(x,)l,. . . , I f ( x , ) l } . Then for all y e 9 we
+
9
9
1 Basic Analysis
10
have If(y)l I (1 - a)Mo + ab. Since, by hypothesis, M o < b, we have (1 - a)Mo + ab < b, contradicting the fact that b is an 1.u.b. of { I f ( x ) l : x E 9}. Hence there is a point xo E 9 such that I f ( x o ) l = b.
1.1.2 Algebras and a-Algebras
Consider an arbitrary set X.A nonempty class V of subsets of X is called an algebra of sets if (i) Sl, S , E V imply that S1 v S , E V,(ii) S E %implies that x - S = ScEV.3 Elementary consequences of this definition are that the empty set 0and the whole set X are in W and S1 n S , , S1 - S , are as well in V if S1, S , E %. That X E V follows from the fact that if a set S E V, then S' E V and S u S' = X E V.It then also follows that X' = 0E V.We finally note that if S1, S , E V, then SC,, S; E W and hence SC, v S$E V and (SC, v S;)c = S , n S, E V. Similarly, it is easily seen that S1 - S , E V. A nonempty class 48, of subsets of a set X is called a a-algebra of sets if (i) S1, S , , . . .,E V, imply that u g Si E V, and (ii) S E V, implies that s'E V,. In particular, this definition implies that if Sl, S , , . . .E V, then Si E V, as well. Let Vo be any collection of subsets of X.By considering the intersection4 of all algebras containing the family Vo, we then generate the smallest algebra containing Vo. This algebra will be denoted by %'(Vo)and will be called the algebra generated by Vo. Similarly, by considering the intersection of all a-algebras containing Vo,we then generate the smallest a-algebra containing V o ,which will be denoted Vu(Vo) and called the a-algebra generated by Wo. We now introduce the concept of a monotone class of subsets of a set X. A set Acof subsets of X is called a monotone class if for every monotone sequence S1, S2,.. . EA',i.e., for which S , c S , c ... or S1 3 S , 3 ..., Si E Acor SiE A', respectively. Let Vo be any collection of subsets of a set X ;then by considering the intersection of all monotone classes containing the family Vo, we generate the smallest monotone class containing Vo, which will be denoted AC(Vo). Let W be an algebra of subsets of a set X (also called an algebra in X).Let S be any subset of X.We define V n S = { A n S : A E V}. Then it is easy to see that V n S is an algebra in (i.e., with complements taken relative to the set S) S. Let @,(V n S) denote the smallest a-algebra in S containing V n S. Similarly let d C ( Vn S) denote the smallest monotone class with subsets in S containing V n S. We note that for S = X,
0:
us
nio.=
The set X will be called a universal set. It is easy to see that the intersection of algebras of sets is an algebra of sets, and that the intersection of a-algebras of sets is a a-algebra of sets.
1.1 Sets and the Heine- Bore1 Theorem
11
GU(%n S ) = WU(%)and Ac(% n S) = &(%) since X E %. Then we have the following important lemma concerning algebras, a-algebras, and monotone classes as direct consequences of the above definitions: Lemma 1.1-2: Let % be an algebra of subsets of a set
X . Then
(i) gU(% n S ) = wU(%)n S , where S is any subset of X , and gU(%) n S is dejined by Wu(%) n S = { A n S : A E Wu(W)},
and the complements are taken relative to the set S, i.e., ‘Xu(%)n S coincides with the smallest a-algebra in S containing % n S ;
Let X and Y be two sets. We define the Cartesian product X x Y as a set given by
x
x Y = {(x, y ) : X E
x,Y E Y } .
Let %: and %: be two o-algebras of subsets of X and Y , respectively. Consider the family LF of subsets of X x Y in the form
Fi:Fi n F j = 0 for all i # j and all finite n i= 1
where the F , are of the form F i= Ai x B, with A, E W;, B, E W;. Let E be any subset of X x Y, we define the x-section and the y-section of E by E, = { x : ( x , ~ ) E Ec} X.
Ex = { Y : ( x , ~ ) E Ec} Y,
We may then state the following lemma. (i) 9 is an algebra. (ii) For any two algebras 6,and bZof subsets of X and Y, respectively, let 6 = bl x J2and %;‘ = %u(61) and WE = %u(6z); then Lemma 1.1.3:
W U ( 4
and i f E c W U ( b ) , then E x E
=
% U ( n
E , E %,,(&I).
(Note that 6 = {A x B : A E bl, BE^^}).
I Basic Analysis
12 1.2
MEASURE, INTEGRATION, AND FUBINI’S THEOREM
1.2.1
Measure Theory
We have already defined the concept of a closed k-cell in Rk by the set W = ( x : a , ~ x , ~ b , , 1j,..., = k},
(1.2.1)
with x = (x,, . . .,Xk)E R’. Here we do not put any restrictions on the real constants a,,.b,. In general we define a k-cell as any set of the form (1.2.1) or any set in this form with some or all of the signs I replaced by < or = as well. Thus in particular the set {x :x, = a,} is a k-cell. We define the family d of elementary sets as the family containing the empty set 0 and any set that is a finite union of k-cells. Thus any nonempty set E in d is of the form E = W, u . u W,, where the are k-cells. It is not difficult to show that 8 is an algebra and any nonempty set in it may be written as the union of a finite number of disjoint k-cells. The measure of a k-cell is defined as its volume and is denoted by p(W) = m = , ( b j - a,). For the empty set 0 we associate the “volume” 0, i.e., p ( 0 ) = 0. We may define the measure of an elementary nonempty set as a sum over a finite number of volumes of pairwise disjoint k-cells. We may also generalize the concept of a measure to an open set, as a subset of R‘, by recalling (see Section 1.1) that an open set may be written as the disjoint union of boxes [see Eqs. (1.1.1) and 1.1.2)], where the latter are particular examples of k-cells. Hence the measure of an open set in Rkmay be defined as the sum of the measures (volumes) of these disjoint boxes. In general, the “measure” of a set, such as Rk,need not be finite, as p(Rk) = to. But R’ may be written as the union of disjoint boxes B’, B2,. . . [see Eq. (1.1.3)] each of finite volume 2-”‘, with nonnegative integers n, i.e., p(B’) = 2-”‘ < m. This particular property of a set as being written as the union over sets each of finite measure is called a a-finiteness property. The above analysis suggests, quite generally, introducing the concept of a measure as an extended (extended means that the value 00 is permissible) real-valued nonnegative set function on a family D (containing in particular the elementary sets) consisting of a suitable class of subsets of Rk such that (i)p(A) 2 0 for any set A E D , (ii) 0 E D and p ( 0 ) = 0, (iii) if A l , A 2 , . . . E D are pairwise disjoint sets, i.e., A, n A, = 125,for i # j , such that A iE D, then A,) = p(A,). The last property (iii) is called the o-additivity property of a measure. Here it is worth recalling that if a family of sets is a a-algebra, then the union of any infinite number of sets in it also belongs to the family. We now study how a family D may be defined and the measures of the
w.
,u(uz
xs
us ,
13
1.2 Measure, Integration, and Fubini‘s Theorem
sets in D are constructed. We already know how to define the measure of the elementary sets in I ;we generalize this to a larger class of subsets in Rk.With this in mind we define a new extended real-valued set function as follows. For any set S c Rk, we define the extended real-valued nonnegative set function Ial
The infimum (or g.1.b.) runs over all coverings {El, E 2 , . ..} of the set S with El, E 2 , . . . € 8 .In particular, we note that if S is an elementary set, i.e., S E I ,then p*(S) = p(S). It is also easy to see that if S =I S’, then p*(S) 2 p*(S’). Some elementary properties of the set function p* are (i) p*(S) 2 0, for any S c Rk,(ii) p*@) = 0, (iii) p*(UE Si)s 1 ; p*(Si) for all S1, S2, . .. as subsets of R’. The set function p*, however, does not necessarily satisfy the a-additivity property of a measure. It becomes a-additive by reducing the family of sets on which it is defined. This leads to the concept of a p*-measurable set. A set A is called p*-measurable if for any S c R’, p*(S) = p*(S n A )
+ p*(S n A‘).
(1.2.3)
Intuitively speaking, a p*-measurable set A is one that does not cut an arbitrary set S into two parts, one part lying “outside” of A and a part “inside” of A , such that the values of the set function p* for the two parts do not add up correctly. By restricting the definition of p* to the p*-measurable sets, satisfying (1.2.3), we obtain a measure, i.e., an extended real-valued set function satisfying all the requirements for a measure, including the cr-additivity property. An important lemma concerning p*-measurability is the following: (i) An elementary set is p*-measurable. (ii) The family M of all p*-measurable sets is a a-algebra. (iii) The set function p* in (1.2.2) when restricted to M is a measure. (iv) Any subset S with p*(S) = 0, is p*-measurable, i.e., S E M .
Lemma 1.2.1 :
Since M is a a-algebra and contains the elements in I ,we then conclude that M 2 W u ( I ) . Now we prove the following uniqueness theorem. Theorem 1.2.1 : There is a unique measure po on the a-algebra W u ( I ) generated by I such char po(S) = p(S) for any S E I .
We note that by the definition of p* as an extended real-valued nonnegative set function, the value + m is not excluded for the expression on the right-hand side of (1.2.2)in the so-called extended system [O, + a].
I
14
Basic Analysis
Suppose that pl and p, are two measures on W U ( b )such that pl(S) = p2(S) for all S E 8. We decompose Rk into disjoint boxes of finite measures [see (1.1.3)]:Rk= Biyp(Bi)< co,B'~b.LetAbeanysetinW~(b)andwrite A = U s l ( A n B'). Then pJ
UFl
uy=
uz,
n
UASn)
=
2 PARi)
i= 1
and n
n
lim z p , ( R i ) = lim x p 2 ( R i ) . n-co
is1
n-co
i=l
Hence by using, in the process, the a-additivity property of the measures we obtain (1.2.4)
i.e.,
Ugl S i e A , . For the case S,
3
Sm+,, we write
where note that the sets (OF=, &)y (S, - S,), ( S , - S3)¶. . . are pairwise disjoint. Thus we may write PAS,) = pj
(7 S k
(kyl
) + lim
Cpj<Si - S i + d
(1.2.5)
"+mi11
,
Now Si = Si+ u ( S , - Si+ 1) and is written as the union of two disjoint sets; hence pASi) = pJjSi+1)
or
+ pASi - Si+ 1)
1.2 Measure, Integration, and Fubini's Theorem
15
where all these measures are finite. Going back to ( 1 . 2 3 , we obtain ~ , @ i ) = pj
n Sk) + pCj(Si)- lim pCj{sn)*
(kyl
Since pj(S1)
-=
00
n+m
and pl(Sn)= p2(Sn),we finally obtain
(1.2.7) & E d , . Therefore dcis a monotone class of subsets of B" and containing d n B". From Lemma 1.1.2 we know that
g,(b) n B" = @,(bn B") = .,kc<s n B").
Therefore dc=I W R , ( 6 )n B" and for any set S E W,(b) n B",pl(S) = pz(S). Hence for any A E W,(b),with A" = A n B" E W,,(d) n B", we may write m
m
because p,(A") = p2(An),or P l U ) = kz(A),
(1.2.8)
which establishes the result of the lemma. It is worth recalling that a box is a k-cell, and hence is in 8,and that any open set S c Rkmay be written as the union of boxes (see Section 1.1). Therefore any open set S is in %,(&), and every closed set S' is in %,,(&), by definition of the latter. It is important to know the relation between the family M of all p*-measurable sets and W,,(b).We already know that M =I U,,(b)and that the set function p* in (1.2.2)when restricted to M is a measure. We also know that the measure defined on W,,(b)is unique. For simplicity of notation, we denote the latter measure by p as well and the former (ie., p* restricted to M ) by p. We define the concept of a measurable cover of a set in M . For any set E E M , S E V,(b) is a measurable cover of E if S 3 E, and for any G E W,(b) such that G c S - E, we have p(G) = 0. The following two lemmas embody ji and p. important results relating M and %?,(b), Lemma 1.2.2: For any set E E M , there exists a set S E %,(b) such that ji(E) = p(S) and such that S is a measurable cover. Lemma 1.2.3: Any subset E in M may be written as the disjoint union of a set B in %,(b)and a subset Eo of a set in %,,(b)with ,u*(EO) = ji(i(E,) = 0, i.e.,
with E , E M . That is, E = B u E , , B E '%,(d), and ji(E) = p(B). Conversely any set B v E,, with BE%#(&) and E , some subset of a set in W,(b) with p*(Eo) = 0, belongs to M .
1 Basic Analysis
16
To distinguish between the a-algebras Wu(b)and M, we call the sets in Wu(b)as Borel sets and the sets in M as the Lebesgue measurable sets. Lemma
1.2.3 states, in particular, that every Lebesgue measurable set may be written as the disjoint union of a Borel set and a Lebesgue measurable set with the latter of Lebesgue measure ii zero. We also note that if we define a measure p on a a-algebra Wo as complete by the condition that if S E Wu with p ( S ) = 0 and that R c S imply R E Wu, then we see that the Lebesgue measure ji is complete. Because if S E M and p ( S ) = 0, then for any R c S we have p*(R) 5 p ( S ) = 0, i.e., R E M [see Lemma 1.2.1(iv)]. Thus we may conclude that the Lebesgue measure p is the completion of the measure p which is restricted to the Borel sets in %#(b). From now on we denote the Lebesgue measure defined for the Lebesgue measurable sets, which include the Borel sets, by p also. We also call the Lebesgue measurable set, simply, measurable. The triplet (R', M ,p) is called a measure space. Finally we say that a certain property holds almost everywhere (a.e.) on a set E if there is a set S c E of measure zero, p ( S ) = 0, such that the property holds for all x E E - S. For example, f ( x ) = 0 a.e. on E if there is a set S c E with f ( x ) = 0 for all x E E - S and p(S) = 0. The following lemma states the translational invariance property of p. Lemma 1.2.4: L e t y E X = Rk and S E M ;then S p(S), where S y = {x y : x E S } .
+
1.2.2
+
+ y ~ M a n d p ( S+ y ) =
Integration
A real-valued function f defined in R' is called a measurable function (or just measurable) if for each real number a, the set {x :f ( x ) I a } is measurable. A complex-valued function f defined in IW' is measurable if its real and imaginary parts are measurable. Lemma 1.2.5: If a real-valued function f defined on a measurable set D is continuous, then f is measurable.
The above result is easy to see. Consider the set {x : f ( x ) I a } for any real number a. Suppose x p is a limit point of this set. Then, by definition, any neighborhood of x p contains a point of {x : f ( x ) I;a}, i.e., contains a point y such that f(y) I a. Since f ( x ) is continuous, it follows that f ( x p ) I a, i.e., x p E {x : f ( x ) Ia}. That is, {x : f ( x ) I a } contains all of its limit points, and therefore it is closed. By using the fact that the measurable sets contain all the open sets and their complements, the closed sets, we conclude that {x : f ( x ) Ia } is measurable. Ifa real-valued function f ( x ) may take on the values f m, we must also require that the sets {x : f ( x ) = + m} and { x : f ( x ) = - oo} be measurable for the measurability off. Iff, fi, and fiare measurable functions
1.2 Measure, Integration, and Fubini's Theorem
+
17
cfz for all c, on some measurable sets D, then f l f z , f 1 max(f, 0), f - = -min(f, 0) are also measurable on D. We define a simple function as any function of the form
Ifl,
f2,
f'
=
n
g(x) =
C ai X S , ( X ) ,
i= 1
for any finite n, for any finite real numbers a i , and for any pairwise disjoint measurable sets S , , . . . , S,. xsi(x) is called the characteristic function of the set Si and is defined by xs,(x) = 1 if x E Si and xs,(x) = 0 if x 4 Si. xsi is obviously measurable. Let S = S i ; then for x 4 S , g(x) = 0. If we want, aixsi(x)+ 0 . xsc(x),where we recall that S' is also we may write g ( x ) = a measurable set. The simple function g ( x ) is said to be integrable on a measurable set E if p ( E n { x : g(x) # 0)) < co. In this case the integral of g(x) on E is defined by
uy=
n
(1.2.9) I f x E { x : g ( x )#O},then wemay findan i ~ [ l..., , n] s u c h t h a t x E S i . T h e condition p ( E n { x : g ( x )# 0)) < 00 is equivalent to stating that g is integrable on E if p ( E n S,) < co,for the corresponding ai # 0, by using the non] ( E n Si) = E n { x : g(x) # O } . negative property of p and the fact that For those sets Si in (1.2.9)for which p(S,) = 0 we may effectively set the corresponding a, = 0. Now we consider the definition of the integral of any nonnegative measurable function f.Consider any sequence {R,} of increasing measurable sets6 R l c R z c ..., of finite measures p(R,) < 00 for n < 00, and such that R , = Rk.For a fixed n, we define the pairwise disjoint sets S, = R , n E n { x :n If ( x ) } ,
{ iinl
S,,, = R , n E n x : -
I f(x)<
,
i = 1,2, ..., n2",
and define the simple functions
where E is some measurable set. We note that the sequence {g,,} is nong,(x) = f ( x ) for all decreasing: 0 I g 1 I g 2 I . If such that
'
One particular example of this family of open (and hence measurable) sets is { R " } ,with R, = {x : 1x1 < n}. n = I , 2. . . . . We shall not, however, make use of any particular realization of
such sets.
I
18
Basic Analysis
x E E. It is easy to see that the g,(x) are integrable on E. To this end note that { X : g n ( X ) > 0) c sn,i u S, and that p(sn, i ) I p(&) < a,p(Sn) < p(Rn)
Ui
< co. We then define the integral off on E by
( 1.2.10)
and if the expression on the right-hand side is finite, then we say that f is integrable on E. An important consequence of the definition of the integral of f is that its value, if it exists, does not depend on the particular sequence of simple functions. More precisely, if {h,} is a nondecreasing sequence of simple functions: 0 I hl I h2 I -< f such that hn(x) = f ( x ) for all X E E, hn(x)integrable on E, and that limn+m h, d p < co,then (1.2.1 1) n-m
We say that a real-valued measurable function is integrable on a measurable set E if its positive f and negative parts f - are integrable on E, and we define +
S,fdP=Sf+dl-p-dl.
(1.2.12)
E
E
Similarly we say that complex-valued function h is integrable on a measurable set E if its real part Re h and imaginary part Im h are integrable on E, and we define
:j dp
=
k R e h) d p
+i
The following results are of importance.
s,
(Im h) dp.
(1.2.13)
Lemma 1.2.6: Iff is a (Lebesgue) measurable function, then there exists a Bore1 measurablefunction g such that f = g a.e. Lemma 1.2.7: Let E be a measurable set. Let f,f l ,and f i below be complexvalued measurablefunctions.
uy=
(i) If E = E j , where El, . . ., En are pairwise disjoint measurable sets, and f is integrable on each of the Ei,then f is integrable on E and c
(ii) For any complex numbers al and a 2 , al f i + a2f2is integrable on E if fland f2 are, and
1.2 Measure, Integration, and Fubini’s Theorem
19
(iii) Let S be any subset of E such that p(S) = 0 (therefore S is also measurable); then isf is integrable on E,
JEf
dP =
J
EnSc
f dP*
This, in particular, means, according to Lemma 1.2.3, that if we write E = B u Eo as a disjoint union, where B is a Borel set and E , (cE ) is a subset of some Borel set with p(EO)= 0, then
(iv) If fl = f2 a.e. on E and fiis integrable on E, then fl is integrable on E
and
(v) Let If 1 5 g a.e. on E, where g is integrable on E ; then f and integrable on E and
I f I are
uYzl
(vi) Let E = D,,where the D j are measurable but not necessarily pairwise disjoint; then if] f 1 is integrable on each D j , 1 f I is integrable on E and
Theorem 1.2.2 : (i) Let {f.} be a nondecreasing sequence of nonnegatioe functions: 0 I fl 5 fi I- . such that limn-ta f.(x) = f (x) a.e. on a measurable set E. If the fn are integrable on E and limn+m fn d p < CQ, then f is integrable on E and
lE
JJ
dp =
lim n-tm
J
E
fn
dp.
(ii) Suppose {gn}is a sequence of complex measurablefunctions such that limn-ta g,(x) = g(x) a.e. on a measurable set E. If there is a nonnegative function f ( x ) integrable on a measurable set E such that Ig,(x)l I f (x) a.e. on E for all n, then g is integrable on E and
n-r m
20
1 Basic Analysis
(iii) Let { f,} be a sequence of nonnegative functions each integrable on a measurable set E. Z f f ( x ) = 1irnN+= I f n ( x )a.e. on E, and Lm! lEf. dp < oo,then
z=
r
m
r
n=l
E
Jfdp= 1 JL~P. E
Part (i) of the above theorem is known as the “Lebesgue monotone convergence theorem,” and part (ii) is known as the “ Lebesgue dominated convergence theorem.” Now we are ready to develop and prove a basic theorem of multiple integrals. 1.2.3
Fubini’s Theorem
We are interested in studying the relation between the measure spaces ( R k l ,MI, PI), (Rk2,M,,p2), and (Rk1+k2, M, p), where R k l + k= z Rkl x Rk2. Let ‘&u(8,),%u(&‘2),and%?u(&‘)bethe Bore1 setsinM,, M2,and M,respectively. 8 , ,8 , ,and &’ are constructed out of kl-, k2-, and ( k , + k,)-cells as defined in the beginning of this section. From Lemma 1.1.3(ii) we know that %‘,(a) = ‘&,( where 9), 9is an algebra, defined in Section 1.1, and the latter consists of all finite disjoint unions of the Cartesian products of sets from %‘,(&‘,) and W,(g2).That is, if F E 9, then F = F i , Fi n F, # 0 for i # j , and Fi = Ai X Bi, Ai E %,(a,) and Bi E %‘u(&‘2). Lemma 1.1.3(ii) also leads to the fact that iff is %‘,(S)-rneasurable, then f, is %‘,(&’,)-measurable and fy is ‘&,(&’,)-measurable, where f,(y) = f ( x , y ) for x fixed in Rkl and f y ( x )= f (x, y ) for y fixed in Rk2. In the following lemma we establish the connection between p ( A ) for any A E W&F) and the measures p1 and p,. To this end we receall that A, E ‘&,(b,) and A y € Wm(&‘,) [see Lemma 1.1.3(ii)].
u?=
u.“=
Let A E W u ( 9 ) . W e write Rkl = B! and Rk2= B: ,where the B! (and similarly the &)are disjoint boxes ofjinite p i (p2) measure. For every x E Rkl let
Lemma 1.2.8:
=:u
m x >= P,(Ax n &),
(1.2.14)
and for every y E RkZlet
440) = P 1 U Y n B!); then &’ is p,-measurable and & is p,-measurable and
(1.2.15)
(1.2.16)
1.2 Measure, Integration, and Fubini's Theorem
21
Consider any set of the form Q n B; x B';, where Q = C x D with C E go(&l) and D E W u ( 8 2 ) .Then for x E B; =
~2(Qx
IXQ
BY x B ~ ( x Y , ) &c,(y)
= XC n B y ( x )
S X D n /I&dp2(y) )
-
- XC n B?(x)p2(D n Similarly for y E BT,
p1(Qy n BY) = XDnB'&)pl(C
n BY)*
Therefore
I B / 4 QnXB';)
= pl(C n &)Pz(D n BY)
and
I,
pi(Qy n 4 )d ~ = 2 pi(C n Bl)p2(D n B';),
and the statements of the lemma are true for all sets Q of the form Q = C x D with C E WU(gl)and D E WU(g2). The latter implies that the lemma is true for all Q E % by the application of the definition of 9(see Section 1.1). We now introduce the following family of sets D"" consisting of all sets in We(%) n B; x B'; such that the statements of the lemma are true for such sets. The family D"" is obviously nonempty as we have just shown that 9 n B; x B'; c D"" and the lemma is true for all the sets in 9 n B; x B';. We first show that D"" is a monotone class in B; x BT. Let S , c S2 c . . , with the Si E D"". Then by definition of the family D"", JBtli
dpi=
IB:2i
dp29
) P1((Si)y)*Also (SJx c where 4 l i ( x ) = ~ 2 ( ( S i ) x )and 4 2 i ( ~ = from the analysis leading to (1.2.4),
)
(SiL .
lim ~ 2 ( ( S i ) x=) ~2 i+ m
(i:,
(1.2.17) (Si+l)x.
Then
(1.2.18)
Finally, the sequence { 4 1 iis} monotonically nondecreasing; therefore by the application of the Lebesgue monotone convergence theorem [Theorem 1.2.2(i)], we conclude that lim / 4 1 i d p l = Slim 4 1 i d p l . i+ m
i-m
(1.2.19)
22
1 Basic Analysis
By using (1.2.18), repeating the analysis for the sequence {&}, using (1.2.17), we arrive at
and finally,
uzl
i.e., S i ~ D " " For . S1 3 Sz 3 ..., with the SieDnm,we have (Si)x 3 ( S i +l)x and the analysis leading to (1.2.7) implies that lim p Z ( ( S i ) x ) = ~z i+w
n
(Sdx
(iI1
)
(1.2.21)
5
where we recall that pZ((Si)J < co since pZ((Si),) 5 pZ(BT)< co. Also, p z ( ( S i ) x ) 2 pz(ni"=l ( S i ) x ) - Thus by the application of the Lebesgue dominated convergence theorem [Theorem 1.2.2(ii)], we have that (1.2.19) is again true, and hence finally arrive at the )~) conclusion that (7g1Si€Dnmby repeating the analysis for P ~ ( ( S ~and upon using (1.2.17). We have thus shown that D"" is a monotone class in BY x BY containing the family 9 n B; x BT, i.e.,
pZ((Si)x) 2 pz((Si+ 1)x) and, in particular,
Dnm3 @,($
n B; x BT)
Dnm3 %',, n( B;9) x&
or
[see (i) and (ii) of Lemma 1.1.23. Hence for any set Q E W B , ( 9 )the statements of the lemma are true, i.e., for any Q E W,,($),
where Q"" = Q n B; x BT. In particular, we note that if Q1, Q2, . . . are pairwise disjoint sets in D"", and hence u Z l Pi= then if we let P, = Ui= Qi, we have PI c Pz c Qi E D"". Now we are ready to define a measure for any Q E Wu(&) by the expression
uz
p(Q) =
1 p(Q n B!
Q"" = Q n B; x BT, (1.2.23)
x BY),
n.m=l
where P(Q"") =
J PZ(Q, n 4')PI
=
JB/l(Q,
n 4 )dpz.
(1.2.24)
BP
It is readily verified that (1.2.23) defines a measure, satisfying, in particular, the a-additivity property of a measure. We note that p(Q) is an extended realvalued set function (i.e., the value + co for it is permissible). From the uniqueness theorem (Theorem 1.2.1), then, the measure in (1.2.23) coincides with the measure p in (Rkr+k2,M ,p ) when restricted to the Bore1 sets, and the
I .2 Measure, Integration, and Fubini's Theorem
23
measure p in (Rk1+k2,M , p), more generally, is the completion of the measure defined in (1.2.23). Now we prove the following lemma. Lemma 1.2.9: f is M-measurable and f = 0 a.e. with respect to the measure p in (Rk1+k2, M , p), then for almost all x E Rkl,f, = 0 a.e. with respect to the measure p 2 . Similarly, for almost all y E Rk2,fy = 0 a.e. with respect to
the measure pl.
Let E = {(x, y ) : f ( x , y ) # 0). Since E E M , it follows from Lemma 1.2.2 that we may find a cover S 3 E, S E W U ( 9 ) ,with p ( E ) = p(S) = 0.' Lemma 1.2.8 implies that
or P(Snm)=
f P2(SF)
dCll
= 0,
B?
written in terms of the boxes B;, BT, i.e., A" = A n B;, if A E Wu(d',) and C"" = C n 4 x BT if C E Wu(.F), etc. Let N" be the set of all points x E 4 for which p 2 ( S 3 # 0. Then ASnm)= fN!2(SF)
dPl = 0,
i.e., pl(N") = 0. For X E & - N", p2(S!3 = 0, by definition. Now SF 3 EF, and thus for x E B; - N", p2(E!3 = 0 and EF E M 2 since the latter is complete. Therefore, for x E 4 - N", f&) = 0 for y E & - EY. Let N = N" and Ex = EF; then for x $ N,f,(y) = 0 for y .$ E x , which is the statement of the lemma since p l ( N ) = p l ( N " ) = 0, and for x # N , p2(Ex) = p2(E!3 = 0. A similar statement holds for fy(x).
u,"=
E=l
Theorem 1.2.3
(Fubini's theorem) :
Iff is integrable on Rk1+k2,then
(1.2.25) are dejned a.e. with respect to the measures p1 and p 2 , respectively, and
(1.2.26)
' Here we write ji(E) in order not to confuse the Lebesgue measure with the Bore1 measure.
24
1 Basic Analysis
Let f be M-measurable; then according to Lemma 1.2.6 there exists a %,($)-measurable function g such that f = g a.e. Then by an application of Lemma 1.2.9, fx = gx a.e. with respect to the p2-measure for almost all x and fy = gy a.e. with respect to the pl-measure for almost ally. Suppose Q E M is of finite p-measure. Let f = xQ; then according to Lemma 1.2.2 there exists a Bore1 set S 3 Q such that
F(Q)
= P(S) =
=
Rkl+k2
t
1
2
IBTpI(SYn B1) dP2 < G o -
n,m=l
=
d~
X S " ~
n,m=l
n,m=l
P2(Sx
BY
Applying Theorem 1.2.2(iii) to the series
F(Q)
=
=
n B?)dPl
c&,=x ~
1
R*i
xs
="
d ~ .
(1.2.27)
xs~ gives (1.2.28)
+kz
Again applying Theorem 1.2.2(iii) to the (absolutely) convergent series Lym=l X B p p 2 ( S x n B?) = p 2 ( S x )< co a.e. with respect to pl, r
(1.2.29) Similarly,
iXQ) = P(S) = lWk;l(Sy) d ~ 2 .
(1.2.30)
From (1.2.28)-(1.2.30) we see that the theorem is true for all f = xQ, where j' = g a.e. with respect to p and g = xs and p ( Q ) = p ( S ) < 00. This in turn implies that the theorem is true for all simple functions. Quite generally, suppose f is nonnegative and let {s"} be a nondecreasing sequence of nonnegative simple functions such that limn+ms" = f a.e. with respect to p. Then we have just shown that
s
(1.2.31)
Rki + k z
where
(1.2.32) (1.2.33)
1.2 Measure, Integration, and Fubini's Theorem
25
By applying the monotone convergence theorem [Theorem 1.2.2(i)] to {s:}, {t,Vi}, {s;}, {$;}, {s"}, we conclude that the statement of the theorem is true for all nonnegative functions. The general result then follows by applying the above to the positive and negative parts of the real and imaginary parts of an integrable function f. Theorem 1.2.4
(Fubini-Tonelli's theorem) :
Let f be a complex M -
measurablefunction; if
(1.2.34) then
As in the proof of Theorem 1.2.3, the above theorem is true i f f = xQ with < 00, or in general iff is a simple function. I f j is nonnegative, then let { f"} be a nondecreasing sequence of nonnegative simple functions such that f" = f a.e. with respect to p. Then
p(Q)
By applying the monotone convergence theorem [Theorem 1.2.2(i)] to the sequences if"), {SWkI f;dpl}, {f;),{ S W z f: dp21r{f3, we obtain the stated result in (1.2.35) if limndmS R k I + k 2 f"dp (= I R k l + k z f d p ) < 00. Now consider, in general, a complex M-measurable function f such that (1.2.34)is true. Since I f I isan M-measurable real-valued nonnegative function, we have just shown that
(1.2.37) From the estimate on the left-hand side of Lemma 1.2.7(v), we have that f is integrable. The stated result in the theorem then follows from Fubini's theorem (Theorem 1.2.3). The importance of Theorem 1.2.4 cannot be overemphasized.*
* An elementary example where the interchange of the orders of integration give different results(!) is the following:
where /(x. y)
= ( y 2 - x2)/(x2
+ y2)?.Many other examples may be also constructed
26 1.3
1 Basic Analysis GEOMETRY IN Rk
We consider subspaces of Rk.We recall that a subspace S of Rkis a nonempty subset of Rksuch that if x and x’ are in S, then ax + x’ is also in S for all real a. If we choose any vectors x,, . . . , x, in Rk,then the set of all linear combinations of these vectors is, by definition, a subspace of Rk called the subspace generated by x l , . . . , x,. The dimension of a subspace S, written dim S, is the maximum number of linearly independent vectors that can be found in S. Suppose the dimension of a subspace S is r. Let {x,, . . . , x,} be any basis of S. Each vector x E S may be then written as x = El= aixi, where the reals ai are uniquely determined. With respect to the basis {xi,. . . ,x,}, we may put each vector X E S in one-to-one correspondence with r-tuple ( a l , . . . , a,) of real numbers, and we denote this correspondence by x 5 (q, . . . ,a,). This correspondence preserves all linear relations if we define for b(bl, . . . ,b,) = @al bbl, . . . ,aa, + bb,) all reals a and b:a(al, . ..,a,) since ax by = E;=,(aai + bbi)xi, where y = El=,bixi. We say that S is isomorphic to R‘, with the latter consisting of r-tuples of real numbers, as there is a correspondence 6 between S and the set of all r-tuples of real numbers that preserves all linear relations. We say that a space I is the direct sum of its subspaces I , and I,, written I = I , 0 Iz, if every vector x E I may be written uniquely as x = x1 + x,, with x, E I, and xz E Iz. A direct consequence of this definition is that the subspacesI and 1, are disjoint, i.e., they have only the zero vector in common. Conversely, if we have three subspaces I , I,, l 2 such that every vector x E I may be written as x = x1 + x2,with x, E 11,x, E I, and the subspaces I , , I, disjoint, then I = Il @ 1 2 . It is readily checked that if I = I, @ 1 2 , then dim I = dim Il + dim I,. Also, since with each vector x E I we may assign a unique vector x1 in I , (and similarly a vector x2 in 12), we may introduce a linear mapping A(1,) that takes every vector x E I into its unique counterpart in 11,i.e., A(1,)x = x,. The latter is obviously a linear mapping; it is called the projection on I, along 1,. Given a space I and a subspace S of I , there is no unique way of choosing a complement of S in I. That is, one may write I = S @ I’ = S @ I”, where I’ and I” may be different. A particular example of a complement of a subspace S of I is the so-called orthogonal complement. Let the standard components of a vector x E Rk be denoted (xl, . . . ,xk) (i.e., the components of x with respect to the standard basis of Wk).For any vectors x, x’ E Rk,we introduce the quantity (inner product) (xlx’) = xix:. Then the orthogonal complement of S c I in I is defined by
,
+
+
+
cf=l
S1 = {xEI:(xIx’) = Oforallx‘~S},
1.3
Geometry in Rk
27
and we may write I = S Q S*. For example, suppose that I c R4and that I is spanned by the vectors (1, 1,0, I), (1, 1, 1, l), and (0, 1,0, 1). Let S c I be a subspace spanned by the vector (0, 1, 1, 1); then the subspace S1 spanned by vectors (0, I, -2, 1) and (1, - 1,2, - 1) is the orthogonal complement of S in I. Let Rk = I Q E, where I is a subspace of Rk and E a complement of I in Rk. We sometimes write I = I, Q 1, and Rk = I , @ E,. Then ifS’ c Rk,we note that A(1,)S’ c E, and A(1)S‘ c E. Lemma 1.3.1 : Suppose that S c E and hence, in particular, that S and I are disjoint. If A(1)S’ = S , then S’ c I 0 S, and dim S I dim S’ I dim I + dim S. A(1)S’ = S means that for any vector x‘ E S’, A(1)x’ E S. Any vector x’ E S’ maybewrittenx’ = x, + x2,withxl ~ I a n d x , ~cSE,i.e.,x’EI Q &which implies that S’ c I Q S. Let {XI,.. .,xr} be a basis of S. For each xi E {xl, . . . , xr} we may introduce a vector xi E S’ such that xi = xi + y i , with y i E I . Consider the expression six; = 0, or A ( I ) xi= six! = xi= a i x i = 0, which implies that a , = . . . = a, = 0. Accordingly the vectors xi, . . . ,x: are linearly independent vectors in S’. Hence dim S I dim S‘. Also, S’ c 1 0 S ; therefore dim S Idim S’ I dim S + dim I . Lemma 1.3.2: Suppose S c E. Then S“ c 1 Q S ifand only ifthere is some subspace S’ c S such rhat A(1)S” = S‘. Suppose there is some subspace S’ c S such that A(1)S” = S’; then we show that S” c 1 0 S. If y E S”, then y is of the form y = y , + y , , where y, E S’ and y , E I with A(1)y = y , , and hence y E I Q S, i.e., S” c I Q S. Conversely, suppose that S” c 1 0 S ; i.e., we may write y = y , + y 2 with y , E S and y, E I . Let {x,, . . . , x,} be a basis of S ” ; then xi = xi’ + xz, where ; aixi, = for some reals x,! E S and xz E I . Any x” E S” may be written x” = I aixi = qx!. We may then choose S’ to be a subspace ai, with A ( I ) generated by the vectors {x:, . . . , xf }. Since any vector x’ E S’ is of the form six,!, and x’ E S, it follows that S’ c S. x’ =
,
x:=,
,
,
,
Lemma 1.3.3 : Let x be a vector in I . Let J be a subspace generated b y the vector x; i.e., any vector in J is of the form ax for some real a. Let S c E. Then A(J)S‘ = S ifand only ifS’ is spanned by the vectors in {x,, . . . , x,, x} or b y the vectors in {x, + a,x, . . . ,x, + a,x}, where {x,, . . . ,x,} is a basis of S and a,, . . . ,a, are some real numbers. If S’ is spanned either by the vectors in {x,, . . . , x,, x} or by the vectors in {x, alx, . . . ,x, a,x}, for some reals ai, then clearly A(J)S’ = S. On the other hand, if A(J)S‘ = S, then from Lemma 1.3.1, dim S’ = dim S + 1, o r dim S’ = dim S, since dim J = 1. Any vector x’ E S’ is of the form x‘ = .;xi Px, where {x,, . . . , x,} is a basis of S and a;, . . . , a:, P are reals.
+
+
+
28
1 Basic Analysis
That is, x’ E S‘ is either a linear combination of the vectors x,, . . . ,x,, x or of the vectors x1 a,x, . . . , x, arx, for some reals a,, . . . , a,, corresponding to the two alternatives dim S’ = dim S + 1 or dim S’ = dim S , respectively.
+
+
Lemma 1.3.4: Let S c E and I = I , 0 I , . Then the condition on S through A(1,)S = S’ with S’ satisfying the condition A(Il)S’ = S is equivalent to the condition on S through A(ll 0 1,)s“ = S.
Any vector XI’ E S is of the form x“ = x’ + x, ,where x‘ E S’ and x2E I , . Ontheotherhand,x’mustbeoftheformx’ = x + x,,wherexESandxl E I , . That is, any vector X ” EisSof the form x” = x + x1 + x2 with X E S , , i=x,+x,. x , E I , , and x , ~ 1 , , or x ” = i + x , with 2 ~ 1 , @ 1 , and Thus if x” E S”, then A(1, 0 1 , ) ~ ”= x E S , i.e., A(Il @ 1,)s” c S. On the other hand, let x E S; then we may introduce a vector x’ E S‘ such that x’ = x + x,, with x, E I , . For such an x’ E S’ we may introduce a vector x” E S” such that x“ = x‘ + x,, with x, E I , , i.e., for any x E S we may introduce a vector x” = x + x, + x2 with x I E I , and x, E I , . That is, if x E S, then
x E A(I,)A(12)S” = A(11 0 I 2 ) S ” , and hence S c A(1, @ Iz)S”, which with the relation A(1, @ 1,)s’’ c S established above implies that A(I, 0 1,)s’’ = S. We shall usually use the notiation x for a vector x E Rk. If L,, . . . , L, are some vectors in Rkspanning a subspace S c Rk,we will often use the notation s = {L,, . . . , L,}. NOTES
Excellent references for Sections 1.1 and 1.2 are Rudin (1964), Royden (1963), Hoffman (1975), Munroe (1959), Berberian (1965), Halmos (1974a), and Rudin (1966); the latter two references are particularly recommended for the topics covered. For Section 1.3 we refer the reader to the classic paper by Weinberg (1960) and the work of Hoffman (1975) and Halmos (1974b).
Chapter 2 / CLASS Bn FUNCTIONS AND FEYNMAN INTEGRALS
Feynman integrands and Feynman integrals will be shown to belong to a special class of functions called the class B, of functions. This class of functions is defined in Section 2.1. In Section 2.2 we study the structure of a Feynman integrand and show that it belongs to this class. This latter property will be important when studying the convergence of Feynman integrals in the next chapter, where it is shown, in particular, that the absolutely integrable integrals as well belong to such a class. Consider two functions f and g of k real variables x,, . . . ,xk, If we can find real positive constants b,, . . . , bk: b, > 1, . . . ,bk > 1, and we can find a strictly positive constant C , independent of xI,. . . ,xk, such that for
1x11 2 b , ,
...
9
2 bk,
(2.1)
we have
then we denote the relation (2.2) symbolically by
If L,, . , ., L, (r 5 k ) are independent vectors spanning a subspace S c Pik, we use the notation S
= {L,, . . . ,L,}. 29
(2.4)
2 Class B, Functions and Feynman Integrals
30 2.1
DEFINITION OF CLASS B,, FUNCTIONS
A function f(P),with P E R", is said to belong to a class &(I) of funciions if for all choices of a nonzero subspace S c R", rn I n independent vectors L,, . . . , L, in R" and a bounded region W c R", such that I f(P)I # 00, there exist a pair of coefficients (real numbers) a(S) and P(S) with the latter [i.e., the coefficients P(S)] nonnegative integers and
S(L,I]~...I]~+L~~~...I],+...+L,I],+C)
where ql, . . . , I], are real and positive such that gl, . . ., I], -P 00 independently and C E W. The sums in (2.1.1) are over all nonnegative integers yl,. . . , y, such that k
1Yi 5 P({L,, i=
* * *
9
(2.1.2)
LJ)
1
for all 1 Ik I rn, and the coefficients P have been arranged in increasing order
LxJ I . * . 5 P({Ll, - * * L7JX (2.1.3) ...,K,,,} is a permutation of the integers in { 1, . . .,m}.As part of where {q, the definition of &(I) we also associate with it some subspace I c R" such B({L,,* *
a ,
9
that the condition If(P)I # 00, for some P E R", implies that I f(P + P)I # 00 for all finite P E I. For the zero subspace we shall use simply the notation B, for B,({O}) in the sequel. As discussed above, the conditions I],, . . . ,I], + 03 mean that there exist real numbers bl > 1, . . .,b, > 1 such that I ] ~2 b,, . . . , I], 2 b,. The constants b,, . . . , b,, and C > 0 as in (2.2), are independent of I ] ~ ,. . . , I],, but may depend on LI, . . . ,L, and W. The coefficientsa(S) and P(S) are, respectively, called power and logarithmic asymptotic coeficients. In the language of Feynman integrands, some fixed subspace I is chosen and is associated with the integration variables. We shall then conveniently introduce the orthogonal complement E of I in Rkand associate it with theso-calledexternalmomentaandthemassesofthe Feynman integrals. 2.2
STRUCTURE OF FEYNMAN INTEGRALS
We are interested in Feynman integrals, which are integrals of the following form and are absolutely convergent : (2.2.1)
2.2 Structure of Feynman Integrals
31
where K = (ky,.. . ,ki),
P = by,.. ., p:),
p = (p’,
. . . ,pp).
(2.2.2)
P denotes the so-called set of external momentum components of a graph, K the set of its internal momentum components (i.e., the integration variables), and p the set of its masses.’ The integrands 9 ( P , K,p, E ) are of the form
11= L
9(PYK , P, 6 ) = g(p,K , pyE )
fl [Q: + p: 1
- i&(Q:+ ,$)I,
> 0, (2.2.3)
where g ( P , K, p, E ) are polynomials in the variables in (2.2.2), in E, and, in general, may even be polynomials in the ( p i ) - as well. The role of the is factor will be discussed shortly. The Ql are of the form n
m
(2.2.4) Each pl coincides with one of the masses piin (2.2.2).It is assumed that pi > 0 for all i = 1, . . . ,p. In our metric Q: = Q: - Qg.Because of the i&factor with E > 0 (and pI > 0), the denominators [Q: + p: - ie(Q: + p:)] in (2.2.3) never vanish for all (Qp,Q,!, Q:, Q:) E R4. Using the relation
CQ: + d
- i4Q:
+ ~?)1-’ = CQ: + d + WQ: + ~ 3 1 x
[(Q:
+ p:)’ + E’(Q: + p;)’]-’,
(2.2.5)
we may rewrite (2.2.3) in the form
JV, K , p, E ) = Y1(P,K , p, E )
+ i9’(P9 K , p, -9,
(2.2.6) where Y1and 9’are real and they are given as the ratio of two polynomials having the same denominator (a polynomial), which never vanishes with the p’ > 0 ( E > 0). In particular, we note that since a polynomial is a continuous function, this implies that Y1,9’,and hence also 9, are continuous and thus measurable as functions of the variables in K with the pi > 0 ( E > 0). The role of the ic factor will become clear. To this end we consider the following expression: Ix(1 - is) - 1 I with x 2 0 and E > 0. As a function of x, (x(1 - i E ) - 1 I attains its minimum when x = (1 + E’)yielding to (1 - x)’ + &’x2 2 eZ(1 + E’)- l , i.e.,
’,
(2.2.7)
’
Details on subgraphs and external and internal variables will be given in Chapter 5. These details are not needed here. The significance of (2.2.1) as a multiple integral and its absolute convergence will be discussed in Chapter 3.
2 Class B,, Functions and Feynman Integrals
32
We also have xlx(1
-
- iE)
I
1I-l
(2.2.8)
(E)-',
and hence from (2.2.7) and (2.2.8) we may write
'
IX(l - i ~ )- 11- I
(X
+ l ) - ' ( l / ~+ Jm).(2.2.9)
On the other hand, Ix(l - iE) - 1 1 = J1
-
2x
+ x'(1 + E')
5 Jl
cJ
+ 2x + x2(1 + E') m ( x + 1). (2.2.10)
From (2.2.9) and (2.2.10)we may then write (x
+ l)-'(JiX?)-'
- 11-1 + l)-'(l/E + J T a p ) .
IIx(1 - iE) I(x
In particular, by choosing x = (Q' following useful lemma ( E > 0):
(2.2.11)
+ p Z ) / Q f we , obtain from (2.2.11) the
Lemma 2.2.1
(QZ + p')-'(J=)-'
+ p' - k(QZ+ p2)1-l I (QE + p')-'(l/~ + ,/-), IIQ'
(2.2.12)
where
Q;
= Q2
+ Qf,
(2.2.13)
and denotes the Euclidean counterpart of Q'. For convenience, thedenominators [Qk+ p: J will becalled the Euclidean counterparts of the denominators [Q: + p: - iE(Q: + p:)]. We define L
s ( P , K9P, E )
= g(P9K , w
E)
/,=
1
(Q?E
+
(2.2.14)
as obtained from(2.2.3)by replacing the denominators in 9by their Euclidean counterparts (Q:E + p:), as given in (2.2.13).From the inequalities in (2.2.12), we may then find constants Ge> 0, Ce > 0, depending only on E , with E > 0, such that E > 0. GeIQ(P, K,P, 6 ) ) II9(P, K , &)ISGelQ(p, K , p, &)I, ~9
(2.2.15)
If we define
2.2 Structure of Feynman Integrals
33
then we conclude from (2.2.15), with E > 0, that the absolute convergence of an integral ge in (2.2.16)implies the absolute convergence of the integral 9, in (2.2.1) and vice versa. The polynomials P(P, K , p, 0) are Lorentz covariant. On the other hand, the presence of the denominators, for example, in (2.2.3), clearly destroys the Lorentz covariant of S t ( P , p ) ; therefore we have finally to take the limit E -,+O to establish the Lorentz covariance of the Feynman integrals in (2.2.1). Finally, we wish to establish that, with E > 0, the Feynman integrands Y(P,K , p, E ) belong to class B4n+4m+p(l). From the right-hand side of the inequality (2.2.15) we conclude that it is sufficient to show that &P, K , p, E ) belong to class B4n+4m+p(l) to arrive at this conclusion. To this end we introduce a (4n + 4m + p)-vector P in a Euclidean space R4n+4m+p such that the 4n + 4m + p independent variables in (2.2.2) may be written as some linear combinations of the components (the standard coordinates) of P. We rewrite the polynomial 9 ( P , K , p, E ) in (2.2.14) [and (2.2.3)]
(2.2.17) where g(P,K , p, E ) is a polynomial in its arguments and the oi are some positive integers. The polynomial @(P, K , p, E ) is of the general form @(P, K , p, E ) = C A:,,,,,kSip“pUi,
(2.2.18)
i
and ,iff
(kyYb1 . . . (Q&,
p“
(pyybl
pw
. . . (p+,
(2.2.20)
.(p”>”s,
with Af,,,,, some suitable coefficients that may depend on E. The sum over i in (2.2.18)goes over a finite number of terms. Let L1, . . .,Lk be any k I4n + 4m + p independent vectors in R4n+4m+p, and consider the vector P having the form
P = Llq1qza’.qk + LzqZ*’*qk + ’*’
+ Lkqk + c,
(2.2.21)
2 Class 8, Functions and Feynman Integrals
34
where q , , .. . ,qk are real and positive and C is confined to a bounded region Win R4n+4m+p such that pi # 0 for all i = 1, . . .,p. We first prove the following important lemma. Lemma 2.2.2 : Consider an expression of the form
a.
+ alx;'
+ + a,(xIx2...x,)-',
i- a 2 ( x I x 2 ) - l
(2.2.22)
where x, > 0, . . .,x, > 0, the ai are real (noninjinite) with arbitrary signs, and a, # 0. Then we mayfind constants b,, . . . ,b,, m,, M , : bl
...,
> 1,
b, > 1,
(2.2.23)
M o 2 mo > 0,
(2.2.24)
such that for XI
2 bl,
...,
X,
2 b,,
(2.2.25)
we have
mo I t [ a o + a l x ; + . where t = sgn a, ,i.e., tao = I a. I.
+ a,(xl -.. xn)- ' 3 I M , ,
(2.2.26)
The proof is partly by induction. Consider the expression (2.2.27) f ( x 1 ) = a0 + alx;', where a. # 0. Let A, and c1 be any two numbers such that A, 2 la, 1, 0 < c1 < lao(,and Al/(laol - cl) > 1. Let bl = A l / ( l a o J- cl). Then
U ( x l )= Iaol
+ talx;'
2 IaOl - lalIx;' 2 laOl - A,x;'
2 c1
>O (2.2.28)
for x 1 2 bl. Now suppose as an induction hypothesis that for an expression
+
<[ao alx;'
+ ... + an-l(xl . . - x , - ~ ) - ~ ] ,
we can find constants bl > 1,. . ., b n - , > 1, and c , - ~ > 0, such that for 2 b,, . . . , x,- 1 2 b,- 1,
XI
+ +
+
a,-,(xl . . - x , , - ~ ) -1 ] 2 c , - ~ > 0. (2.2.29) <[ao alx;' Accordingly, we have the following chain of inequalities: <[ao a1x;' a,-,(xl . . - x , , - ~ ) - ' a,(xl x,)-l] 2 c , - ~ ta,(xl .-.x,)-' 2 c , - ~ - la,l(x,...x ,-,)- 1x;'
+
+
+ +
2 c , - ~ - la,J(b, > c,-, - la,Ix;'.
+
b,-,)-'x~' (2.2.30)
2.2 Structure of Feynman Integrals
35
Therefore we may use the chain of inequalities in (2.2.28), find two numbers A, and c, and that A, 2 Iu,,~, and 0 c c, c c , , - ~ ,and A,,/(c,-, - c,) > 1, and choose b, = A,/(c,- - c,), where we recall by the induction hypothesis that c,- > 0, to conclude from (2.2.30) quite generally that for x, 2 b , > 1, ..., x, 2 b, > 1, t[ao
+ a,x; + + a,(x, . * * *
x,)-
* *
'1 2 c,,
(2.2.31)
with c, > 0. This establishes the left-hand side of the inequality (2.2.26) by choosing mo = c,. On the other hand, for xi 2 bi > 1, i = 1,. . . ,n, where b,, . . ., b, have been introduced above, we may choose M o any finite constant such that
+ Iall(bl)-' + + la,l(bl . * * b , , ) - ' ,
Mo 2
(2.2.32)
which establishes the right-hand-side inequality in (2.2.26). This completes the proof of the lemma. The important thing to note in this lemma is that no matter what the values of the real finite constants a,, . . . ,a, are, we may always find constants bl > 1, . . . , b, > 1, with the conditions in (2.2.25) on x,, . . . , x,, such that the expression in (2.2.22) may be bounded as in (2.2.26). The 4n + 4m + p independent variables in (2.2.2) may be written as some linear combinations zi, i = 1,2,. . . ,(4n + 4m + p ) of the components of P and we note from (2.2.21), in particular, that these zimay, in general, depend on the parameters q r , . . . ,qk, i.e., zi = zi(ql,. . . , qk). We write
ky
= z,,
...,
k:
py = ~ 4 , + 1 ,
=~4,,
. . .,
P:
=
~4n+4m,
(2.2.33) p1 = z 4 n + 4 m +
pP = ~ 4 n + 4 m + p ,
* * * 7
1,
and hence we may write for the polynomial $(P, K , p, E ) in a compact and suitable notation, 4n + 4 m + p
g ( ~K , , P, E)
=
C A L ~ fl
(2.2.34)
(zjYi',
j= 1
i
where the mij are some nonnegative integers and the i are restricted over those for which ALl # 0. Finally, we introduce vectors V,, . . . ,V4n+4m+pin ~4n+4m+p such that
Vj.P
= zj,
j = 1, ..., 4n
+ 4m + p.
(2.2.35)
Now suppose that for j fixed,
Vj*L1 = 0,
...)
Vj *
Lru)- 1
= 0,
Vj * L,,
= cjr #
0. (2.2.36)
36
2 Class B, Functions and Feynman Integrals
Then we have
v' ' p = c j r q r ( j ) J
* * '
qk
+ '. + '
Cjkqk
-k
Cjr
Cjr
# 0,
(2.2.37)
where Vj * C = c j . The expression (2.2.37) may be rewritten = vr(j) *
vj '
* '
?k(Cjr
+
' ' '
+ cjk(qr(j)
' ' '
qk-
I)-'
+ cj(qr(j)
' * qk)-
'1,
(2.2.38) with cjr # 0. Therefore we may apply the inequalities (2.2.26) of Lemma 2.2.2 to conclude that we may find constants b,(j)> 1,. . . , b k U ) > 1, M#) > 0, rn#) > 0, such that for gr(j) 2 b r w , .. . ,qk 2 bkci,,
(2.2.39) m#'q,(j) * ' q k 5 tvj * P = I v, * PI 5 hf#'qr(j) . ' ' q k r where = sgn cjr. For 1 I s(j) c r(j) we introduce any finite fixed constants b,(,) > 1. With such a notation, we may apply the right-hand side of the inequality (2.2.39) to (2.2.34) with (2.2.35) to conclude, after summing over i in (2.2.34), that we may find a strictly positive constant C such that
<
I@(P,KcL,c)I I cq+-).Ip,
(2.2.40)
for I],2 b:
r = 1, . . ., k,
= max b,,j),
(2.2.4 1)
j
where j
E
[l, . . . ,4n
+ 4m + p ] . The exponent 6, is of the form (2.2.42)
where the index i was introduced in (2.2.34), and the summation in (2.2.42) is over all j E [l, . . . ,4n + 4m + p] for which
Vj.L, # 0
Vj.L, # 0
and/or
and/or and/or
-
Vj L, # 0.
(2.2.43)
The condition in (2.2.43) may be equivalently stated as requiring that Vj not be orthogonal to the subspace { L l , .. . , Lr}.Therefore we may write
6,
= d({L,,
* * *
9
Lr})
(2.2.44)
In obtaining the estimate (2.2.40) we have not paid attention to the signs ofthe coefficients Ah, in (2.2.24). Such an estimate for the numerator of a Feynman integrand will be sufficient in all our applications in this book, including the convergence proof of renormalization in Chapter 5. We shall denote tke estimated degree 6, of the numerator @ with respect to q, simply by degr,,r P.
2.2 Structure of Feynman Integrals
37
Now we consider the denominators in (2.2.14).To this end we may use the labeling in (2.2.33)and we may rewrite (2.2.4) in the form n
QP
=
1
m
+ 1b , j z 4 n + 4 ( j -
aljz4,j- I)+"+ 1
j= 1
l ) + n + 1.
(2.2.45)
j= 1
By writing au = a(/, j ) , b,, = b(1, j) and redefining the indices of the summations, we may rewrite (2.2.45): 4n-3+a
QP
=
1
i=a+ 1
a(,,
i-0-1
i-4n-a-1
4 n + 4 m - 3 +a i=4n+a+l
(2.2.46) Define
+ 1). c(l, i, a\ = 4n+a+l
I
+ 4m - 3 + a (2.2.47)
and let c(1, i. a) = 0 for i = 4n - 2 notations we may rewrite (2.2.46)as
+ a, 4n - 1 +
O,
4n
+ a. With
these
4n+ 4m - 3 +a
QP
1
=
c(l, i, a)zi.
(2.2.48)
i=a+ 1
For any 1 E [l, . . . , L] there exists from (2.2.33) an integer t ( l ) E [l, such that
.. . ,p]
4n+4m + p
p, =
1
=
S(i - 4n - 4m, t(l))zi,
(2.2.49)
i=4n+4m+l
where S(i,j) is the Kronecker delta: S(i, i ) = 1 and S ( i , j ) = 0 for i # j . Let A(a) = A'(a) = 1
1 0 -
if a E [ O , 1, 2, 31 if a = 4 ,
(2.2.50) (2.2.51)
A@),
and define Qf = p i . We may then combine (2.2.48)and (2.2.49)to write 4n+ 4m- 3 +a
QY
=
1
i=o+ 1
A(n)c(l, i, a)zi
+
4n+4m
+p - 4
1
+a
A'"(a)S(i - 4n - 4m, t(l))zi
i =4n+4m- 3 +a
(2.2.52)
2 Class Bn Functions and Feynman Integrals
38
for a = 0, 1, 2, 3,4. Finally, upon writing d(l, i, a) = c(l, i, a), d(l, i, 4) S(i - 4n - 4m, t(l)),we obtain the convenient expression
=
4n+4m+p-4+o
Qf
=
C
41, i, a)zi,
(2.2.53)
i=o+ 1
or by the understanding that we restrict in (2.2.53) only those i for which d(l, i, a) # 0, we write
-
d(l, i, a)zi = C d(l, i, a)Vi P = V(al) P,
Qf =
i
i
(2.2.54)
where V(al) = C d(l, i, a)Vi.
(2.2.55)
i
Suppose from (2.2.36)that
V(aO L,
= 0,
.. .,
V(al) Ls(a,)- = 0,
h(s(a0, a, 1) # 0,
(2.2.56)
h(s(al),a, l ) = c' d(l, i, a)cis,
(2.2.57)
v(a0
Ls(a,) =
where i
with the sum, in general, over some subset of the set of the i contributing in (2.2.54) and V(aO C = h(aZ).
(2.2.58)
Equations (2.2.54)-(2.2.58) then imply
+ (qs(al)
QP = qs(,,1) . .. qkCh(s(al),a, 0 +
4 k ) - 'h(al)I,
(2.2.59)
with h(s(al), a, I> # 0. Let M a O , a, 0 = sgn h(s(al), a, l ) ;
(2.2.60)
we may then apply the left-hand side of (2.2.26) of Lemma 2.2.2 to the coefficient of qs(al)- . V k within the square brackets in (2.2.59)to conclude that we may find constants &(a/), a, I> > 1 , . . . ,b(k, a, 0 > 1, mo(ar>> 0, such that for q s ( a l ) 2 b(s(aO, a, 0, . . . , t l k 2 b(k, a, 0, Hs(aO, a, OQ; 2
Vs(a1) ' ' *
qkmO(al),
(2.2.61)
or
( Q Y 2 V&) with rni(al) > 0.
* *
~tmi(al),
(2.2.62)
2.2 Structure of Feynman Integrals
39
Let J(I) be that subset ofelements in the set J = (0, 1 , 2 , 3 , 4 } such that for any j E J(0, s(jI) = min s(aI) = s(l).
(2.2.63)
acJ
For future reference, we write
JU)
= {jI(0,j2(O9 . .
.I.
(2.2.64)
For a E J - J(I) = JJI), we introduce any finite fixed constants b(s(aI) - 1, > 1 , . . ., @(I), a, l) > 1, and we define
a, l)
b(i, I ) = max b(i, a, I),
(2.2.65)
a€J
with s(l) I i I k. Then for with i = $I), s(I)
+ 1,. . . , k, we have from (2.2.62) (2.2.67)
where
&;(I)
=
C m&I)
> 0.
(2.2.68)
aeJ(1)
We may again apply the left-hand side of (2.2.26) now to the coefficient of q$, . . . qf within the square brackets in (2.2.67) to conclude that we may find constants b2(s(I),I ) > 1, . . . ,b2(k, I ) > 1, which may be so chosen that (b(i,01 2 b(i, I>, s(I) Ii Ik, and mi(I> > 0, such that for qz 2 b2(i,I),
i = s(I), s(I)
+ 1,. . .,k,
(2.2.69)
we have from (2.2.67) that Q~E
+d 2
&)
* * *
(2.2.70)
tlhi%O.
We introduce any finite fixed constants J b(i, 01 > 1 for 1 _< i < s(I). With this notation let b:' = maxIb(r, 01,
r = 1,
..., k, I
= 1 ,..., L.
(2.2.71)
1
Then from (2.2.70) and (2.2.71) we have for qr 2 b:', r = 1,.
. ., k,
L
n(Q& + p;)-' IC ' ~ " ' * * . ~ ~ y k ,
I=1
(2.2.72)
40
2
where C =
Class BnFunctions and Feynman Integrals
nk= (mi(l))- '. The exponents y, are of the form L
y, = 2 C Af),
(2.2.73)
I= 1
where Af)
=
1, if and only if, for the fixed I in question,
V(jl(I)l) L1 # 0 and/or . and/or V(j2(l)I)*L2# 0 V(j2(I)l) L, # 0
.
and/or and/or and/or
-
-
V(j,(l)I) L, # 0 and/or ..- , (2.2.74)
with jl(I), j2(l), . . . ,the elements in the set J(I), defined in (2.2.64),and depend on I, and Af) = 0 otherwise. The condition in (2.2.74) may be equivalently stated as having at least one of the vectors V(jl(l)l), V(j2(l)l), . . . not orthogonal to the subspace { L l , .. . , Lr}. These vectors are defined in terms of the vectors Vi in (2.2.55). If for a particular I , the corresponding expression (Qf,+ p f ) is independent of the parameters ql,.. . ,qk, then Aji) = 0 for i = 1 , . . . ,k. In such a case we may write the bound Qf, + pf 2 p: and take mi(/)= pf for the corresponding I in C'. From (2.2.74) we may also write Yr
= Y ( { L ~ ,. . ?
LrIl
(2.2.75)
+
y, coincides with the degree of (Q: pf), with respect to q r , and will be denoted by degrqP (Qf pi). We may repeat an analysis for the factor
+
n
n
4n+4mt o
D
-01 E
i=l
(Zj)-aj-4n-4m
j=4n+4m+ 1
in (2.2.17)in a similar way to the one leading to (2.2.70)and (2.2.72).We may then find constants b:" such that for w, # 0 [see (2.2.78)] by > 1, and for w, = 0 [see (2.2.78)] we introduce any finite fixed constants b:' > 1 , r = 1, . . . k, such that for q, 2 b:", (2.2.76)
n
4n+4m+p
< C q F U 1 . . . qk-Uk,
(zj)-01-4.-4m
(2.2.77)
j=4n+4m+ 1
with c" > 0, w, =
y)
fsj-4n-4mr
(2.2.78)
j
m=
and w, coincides with the degree of (pi)+atwith respect to the parameter q,. The sum in (2.2.78)is over all those j E [4n + 4m + 1, . . ., 4 n + 4m + p ] for which (2.2.43) is true. If (2.2.43) is not true, then w, = 0. Again we may write (2.2.79) o r = u(IL1, . * * Lr}). 9
2.2 Structure of Feynman Integrals
41
Finally, from (2.2.14), (2.2.17), (2.2.40), (2.2.44), (2.2.72), (2.2.75), (2.2.77), and (2.2.79), 1 $(P, K,p, 6 ) I 5 Coq:((LI)) . . . q$(L1. ...PLk)) (2.2.80) for
= max[b:,
(2.2.8 1) b:, &"I. The parameters b;, b:, by are defined, respectively, through (2.2.41), (2.2.71), and (2.2.76), and C, = CC'C" > 0, q, 2 b,
a(&,,
..
. 9
Lr)) = 6({L1,. -
9
Lr)) - y({L,,. - . Lr}) - u((L1,. . . ,L,)).' 9
(2.2.82) The latter is identified with degr,,, $(P, K,p, E ) and degrqrY ( P , K,p, E).' With the constraints p i > 0, i = 1, . . . ,p, one cannot consistently find vectors L,, . . . , L, and C in (2.2.21) to make any of the denominators in the expression for 9 in (2.2.14) [and any of the ones in (2.2.3) for E > 01 vanish identically and thus possibly obtain that 131= 00 (and 1 9 1 = 00). Also, if I is a 4n-dimensional subspace of R4n+4m+p associated with the integration variables in K, then Id(P)I # 00, with P ER4n+4m+p, implies that for any finite vector P E I, I$(P + P)(# 00 [and I9(P + P)I # 00, E > 01 for pi > 0,i = 1, . . . , p . Thus we have established that Q(P, K,p, E), in (2.2.14), and hence from K,p, E), in (2.2.3) for E > 0, belong to the class B4,,+3m+p(I) (2.2.15) also Y(P, with zero logarithmic asymptotic coefficient (/.? = 0). Application of this useful result will be made later in the book. In particular, we shall learn in the next chapter that the integrals 2Fc, if absolutely convergent, also belong to such a class with, in general, nonzero logarithmic asymptotic coefficients as well. Finally, by replacing the Minkowski metric grv by the Euclidean metric qpv,diag qpv = [l, 1, 1, 11, in the expression for 9 and setting E = 0 in the latter we obtain what we call the Euclidean version of a Feynman integrand Y E:
I,= L
yE(P, K,
= PdP, K 9
fl CQ& + PKI.
(2.2.83)
1
Again the same analysis as for d establishes the fact that YEalso belongs to class B4n+4m+p(I) with power asymptotic coefficients, with the q, 2 b, for some constants b, > 1, that may be identified with degr,, YE,and havingzero For any two polynomials Pl(x) and P2(x), we define degr,[Pl(x)/P2(x)] = degr, Pl(x) degr, PAX). Note that degr,,[Q: + p f - k(Q: p:)] = degr,,[Q& p f ] , with E > 0.
+
+
42
2 Class B, Functions and Feynman Integrals
logarithmic coefficients. The Euclidean version of a Feynman integral is (up to a multiplicative constant) given by 9E(p, p) =
IR4dK
K,
(2.2.84)
What should be particularly noted in this chapter is that no matter what the -values of the coefficients alj,blj in (2.2.4), we may always find constants b, > 1, r = 1, . . . ,k,and introduce parameters g,, r = 1, . . .,k, as in (2.2.21), with I], 2 b,, such that we may bound the absolute value of 3 (and similarly for 9,E > 0, and 9,)as in (2.2.80), with the coefficients a({L,,. . . , L,}) denoting estimated degrees of 3 with respect to the parameters g,. This basic result follows essentially from the key lemma, Lemma 2.2.2. The greater generality of the definition of class B4n+4m+p(I) involving logarithmic coefficients will be important when studying Feynman integrals. NOTES
The class B, of functions was introduced by Fink (1967,1968), as a subclass of functions introduced by Weinberg (1960), to deal not only with the power growth of Feynman integrals but also with their logarithmic growth. The iE factor in the denominators of Feynman integrands [see (2.2.3)] was first introduced by Zimmermann (1968), and the basic inequalities in Eq. (2.2.12)are due to him as well. The analysis in this chapter, and in particular Lemma 2.2.2, is based on Manoukian (1982b), and we have slightly modified the original definition of class B,. The unification of the masses with the momenta as components of a vector P was also considered by Slavnov (1974).
Chapter 3 / THE POWER-COUNTING THEOREM AND MORE
In this chapter we discuss and prove a powerful theorem in quantum field theory that will be used frequently throughout the book. The theorem gives a convergence criterion for Feynman integrals and establishes that the integrals themselves belong to class B,-functions. It also yields the expression for the power and the logarithmic asymptotic coefficients of the integrals in terms of the corresponding ones of the Feynman integrands.
3.1
STATEMENT OF THE THEOREM
Let f(P) be a class &(I)-function with the latter as defined in Section 2.1, with P E R". Let I be a k-dimensional subspace of R". Suppose L,, L2, . . . ,L, constitute a set of orthonormal vectors spanning the subspace I . We consider iterated integrals of the form
1
03
fL,
...L*(P)=
-m
m
4 9 J-,dy*
m * * *
S__dYkf(P + b y ,
+ ... + L,y,). (3.1.1)
The Fubini-Tonelli theorem (Theorem 1.2.4) then states, in particular, that theabsoluteintegrabilityofameasurablefunctionf(P + L,y, + . . . + Lkyk) in any order, in particular, in the order corresponding to the integrations over y,, . . . , y , as shown in (3.1.1), means that the value of the integral is 43
3
44
The Power-Counting Theorem and More
independent of the particular vectors L , , . . .,Lk and depends only on the subspace I. That is, in an obvious notation, we have (uniquely) f L 1 ...Lk(p)=
Im Im dyl
-m
=
W
dyZ
dykf(P
* * '
-m
d k P f(P
+ Lly1 +
+ P ) = fi(Pi.
. * *
+ Lkyk) (3.1.2)
Furthermore, by the application of Lemma 1.2.4, we conclude that if we translate the vector P in f,(P) by a vector P E I , then the value of the integral h(P) does not change: fi(P + P ) = f,(P). The latter means that fi(P) depends only on the projection of P along the subspace I . For concreteness and in view of applications, we introduce a complement of I in R", which we may conveniently take to be the orthogonal complement of 1 in R", and we denote it by E: R" = I @ E and restrict the vector PEE. With f taken to be a renormalized Feynman interand (Section 5.2) and with the absolute convergence of fi (Theorem 3.1.1 and Section 5.3) established, we may then rigorously justify the usual computations of renormalized Feynman amplitudes through iterated integrals of the form (3.1.1). We may identify the components of the independent external momenta and the masses associated with a (proper and connected) graph G,' in general, as some linear combinations of the components of the vector P E E. I will be the subspace associated with the class &(I). We suppose that f in (3.1.1) is locally (i.e., over any bounded region in I) absolutely integrable. We denote the power and logarithmic asymptotic coefficients off by a(S) and fl(S), respectively, with
s c R".
Theorem 3.1.1 : For all Jinite for f(P P),P'E 1,'
+
P such that A(1)P = P and If(P)I # co, if
[A1 DI = max[a(S)
+ dim S'] < 0,
(3.1.3)
S'C I
for all nonzero subspaces S' c I , then f,(P) is absolutely convergent. If [A] is true, then [B] fi(P) E Bn-k with power asymptotic coefficients aI(S), S c E, gioen by q ( S ) = max [a(S')
+ dim S' - dim S].
(3.1.4)
A(I)S'=S
' Definitions of internal and external momenta and proper and connccted graphs will be given in Chapter 5. These definitions will not be needed in this chapter. Recall by definition ofclass E,(I) that If(P + P)l # rn for all finite P E I. Also note that for finitc P , l f ( P ) l # zif and only if If(A(I)P)(# m.
'
3.1 Statement of the Theorem
45
let A be the set of all the maximizing subspaces S’ in (3.1.4), i.e., for any S’ E A, a,(S) = cr(S’)
+ dim S’ - dim S,
(3.1.5)
then A is afinite set. A subspace S’ E A will be called a maximizing subspace for the I integration relative to the subspace S. [C] The logarithmic asymptotic coeficients of fi are given by k
PAS> = max P(S) + S’E ”4f
1 Pi,
(3.1.6)
i=1
where A is the set of all maximizing subspace as defined in [B]. The parameters p i , for i = 1, . . . ,k, are called the dimension numbers and may take on the values of 0 or 1 . They are defined inductively as follows. Let I , , 1 2 , . . . , 1, be one-dimensional subspaces of I such that I = I , 0 l 2 Q . . . 0 Ik. Then ifall the maximizing subspacesfor the I , integration relative to S c E, after performing the I 2 Q . . . 0 l k integration, have the same dimension, then p , = 0, otherwise p , = 1. A maximizing subspace S‘ for the 1 , integration relative to S, after performing the l2 0 . . 0 I k integration is dejined by
-
a I ( S ) = ay,$ ... eIk(S’)
+ dim S’ - dim S,
(3.1.7)
where a,,$ ...e,k(S’) is defined similarly to a,(S) in (3.1.4) with A ( I ) replaced by A(I2 0 .. . 0 Ik). Similarly, if‘ all the maximizing subspaces for the I j integration, after performing the I j + 0 . 0 I k integration, relative to any one of the maximizing subspaces for the I j - integration, after performing the I j 0 . . . 0 I k integration, have the same dimension, then p j = 0; otherwise p J. = 1. The dimension numbers p j , j = 1 , . . . , k, may be determined with respect to any decomposition of I into one-dimensional subspaces. That is, if I = I, 0 . . . 0 I , and I = 1; 0 . . . Q r k , dim I i = dim 1: = 1, with the li and I:, in general, diflerent, then
,
k
,
k
1 Pi = 1 P:,
i=1
(3.1.8)
i= 1
where the pi correspond to the dimension numbers of the decomposition: I = 1; 0 . 0 I ; . In the light of Feynman integrands [i.e., YEin (2.2.83), 3 in (2.2.14), and with E > 0 for 9 in (2.2.3)], the condition If(P)I # 00 is automatically satisjied as long as P is restricted so that p i # 0 ,for all i = 1, . . . , p , and the theorem is then immediately applicable.
This completes the statement of the theorem. We first prove the theorem in one dimension, i.e., when dim I = 1 ; we then generalize the proof to an arbitrary finite number of dimensions by induction.
3 The Power-Counting Theorem and More
46 3.2
PROOF OF THE THEOREM IN ONE DIMENSION
In this section we prove Theorem 3.1.1 in one dimension for I , i.e., for the integral (3.2.1) when I = {L}, dim I = 1, where If(P)I # 00. The absolute convergence criterion [A] of the theorem is easy to prove. We recall that f is assumed to be locally absolutely integrable. Let P be confined to a finite region Win R".By definition of class &(I) (Section 2.1), we may find a constant b > 1 such that for Iyl 2 b (3.2.2) y=o
for some finite constant M > 0. Obviously, if
+ 1 < 0,
a(&))
(3.2.3)
then the integral in (3.2.1) is absolutely convergent. As the only nonnull subspace of I is I itself, we may write for (3.2.3)
DI = LY({L})+ 1 < 0,
(3.2.4)
which is the statement of the criterion [A] in the theorem. Now consider the vector P + Ly in the form
P + Ly = L l ~ 1 ~ 2qm *
a
*
+ Lzqz
a
*
*
q,
+ + Lmvm+ Ly + C,
(3.2.5)
where C is confined to a finite region in R", and q i , i = 1,. . . ,m,0 I rn -< n - 1, are real and positive. By definition of class B,,(I), given the subspace I associated with B,,(I),the condition 1 f(P)I # co implies 1 f(P + Ly) I # co. The basic idea of the proof of the remaining part of the theorem is as follows. We replace the interval (- co, 00) of integration over y by a finite number of subintervals over each of which the class &(I)-property o f f may be applied and the integrations over the bound of I f 1 on each subinterval may be then explicitly carried out. Let Y = z ~ 1 ~ 2V m ;
(3.2.6)
then we may find constants bo > 1, bl > 1, . . ., b, > 1, such that with
3.2 Proofof the Theorem in One Dimension
47
fsatisfies an inequality as in (2.1.1) with asymptotic coefficients
a({L)),
a(& Ll I),
. . .,
d{L, L,, * . ., L,,,}),
P({L}),
P({L Lll)?
.
K{L, L1,
*,
*.
9
(3.2.8)
L,,,}),
and where
P + L y = k L l Z 1 q 1 q 2 * * * q mL1q1q2***qm + + . . . + L,,,q,,,+C. (3.2.9) Accordingly, we introduce the following pair of intervals to which y, as defined in (3.2.6), and with the restrictions in (3.2.7), may belong: J * = { y : y = Zq1 . . . q m , IZI = + Z 2 bo}.
(3.2.10)
Now we consider the region with l y l 5 bOqlq2 ... q,,,. To this end we may use the Heine-Bore1 theorem (Section 1.1) to cover the closed and bounded set [-b,, b,] by a finite number of open intervals ( V i , ,Iil, U i , - Ail), where [ Ui, [ 5 bo and we determine the Ail consistently. The index il runs over a finite number of set of integers (Section 1.1). We write (3.2.11) Y = Ui1~1~12 * . . q m + Z V * ~* * q m ,
+
and hence P
P
+ Ly may be written in the form
+ LY = (L1 + Vi,L)('11/IzI)IZlrt2 . . * ~ ] mf LIZlrl,
+ L2q2 " ' q m + . . * + Lmqm+ c.
Vm
(3.2.12)
Thus we may find constants bo(il) > 1, bl(il) > 1, . . .,b,,,(il) > 1 such that for
(3.2.13)
f(P
+ Lq) satisfies an inequality as in (2.1.1) with asymptotic coefficients (3.2.14)
+
for j = 1 , . . . ,rn, where we have used the notation a(il) = a((Ll U i , L } ) and p(il) = p({L, + Ui,L)). From the conditions (3.2.13) we may take l i ,such that 0 < Ail < b;l(il). (3.2.15) Hence we may introduce another pair of intervals:
J t = { y : y = Ui,qlq2...q,,, + Z q 2 . . . q m r b o ( i lI ) IZI = + Z I q l l i , } , (3.2.16) where Ail satisfies (3.2.15) and I U i ,I I b,.
The Power-Counting Theorem and More
3
48
We may again consider the closed and bounded set [ -bo(il), bo(il)] and cover it by a finite number of open intervals, as done above, and thus continuing in this manner we may generate the following pairs of intervals: J:
,..., ir = { Y : Y =
+ ... + uil ,...,i , V r * * . V m + Z V , +*~* * ~ r n ,
ui1Vl * * * ~ m
b O ( i l , * * . ,ir)
(3.2.17)
IIZI = f Z I V r 4 1 , . . . , i r } ,
for 1 Ir I m,where
I ui,,....i,I
I bo(i19
* * *
7
0 < A i l ,..., i, < b i l ( i l , . . . , ir), (3.2.18)
ir),
with the bo(il,. . . , ir) > 1 and b r ( i l , ..., ir) > 1 some suitable constants. and Finally, we consider the interval [ -bo(il, . . ., i,), bo(il, . . . , i,)] introduce the following pair of intervals: Of Ji1
,..., i, =
{ v :Y = UilVl . . . V m + * * + ui1....,i,Vm 0 I)Z1 = + Z I bo(il,. . . , i,)}. 1
Thus we have shown that y intervals:
+ Z, (3.2.19)
may fall in at least one of the following
E (w'
J f = { Y : Y = Z V ~ . . . V , , I Z I) Z = >b,},
+ ui1,..., i r V r * * . V r n + Z V r + 1 r = 1, . . . ,rn, x bo(il, . . . , ir) I IZI = + Z I v r A i l , ..., i,}, Of Ji, ,....i, = {Y : Y = uilV1 tlm + + ui1,...,i,Vm + Z, (3.2.20) x 0 I JZI= & Z Ibo(il, . . . , i,,,)}, Jif,....i,
=
{ Y : Y = uirq1
*..rtrn
*
*
a
+
* * * ~ r n r
m
.
1
where the indices i l , . . . , i, run over a finite number of integers. We consider the integration of I f 1 over y in each of the intervals in (3.2.20). We may then use the bound F
rn
F
(3.2.21) where the indices i l , . . . , irnrun over afinite number of integers, to complete the proof of the remaining part of the theorem in one dimension. To this end note that the argument P + Ly o f f corresponding to the terms in (3.2.21)is of the general form (L1 + ~ ~ L ) v ~ . . . v ... ~ v~ l o rnv + . .~. ++ ( LI, + U r L ) V r V o V r + l . . . V m m . * * + L m V m + C, (3.2-22) f LlIOVr+l " * ~ + m Lr+lVr+l . * ~ +
3.2 Proof of the Theorem in One Dimension
49
where u l , u 2 , . . . , ur are real numbers and take on the finite set of values U i l ,. . . , Uil,...,i,, where 1 I r < m. Also note that L, u,L, . . . ) Lr urL, L, L,+l, . . . , L, (3.2.23) are (m + 1) independent vectors. Without loss of generality and for simplicity of notation we assume that we have the following ordering of the logarithmic asymptotic coefficients corresponding to a typical term in (3.2.21):
+
+
+
P({L, <+UlL)) I. .. IP({L, + UlL, . . . , Lr urL}) I;P({L, L1,. . . , L,}) (3.2.24) I* * IP({L, L,, . . . , Lm}). Now we consider the contribution of each interval on the right-hand side of (3.2.21). [I] Let y c J * . The vector P + Ly is then of the form given in (3.2.9). Thus we may find constants br > 1, r = 1, .. ., m, M > 0, such that for V r 2 br,
s,*
m
dy I ~ ( + P L ~ )=I
dlzt If(kLIZlV1
a
*
*
Vrn
bo
+
+ LmVm + C ) l ~ *l* * t l m < M~;(”L.LI)) . . . V a((L.LI. ....Lml) m x C (In q l ) y l . (In qrn)Ym,
(3.2.25)
YI.....YnI
where the sum is over all nonnegative integers yl, . . . , ym such that k
yi
< fi({L, L1, . . .,Lk}),
1 Ik
(3.2.26)
m.
i= 1
In writing the expression on the extreme right-hand side of (3.2.25) we have used the fact that the integral
[1 z I”((L))(lnIz I
170
d Iz I
(3.2.27)
is, with the hypothesis a({L}) + 1 < 0, a finite constant which we have absorbed in the constant M .
[I13 Let y E Jif, ...,i,, 1 Ir Im. The corresponding P + Ly may be written in the form P + LY = (LI + Ui1L)Vl . . . ~ r - l ( ~ r / l Z l ) l Z l ~.r *+*l ~
m
+ . . * + (Lr + Uil ,..., i~L)(Vr/lZl)lZlVr+l* * * ~ m k LIZIVr+l . . * V m + Lr+lVr+l * * ‘ V m + + LmVm + C. . * *
(3.2.28)
50
3
The Power-Counting Theorem and More
Accordingly, with q, 2 bf(il,. .,, i,) > 1
( I # r),
. . . , i,) > 1, IZI 2 bo(il,. . . , i,) > 1, M(il, . . . , i,) > 0, qr/lZl 2 b,(il,
(3.2.29)
0 < A i l , ...,i p < K 1 ( i 1 ,. . ., i,)
[see definition of Jif,...,i, in (3.2.17)], we may write r
i= 1
yi I p(il,. . . , ik),
1I k
-< r, (3.2.3 1)
k
k
2 Yi -k yo + 2
i= 1
yi i=r+ 1
I p({L, L1,. ..,Lk}),
I
+ 1 I k I m.
In evaluating the integral on the extreme right-hand side of the inequality (3.2.30), three possibilities arise. To see this let 1 > 1 > 0, b > 1, and fl, j'nonnegative integers; then for q -P 00 we have
O(qa(ln q ) B + @ ' + o(qa'+'(In q ) P ' )
l)
if a' + 1 = a if a ' + 1 < a if a' + 1 > a.
(3.2.32)
51
3.2 Proojof' the Theorem in One Dimension
Therefore we may find constants c(il, . . . , in) > 1 and N(il, . . . , i,) > 0 such that n
, l ; v , , ; t $ . L ~ ,.... L . + i I )
x YI.
. . . vlna((L,Li .....Lml)
1 (In ql)yl . . . (In qr)Yr . . . (In q,JYm, .... Y" (3.2.33)
for q r 2 br(il,.. . , I,), I # r, and q, 2 c(il, . . . , i,) > 1. The coefficient a, and the sums over the relabelled nonnegative integers yl,. . . , ym will now depend in which of the three categories in (3.2.32) we are. We spell out each case separately. (a) If a({L, L 1 , .. . ,L,}) + 1 = a(il,. . . , i,), then we have from (3.2.32) and (3.2.33), a, = a(il, .. ., i,) = a({L, L 1 , .. ., L,})
+ 1.
(3.2.34)
The sum in (3.2.33) is, from (3.2.3 1) and (3.2.32), over all nonnegative integers . . ,y m such that
yl,.
k
yi IB(il,. . . , ik),
1 I kI r
-
1,
i= 1 r
1 yi Imax[P(il, ..., i,), P({L, L 1 , .. ., L,})] + L3
(3.2.35)
i= 1
k
(b) If a({L, L,, . . . , L,})
+ 1 < a(il,. . . , i,), then from (3.2.32) we have a, = a(il, . . . , i,),
(3.2.36) and from (3.2.31) and (3.2.32), the sum in (3.2.33) is over all nonnegative integers yl,. . . , ym such that k
' Note that because of the assumption on the ordering in (3.2.24) we could have written Yi
Ib({L,L,. . . , L,))
+ 1 in (3.2.35) since
max[/j(i,. . . . , i,),b({L, L , , . . . *L,})] = /{({I+ L , , ..., L,}).
xi=,
The form as given in (3.2.35), for yi, is. however, more useful for generalization when the ordering condition in (3.2.24) is relaxed.
3 The Power-Counting Theorem and More
52 (c) If cr({L, L1, . . . , L,})
+ 1 > a(il, . . . , ir),
a, =
a({L, L,, . . ., L,})
then from (3.2.32)
+ 1.
(3.2.38)
From (3.2.31)and (3.2.32), the sum in (3.2.33)is now over all nonnegative integers yl, . . ., y, such that k
[III]
Let Y E J ~ l ~ . . . Then , i m . the vector P
P + Ly
=
(L,
+ Lli,L)Vl . . . q, +
* * *
+ Ly may be written
+ (Lm + uil ,..., i,Lhm + C‘, (3.2.40)
where
C
=
0 IIZI 5 bo(il, .. ., ,)i
C f LIZI,
(3.2.41)
and thus C’is confined to a finite region in R”. Accordingly, we may find constants M‘(il,.. . , i,) > 0, bi(il,. . . , ),i > 1, 1 = 1,. . ., m, such that for q/ 2 b:(i1, . . . *,)i
If(P
+ Ly)I 5 M’(il, . . ., i,)q:(il) . . . q ~ i l ~ . . . * 1 i m(In ’ q1IY1.
*
. (In q m ) Y m ,
Y
(3.2.42) where the sum is over all nonnegative integers yl, . . . ,y, such that k yi
5 p(i1,. . . , ik),
1 5 k 5 m.
(3.2.43)
. (In q,)Ym,
(3.2.44)
i= 1
Hence
x
1 (In
ql)yl
Y
where the sum is over all nonnegative integers yl,. . . , 7, satisfying (3.2.43). Now we use the inequality (3.2.21),and the estimates in [I]-[III] to sum over thejnite number of terms in (3.2.21)as i l , . . . , i, are made to vary over their corresponding sets of finite integers. Thus we may find some new constants M > 0, b , > 1,. . ., b, > 1, and take q I 2 b,, ...,?,I, 2 b,. The
3.2 Proof of the Theorem in One Dimension
53
power asymptotic coefficients a,({L,, . . . , L,}) of (3.2.21), (3.2.25), (3.2.33)-(3.2.44): a,(&, . . . ,L,}) = max{a({Ll
are then given from
+ ulL,. . .,L, + UrL}), a(&
L 1 , ... , L,}) + l}, (3.2.45)
where u , , . .., u, take on the finite set of values U i l ,..., U i ,,.,,,i,, and the . . . , i, vary over a finite set of integers. Accordingly, the set of maximizing subspaces for a,({L,, . . . ,L,}) in (3.2.45) is a finite set. According to Lemma 1.3.3, (3.2.45) may be rewritten i,,
a,(S,) = max [a(S’)
+ dim S’ - dim S,],
(3.2.46)
A(I)S’=S,
where S, = {L,, . . . , L,} and we note, in particular, dim S’ = dim S, or dim S’ = dim S, + 1 in (3.2.46), as {L, + u,L, . . . ,L, + u,L} has dimension equal to dim S, and {L, L,, . . . ,L,} has dimension equal to (dim S, 1). Also the set A [see (3.1.5)] of the maximizing subspaces for the I integration, i.e., over y, relative to S , is afinite set. Now we turn to the logarithmic asymptotic coefficients b({L,, .. . ,L,}), 1 Ir Im. From (3.2.26), (3.2.33)-(3.2.35), (3.2.37)-(3.2.39), and (3.2.43) and the inequality (3.2.21), we obtain
+
(i) P , ( S r )
=
max{max[P(il,.
..,i,)
P({L, L1,.. .,Lr})])
+1
(3.2.47)
+ 1P
(3.2.48)
if a,({Ll,. .., L,}) = a(i,, .. ., i,) = a({L, L,, .. ., L,})
for some u i , . . . , u,, i.e., if all the maximizing subspaces in A are of the form {L, + ulL, . .., L, + u,L} and {L,L1,..., L,}, for some u1,..., u,. We recall with the definition of the dimension number p in Section 3.1; p = 1 in this case, as the dimension of {L, + u,L, . . . , L, u,L} and {L, L,, . . . ,L,} are different (they differ by one). The u,, . . . , u, take on values from the finite set of values U i , ,.. ., Uir and i,,. .., i, vary over a finite set of positive integers, all corresponding to the maximizing subspaces for the I integration relative to S,, i.e., corresponding to the subspaces in A. (ii) P , ( S r ) = max{b(il, . . . , i r ) } (3.2.49)
+
if a,({L1,. . . , L,}) = a(il,. . . , i,)
(3.2.50)
for some u l , . .., u, taking on values from the set Uilr..., U i ,,...,i m , and a({L, L,, . . . , L,}) + 1 < a(il,. . ., i,) for the corresponding u,, . . . , u,. That Recall that a(i,, . . . , i,) is of the form a({LI
+ u , L , . . . , L, + u,L)).
3
54
The Power-Counting Theorem and More
+
is, all the maximizing subspaces in A are of the form {L, u,L, . . . , L, + u,L} for some u , , . . . ,u,. They all have the same dimension and hence p = 0, by definition, in this case. (iii) P d S r ) = P({L,L1, * *
3
3
(3.2.51)
Lr})
if (3.2.52) Lr}) = N{L, Li, . . ., Lr)) + 1 ar({L, and a ( { L , + ulL, . . . , L, + u,L}) < a((L, L,, . . . ,L,}) + 1 for all u l , . . . , u, taking values from the finite set of values U i l , .. . , Uil,..., i, with i l , . . . , i, varying over a finite set of positive integers. That is, {L, L,, . . . ,L,} is the only maximizing subspace in A, and p = 0, by definition, in this case. From (i)-(iii), with the corresponding expressions (3.2.47), (3.2.49), (3.2.51), and the dimension numbers worked out above for each case, we may then write - 9
(3.2.53) which coincides with (3.1.6) for k = 1. This completes the proof of Theorem 3.1.1 in one dimension, i.e., when k = 1. 3.3
PROOF OF THE THEOREM I N AN ARBITRARY FINITE NUMBER OF DIMENSIONS
In this section we prove Theorem 3.1.1 in an arbitrary finite number of dimensions for I by induction. 3.3.1
The Absolute Convergence Criterion and t h e Power Asymptotic Coefficients
We first define the following class of B:-functions. Definition of class B:: A function f(P), with P E R", is said to belong to a class B f ( 1 ) if for all choices of a nonzero subspace S c R", m I n, independent vectors L,, . . . , L, in R" and a bounded region W c R", such that If(P)l # co,there exist coefficients a(S) and
S ( L ~ V I V ~+ ' .L 2.~V2 ~ * * * 2 1+m L m V m + C) NI,
1 2...,N m ....,
( ~ nq 1 ) ~ 1* . * (ln qrn)Y->,
YI
ym 0 SYib N ,
(3.3.1)
3.3 Proof in an Arbitrary Finite Number of Dimensions
55
where q , , . . . , q, are real and positive such that q l , . .., q, -+ 00, independently, C E W,and each of N , , . . . ,N , is made to vary over any arbitrary finite number of nonnegative integers. We also associate with B:(I) a subspace I c R" such that the condition If(P)l # co, for some P E R " , implies that If(P + P)I # 00 for all finite P E I . As before a(S) will be called a power asymptotic coefficient. Obviously we have &(I) c B;(I). To simplify the notation we shall denote B:({O}) simply by Bz. Suppose that f is locally integrable in I. We first prove the following theorem. Theorem 3.3.1 : Let f(P) E Bn(I), and hence also f(P) E B:(I), with asymptotic coeficients a(S) and /3(S), S c R". For allfinite P such that A(I)P = P and I f(P)I # co,iffor f ( P P), P E I ,
+
DI = max[a(S)
[A]
S'CI
+ dim S ] < 0,
(3.3.2)
thenh(P) is absolutely convergent. If [A] is true, then
[B] fI(P) E B:-k, dim I = k, with power asymptotic coeficients a,@), S c E: al(S) = max [a(#) A(1)S' = S
+ dim S - dim S].
(3.3.3)
The proof of the theorem is by induction. Suppose first dim I = 1. Since f E &(I) c B:(I), we may use the definition of the former class and apply the proof of Theorem 3.1.1 for dim I = 1 as given in Section 3.2, word for word, corresponding to each of the terms in the summand in (3.3.1), i.e., to ~ ; ( ( L I ) ). . . q?R~....aLrn))(ln ~ , ) Y I . . . (In qm)Ym for fixed integers y,, ..., y,, to conclude after summing on y1,..., y, and N , , . . . , N, that f,is absolutely convergent if DI < 0, as given in (3.3.2), and that fr E B:- with power asymptotic coefficients, as given in (3.3.3). Finally, we note that if dim I > 1, then we may write I = I' @ I", with dim I' = 1. Let f~ Bz(I). Upon writing
f m=
Jf(P +PI I'
symbolically, we note by definition of f(P) E B,O(I)that If(P)I # 00 implies If(P P)I # co with P E I' c I and If(P + P + P)I # co with P E I" c I . Accordingly, if (3.3.2) is true with I in the latter replaced by 1', we have, by repeating the proof in Section 3.2, with dim I' = 1, If(P)I # a,that lfI,(P)l # coandlfl.(P P)]# c o w i t h P " ~ I " ( cI),andf,,(P)EB,O-,(I").
+
+
3 The Power-Counting Theorem and More
56
Now we generalize the proof of Theorem 3.3.1 for dim I > 1. As an induction hypothesis, suppose that Theorem 3.3.1 is true whenever a subspace of integration of I has dimensionality I k', and then we prove the theorem for dim I = k' + 1. Let I be a (k' + 1)-dimensional subspace of R".We decompose I as follows: (3.3.4)
1 = I, CD 1 2 ,
where dim I, = k, and dim l2 = k2, with k l = 1, k2 = k', and hence k l + k2 = k' + 1. Let f~ B,(l) c B:(I); write (3.3.5) and we now justify the validity of the definition of fi(P), given in ( 3 . 3 3 , as an absolutely convergent integral by induction. According to our induction hypothesis, h, is absolutely convergent (from [A]) if
D I , ( f ) = max [a(S)
+ dim S ] < 0.
(3.3.6)
S"cI2
From [B], if D I , ( f ) < 0, then f r , ~ B : - ~ , ( l[with ~ ) B:-k replaced by B:- k i ( l l ) I and a,,(S') = max [a(S")
+ dim S" - dim S'].
(3.3.7)
A(I2)S" =S'
Since fr, E B:-kz(ll) and dim I, Ik', we may again use [A], by our induction hypothesis, to conclude that fi as given in (3.3.5) is absolutely convergent if D I , ( f i z )= max[aI,(S)
+ dim S] < 0.
(3.3.8)
S'CI,
Accordingly, the integral
s,, s,, dklP
dk2Pf(P + P
+P)
(3.3.9)
is absolutely convergent if both (3.3.6) and (3.3.8) are true. Hence by the Fubini-Tonelli theorem (Theorem 1.2.4), the integral (3.3.10)
3.3 Proof in an Arbitrary Finite Number of Dimensions
57
is absolutely convergent, and its value coincides with the value of the iterated integral in (3.3.9) if DI = maxC~,,(f), DI1(h2)I< 0,
(3.3.1 1)
i.e., when both (3.3.6) and (3.3.8) are true. Now from (3.3.7) and (3.3.8) we may rewrite for the latter
Dll(h2) =
max A(l2)S”
[a(S”)
+ dim S”].
(3.3.12)
IS’c I 1
We may apply Lemma 1.3.2 to (3.3.12) and (3.3.6) to rewrite the condition (3.3.1 1):
D,
=
max[a(S”)
+ dim S ] < 0,
(3.3.13)
S” c I
where we have used the fact that I = I, @ 1 2 . Equation (3.3.13) is nothing but the criterion [A] of the theorem for dim I = k’ + 1. Again, since dim I , = 1 Ik’ and f12E B:.-k2(II), our induction hypothesis implies a,(S) = max [aI,(S’)
+ dim S’ - dim S],
(3.3.14)
A(f i)S’=S
where a,,(S’) is given in (3.3.7). Accordingly, a,(S) may be rewritten a,(S) = max
{
max [a(S”)
I
+ dim S” - dim S l .
A ( I l ) S ‘ = S A(Iz)S“=S’
(3.3.15)
By an immediate application of Lemma 1.3.4, we then have from (3.3.15) a,(S) = max [a(S”) A(I)S” = S
+ dim S” - dim Sl.
(3.3.16)
Therefore we have proved parts [A] and [B] of the theorem for dim I = k‘ + 1 as well, with f, E B:-kr- where the logarithmic factors are treaded as in Section 3.2 since k , = 1 and h2E B:-k,(I,), thus completing its proof for all finite k = dim I > 1 by induction. We complete this subsection by showing that the set Jl of all the maximizing subspaces for the I integration relative to S in (3.3.3), as defined through (3.1.5) is a finite set; i.e., there is a finite number of subspaces in A. To this end, as before, we decompose I into two arbitrary disjoint subspaces I , , I,: I = I , @ Z2. We reconsider Eqs. (3.3.7), (3.3.14), and (3.3.16): a,,(S’) = max [a(S”)
+ dim S” - dim S],
(3.3.17)
A(Iz)S”=S’
+ dim S’ - dim S] [a(S”) + dim S” - dim S],
al(S) = max [a,,(S’) A(I1)S‘=S
=
max A(I)S” = s
(3.3.18)
3 The Power-Counting Theorem and More
58
where S c E,
R" = Z @ E,
YcE2,
R"=12@E2,
s
(3.3.19)
c R".
Since we have already established the absolute convergence of fi in (3.3.10), under the condition (3.3.2), we may in evaluating it use the iterated integral (3.3.9), and we may further change the orders of integration given in the latter at will. We have already established in Section 3.2, for dim I = 1, that AV consists of only a finite number of (maximizing) subspaces. As an induction hypothesis suppose that the latter is also true whenever we have a subspace I' c Z with dim I' < k. In particular, we suppose there are only a finite number of maximizing subspaces for the I , integration relative to S after performing the Z2 integration. Denote the set of these maximizing subspaces by {S,,},,, the latter means that a,(S) = a,,(S,,)
+ dim SL - dim S ,
A( 11)Sl = S.
(3.3.20)
For each Su,suppose, again by the induction hypothesis, that there is only a finite number of maximizing subspaces for the Z2 integration relative to Sue {S,,},,.Denote the set of these maximizing subspaces by {Sa,},;then the latter means that a,,(Su) = a(Su,)
+ dim Suv- dim Su,
A(I,)S&, = S&.
(3.3.21)
In the above, dim II < k and dim l 2 < k. We then prove the following lemma. Lemma 3.3.1 : The set {Spv},,v is precisely the set A of all maximizing subspaces for the Z integration relative to S , i.e.,
+
a@) = a(Sup) dim Sup- dim S,
A(Z)Sup= S,
(3.3.22)
with SapE {SNV},,,, and any maximizing subspace for the Z integration relative Since p and v run to S, i.e., satisfying (3.3.22), necessarily belongs to {S,,v}v. over afinite set of integers, we establish, in particular, thefinite property of A for the I integration as well by induction with dim Z = k.
3.3 Proof in an Arbitrary Finite Number of Dimensions
59
The proof of the above lemma is elementary. First we note that any S&, c {SLv},,vis a maximizing subspace for the I integration to S, as from (3.3.20) and (3.3.21) we have
+ dim Sup- dim S&)+ dim Su- dim S = a(SA,) + dim Sup- dim S,
a,(S) = (a(Su,)
(3.3.23)
and
or A(I1 @ Z2)So,= S.
(3.3.24)
Now suppose S A is a maximizing subspace for the I integration relative to S, i.e., a I ( S ) = a(Sg) + dim S; - dim S,
(3.3.25)
Sb = A(Z2)Sg ;
(3.3.26)
A(Z1)So = S = A(Z)Sg.
(3.3.27)
and let
then
On the other hand, we note that a,(S) = max [a,,(S) A(Il)S‘=S
=
max [a($’)
+ dim S‘ - dim S] 2 a,,(Sb) + dim So - dim S + dim S” - dim So] + dim So - dim S
A(I 2)s‘‘= Sb
2 a(Si)
+ dim Sg - dim S = a,(S),
(3.3.28)
where in writing the last equality in (3.3.28) we have used (3.3.25). The chain of inequalities in (3.3.28) implies
+ + dim Sg - dim So,
aI(S) = aI,(So) dim So - dim S ,
aI,(Sb) =
(3.3.29) (3.3.30)
as the first and the last terms in this chain are the same quantities, and hence all the inequalities in it are reduced to equalities. Equation (3.3.29) implies that So is a maximizing subspace for the Z l integration relative to S after performing the Z2 integration. The latter then implies that SoE {S;},, by
3 The Power-Counting Theorem and More
60
definition of {Y,}, as the set of all the maximizing subspaces for the II integration relative to S after performing the Z2 integration. O n the other hand, for each S,,E {S,},,,we have defined {SIv},,v as the set of all the maximizing subspaces for the I , integration relative to S, and consequently from (3.3.20) S%must be in {S,v},v, as So is in {S,,},,. This completes the proof of the lemma. 3.3.2
The Logarithmic Asymptotic Coefficients and the Class B,,+AProperty of ./,
In Section 3.3.1 we have established that if ~ E B , ( I () c B:(l)), and
D, < Oas defined in (3.3.2), thenfi E B:-k, with power asymptotic coefficients as given in (3.3.3). In this subsection we complete the proof of Theorem 3.1.1 by showing that f, also belongs to B,,-k with logarithmic coefficients as given in (3.1.6). Let I = I, @ I, @ . . - @ I,, with dim I, = 1, j = 1, 2, . . . , k, let {S,,},, be the set of all the maximizing subspaces for the I, integration relative to S c E, after performing the I2 @ - - @ I, integration. Recursively, we define {Sp,,...,p,}cjto be the set of all the maximizing subspaces for the Z j integration, after performing the I,, @ . . . @ 1, integration, relative to S,,,,..,,,-, as one of the maximizing subspaces in { S o ,,...,,,,-,},,, ,...,",- , for the integration, after performing the I , 0 . . . @ I k integration, with j = 2, .. . , k. For each fixed j , j = 2, . . . , k, choose any5 subspace S,,,,..., , E {S,,,,.,,,uj- ,} ,,j- ,. We may then define the dimension numbers computed relative to the chosen maximizing subspaces as follows: Definition 3.3.1 :
,
,,-
I
0
' j =
1
if all the subspaces in {S,,,,,,,,,,},,have the same dimension otherwise (3.3.31)
f o r j = 1, 2, .. ., k. Obviously p , is computed relative to the subspace S c E. We now prove the following important lemma. Accordingto Fink(1967,1968), thedimension numbersplin(3.3.31)areindependent ofthe maximizing subspaces for the 1,- integration relative to which they are computed. Accordingly, the definition of the parameters p , as given here is consistent. Refer to Fink (1967, 1968) for a proof of this statement.
,
61
3.3 Proof in an Arbitrary Finite Number of Dimensions
Lemma 3.3.2: If I = 1; Q 1; Q . . Q 1; is any other decomposition of I , dim 1'. I = 1, j = 1 , . . . , k, with corresponding dimensions numbers p i , . . . ,p i , then p i = Z:= p i , and the logarithmic asymptotic coejicients are given as in (3.1.6).
xf=
The proof is by induction. Let { S , } be the set of all maximizing subspaces for the I integration relative to S after performing the l2 Q - - Q I k integration. Let {s"vfl}v be the set of all maximizing subspaces for the I, Q . . . Q I k integration relative to S,. According to Lemma 3.3.1, the set {~,,}, constitutes all of the maximizing subspaces for the I integration relative to S . As an induction hypothesis, suppose that f r Z e ... E & k + 1(11)with logarithmic asymptotic coefficients:
x k
=
"'@fk(s,)
max
P(Svp)
+
V
Since dim I l
=
(3.3.32)
Pi-
i=2
1, we conclude from (3.2.53) that Pl(S) = max PI**...eI,(S,)
,
+
(3.3.33)
P 1 9
or from (3.3.32) k
k
P,(s) = max p(SV,) + i1 p i = max P ( S ) + 1p i , v. P = 1 S E A i=1
(3.3.34)
and E B"-k. Finally, let I = 1; Q . . . Q 1; be another decomposition of I into onedimensional components. Let { TU],be the set of all maximizing subspaces for the 1; integration relative to S after performing the 1; Q . .. Q integration. Let { T,,}, be the set of all maximizing subspaces for the r, Q Q l ; integration relative to T,. Then according to Lemma 3.3.1 we have again that { TPa}ps = A = {s"v,}v,. With DI c 0, we have from Theorem 3.1.1 [A], by the application of the Fubini-Tonelli theorem (Theorem 1.2.4), that JI .._e I k f = Jlie... e,Lf,written in a symbolic notation. Hence we conclude (by induction as above) that k
k
Pl(s)= maxfr P(T,,) + i1 p: = max p(SV,) + C p i , =1 i=1 P.
If=
V9
xf=
(3.3.35)
0
or from (3.3.34) that pi = p i . This completes the proof of the lemma. By completing the proof of the above lemma we have thus finally completed the proof of Theorem 3.1.1.
62
3
The Power-Counting Theorem and More
NOTES
The power-counting theorem is due to Weinberg (1960), where the convergence criterion and the expression for the power asymptotic coefficients are obtained. This work was then generalized by Fink (1967, 1968) to obtain the expression for the logarithmic asymptotic coefficientsby defining in the process the class B, of functions. We have followed their treatment quite closely, but have slightly modified the original definition of class B,.
Chapter 4 /
E
+
+O LIMIT OF FEYNMAN
INTEGRALS AND LORENTZ COVARIANCE
The purpose of this chapter is to establish the E 4 +O limit of Feynman integrals of the type defined in Section 2.2,(2.2.1)-(2.2.4), which are absolutely convergent for E > 0. This result is essential, as the presence of the factors ie(Q: + p;) in the denominators of, for example, the integrands in (2.2.3) clearly break the Lorentz covariance of the integrals. We show that the limit E -, +O of absolutely convergent Feynman integrals, (2.2.1), exists in the sense of distributions in Minkowski space. To this end we give a few definitions. 4.1
CLASS 9 ’ ( R k ) FUNCTIONS (TEST FUNCTIONS)
A function f ( x ) , x E Rk,belongs to class (or space) 9’(Rk) of functions, if f ( x ) is infinitely continuously differentiablein x,l andf(x) together with all its derivatives of arbitrary order vanish for 1x1 + co faster than any power of 1/1 x 1, where I x I = (If= xf)l/’. This class of functions is usually called the class of test functions and was introduced by Schwartz (1978). The property of an 9’(Rk)-function f may be precisely stated as follows. For any pair of k-tuples of positive integers n = (nl,n2,.. ., nk), m = (ml,m2,.. . ,mk), we can find a real positive constant C{, depending on n, m,and f such that
I X ” Dmf(x)I < CL,m
’ Functions with the property given so far are called %“-functions. 63
(4.1.1)
64
4
c
+
+ O Limit of Feynman Integrals and Lorentz Covariance
where
let Fc,(P,p) be an absolutely convergent Feynman integral of the general type defined in (2.2.1) with E 7 0. P = (py, . . .,p:) denotes the components of the independent external momenta of a graph. We show that for any f(P)E Y(R4"'), with
Z(f)= J
d P f ( P ) F e ( P ,PI,
(4.1.3)
IW4m
the limit E + + O of T,(f) exists. The existence of such a limit is called convergence of Fz(P,p) for E -, + O as a tempered distribution.' We first give a generalization of the so-called Lagrange interpolating formula. Consider a polynomial of degree n in the real x E R', n
P(x) =
2 a,(x)J.
(4.1.4)
j=O
Let xo, xl, .. . ,x" be n + 1 distinct values for x. We may then introduce new coefficients b o , b,, . . , ,bn to rewrite (4.1.4) in the form
P(x) =
n
n
j=O
i=O
C b j n (X
- xi).
(4.1.5)
i#j
We may solve for b, as follows. Let x = xj in (4.1.5) for somej; then n
bj
=P
( x ' ) ~(x'
- xi)-,.
(4.1.6)
i=o i# j
We shall not go into the details of tempered distributions T as the class of all continuous linear functionals on Y(R'). By linearity it is meant that for any complex number a, T ( u f ) = or(/),etc., and by continuity it is meant that T ( J )-+ T(f)whenever 1l.f- J l l n , m -, 0,i -+ m, for all n, and m, where
In'l~n.Im'lsm
The class (or space) of tempered distributions is denoted Y ' ( R k ) ,to which the limit of T, for E -, +O in (4.1.3), if it exists, belongs with k = 4m. For such details see Schwartz (1978). Here it suffices to establish the existence of the limit E -+ +O of T , ( f ) for all ~ ( P ) Y(R4"). E
4.1
Class 9'(Rk) Functions (Test Functions)
65
Hence we may equivalently write for P(x)~
i# j n
n
(4.1.7) The last equality defines the coefficientsA!;). The superscript (x) on A$) is just to remind us that these constants depend on the values xo, . . .,x" given to the variable x. Equation (4.1.7)also defines the coefficientsajin (4.1.4)in terms of the A!;) as follows:
(4.1.8) Now consider a function Q( Y; xl, . . . ,x,) that is a polynomial of degrees d , , . . , , d , in the real variables x,, . . . ,x , ; x i E R', respectively, and also depends on a variable Y E Rk:
Applying formula (4.1.7)to (4.1.9)for the variable x,, we obtain dm
Q(Y;xl ,..., x,)=
1 Q(Y;xl ,..., x,-~,xA~) j,=O
dm
C(x,,,)lm.4bl,
(4.1.10)
1,=0
where x i , . . . ,xim denote d , + 1 distinct values for x,. Applying (4.1.7) m - 1 more times to (4.1.10)for the remaining variables x,- l,. . .,xl, we obtain for the coefficients a,, ...,,,,( Y) the expression
+
+
where (xy, . . . ,x:), . . . ,(x:~, . . . , x k ) denote ( d , + I)(& l).-.(d, 1) distinct points in R". This gives a direct generalization of (4.1.8)in R". The first equality in (4.1.7) is known as Lagrange's interpolating formula (cf. Isaacson and Keller, 1966).
4
66
E +
+O Limit of Feynman Integrals and Lorentz Covariance
Equation (4.1.11) leads to a very useful result. It says, in particular, that if Q(Y;x l , . . .,x,) is absolutely integrable in Y at (d, 1) x . . - x (d, 1) distinct points for (xl, . .. ,x,) in Iw", then from
+
s..
dYlQ(Y;xj,',..., xl,)l < 00,
+
(4.1.12)
a',,...,lm(Y) is also absolutely integrable.
4.2
BASIC ESTIMATES, COVARIANCE
E-+
+ O LIMIT, A N D LORENTZ
We rewrite a Feynman integral as
(4.2.1)
(4.2.2) (4.2.3)
(4.2.4) where K = (ky, .. .,k;), P = (p:,
. ..,pi).
We use the so-called Feynman parameter representation4: L
(4.2.5)
nDL1=(L-l)! I= 1
where L
a = (a1,..., aL):al2 0, C a l= 1 I= 1
Cf. Feynman (1949b).
(4.2.6)
4.2 Basic Estimates, E 4 +O Limit, and Lorentz Covariance
67
which is easily proved by induction in L. Accordingly, we may rewrite 9 & ( P P, I :
.
Fe(P,p ) = ( L - l)!
(4.2.7)
We note that (4.2.8) L
1alQ: =
I= 1
n
+
n
k j * (AjYkJ) 2 j,j’=l
j= 1
B j k j + Co,
(4.2.9)
where L
BY
aIaljq1”,
=
(4.2.10)
I= 1
L
(4.2.11) (4.2.12) (4.2.13) If the Feynman integral FC,(P, p), E > 0, in (4.2.7) is absolutely convergent, then we may interchange the order of the K and a integrations. To see that this may be done, consider the inequality (2.2.9), ~ x ( l- ic) - I [ - ’ I (x
+ I ) - ~ ( I / +E J-1,
x 2 0,
E
> 0,
and set
to obtain
where
4,= Q?E + cc: = Q? + (Q?)’ + d .
(4.2.16)
4
68
6+
+ O Limit of Feynman Integrals and Lorentz Covariance
Accordingly, we have
(4.2.17)
where we have used (4.2.5), with D , replaced by DIE,to integrate over a, and the definition of the integrand $(P, K, p, E ) given in (2.2.14), and where G, > 0 is a constant. Now we know from the right-hand side of inequality (2.2.15)that the absolute convergenceof &(P, p), E > 0, implies, in particular, the absolute convergence of S e ( P , p ) . Hence the right-hand side of the inequality (4.2.17) is bounded, i.e.,
Jg l il I L
-L
( L - 1 ) ! 1 d K ( P ( P , K , p , E ) I da ~ a l D 1 < a, (4.2.18) W4"
and by the Fubini-Tonelli's theorem (Theorem 1.2.4)we may interchange the orders of integrations over K and a in (4.2.7) and integrate over K first if we wish. To this end we assume that the matrix a = [alj]defined in (4.2.4) is of rank n.' Consider the situation when u1 > 0,.. ., a L > 0. Then in this case the matrix [ A j j . ] with elements defined in (4.2.13) is nonsingular. Let k' be the solution of the linear equations i A j j . k j ! ' = -BM I '
(4.2.19)
j= 1
where the Bf have been defined in (4.2.10).The solution (4.2.19) corresponds to the stationary value of the expression alQ: = D(k) in (4.2.8), i.e., (d/dk') D(k)I,=,. = 0. From (4.2.19) we may then write
c;"=l
D(k) 3 Do(k) 3
L
n
1=1
j,j'=l
L
n
I= 1
j.I'=1
1 alQ; = 1 ( k j - k>)Ajj.(kr- kj,) + D(k'), 1alQ; = 1(k, - k J ) A,(k, - k;,) + Do(k).
(4.2.20) (4.2.21)
'
In Chapter 5 we shall see that n of the k', for example, could be consistently chosen to coincide with the integration variables k l , . . . ,k,, i.e., k' = k l , 1 I I 5 n, when properly labeled, and hence all = Slj for 1 S I, j < n. More generally, n of the k' may be chosen to be of the form k' = aljkj,for I I 2 n, with [arj].I 5 I, j < n,a nonsingular matrix, which reduces to the I aljkj case just mentioned ifwe choose all = S l j .The remaining k' would be of the form k' = with I > n, for some a',. The rank of the matrix [au]. 1 s I I L, 1 Ij In, is then clearly of rank n. Note that we always have L > n in (4.2.1).
I;=
z=
4.2 Basic Estimates, E
+ O Limit, and Lorentz Covariance
-P
69
We note, in particular, from the very definition of D,(k), that (4.2.22)
D&) 2 0.
Let 0 = [OjY] be a matrix such that
ky - kifl = 10.. JJ’ k .I”
(4.2.23)
i’
OTAO = I,
(4.2.24)
where R = (i?, . . . ,ki) will define new integration variables. We have established, through (4.2.17)and (4.2.18),by the application of the Fubini-Tonelli theorem, that we may interchange the orders of integration over K and a E 9 in (4.2.7);hence with I? as new integration variables we obtain from (4.2.7)
where R = (k?, . . . ,I;),d (a set of measure zero) is the set of all a E 9 for which at least one of the aIE 9 , 1 I i I L,vanishes, and from (4.2.2),(4.2.5), (4.2.20)-(4.2.24) V = (1 - iE)
n
n
L
j= 1
j= 1
I= 1
1gf - C (KY)’ + D(k’) - ieD,(k’) + (1 - ic) C a l p f . (4.2.26)
From (4.2.19), (4.2.Q (4.2.9), and definitions (4.2.10)-(4.2.13), we see that D(k’) and Do(k) are some quadratic forms in p and p respectively,6 i.e., m
(4.2.27) m
DO(k‘) =
1 pj
*
(ujY(a)Pf)
P * (UP),
(4.2.28)
j, j‘= 1
where the UjY(a) are rational in a. We note, in particular, directly from (4.2.22)that p*(Up) 2 0.
(4.2.29)
There is a very long history in the literature on quantum field theory of matrices of the form in (4.2.27)and (4.2.28); cf. Todorov (1971). and references therein.
4 c
70
+
+0 Limit of Feynman Integrals and Lorentz Covariance
The matrix U = [ U j , , ] ,with U j j .as a rational function in a, originally defined for a E 9 - d may be extended to all a E 9.To this end we introduce real variables xl,. . . ,x,, y , , . . . ,y,, and by analogy with Q1in (4.2.3) define n
m
(4.2.30)
with the a l j ,b,, introduced in (4.2.4).Consider the quadratic form L
G(x, Y , a ) =
C
(4.2.31)
1=1
defined by analogy with Do(k) in (4.2.21). We may define a matrix U = CUiJ@)l by (4.2.32) which is continuous in a for all a E 9 and coincides with the coefficients in (4.2.27) and (4.2.28) for ai > 0, i = 1 , . . . ,L. Accordingly, the matrix U = [UiJ{a)] in (4.2.27) and (4.2.28) may be extended to one with continuous elements for all a E 9, Since B(P, K' + OR,p, E ) is, in particular, a polynomial in the f;i", the integration over R in (4.2.25) may be explicitly carried out in a standard manner,' which yields an expression of the form (up to overall multiplicative constants):
- ~ E ) - ( ~ ~ / ~ ) + ' O ) NP,( ~p,, E)F,(a, P)-',
(4.2.33)
const (1 - i ~ ) - ( ~ " / ~ + 'da~ )N(a, P , p, E)F,(a, P)-',
s,
(4.2.34)
+ M Z - ie(p
(4.2.35)
= const (1
or
e(P, p)
=
where F,(a, P ) = p u p
U p + M'),
L
(4.2.36)
ro is the largest integer sdp/2, where d p is the degree of the polynomial B in R,and t = L - 2n - ro. (4.2.37)
' Cf. Jauch and Rohrlich (1976). Appendix AS.
4.2 Basic Estimates, E
+
+0 Limit, and Lorentz Covariance
71
N(a, P, p, E ) is a polynomial in P, p, E and in the p; ’ ,in general, as well and is rational in a. We note, in particular, that 9 ( P , K, p, 0) is Lorentz covariant, and hence so is N(a, P, p, 0). Lemma 4.2.1
(4.2.38)
and (4.2.39)
for all f ( P ) E .4”(R4“‘),are absolutely convergent for
E
> 0.
From the elementary inequality (2.2.10), with x = (p Up + M2)/(p0Upo), we have from (4.2.35)
I Fe(a, I I Jm ( P E UP, + M’),
(4.2.40)
where
) continuous on 9, we may find a strictly positive constant Since the V j j , ( a are
H (cf. Lemma 1.1.1), independent of a, such that (4.2.42)
where
p
= max p l .
(4.2.43)
1
From (4.2.40) and (4.2.42), we then have
Accordingly, we have for (i) da IN(a, P, p, &)I IH‘
I¶
da IN(a, P, p, E ) ( IFe(a,P ) ( - ‘ .
(4.2.45)
12
4
+O Limit of Feynman Integrals and Lmentz Covariance
E +
The integral on the right-hand side of the inequality (4.3.45)may be majorized from (4.2.33) as da IN(a, P , p, &)I IF,(a, P)l-' I const x (ilalDIE)-'- <
*,
(4.2.46)
as the absolute convergence of (4.2.1) implies, from (2.2.15), the absolute convergence of the integral on the right-hand side of (4.2.46). Equations (4.2.45) and (4.2.46) then establish the absolute convergence of the integral in (i). To establish the absolute convergence of the integral in (ii) we write P, CC, E) =
1 EaPm"a,m*(a, PI,
(4.2.47)
a. m'
in a notation similar to the one in (2.2.18). The generalized Lagrange interpolating formula, in particular, inequality (4.1.12),then establishes from (i) that j 9 da Na,,,,(o!,p ) is absolutely convergent. Finally, we use the inequality
I F,(a, P ) I 2 E(P u p + M 2 )2 &p2,
(4.2.48)
-p = minpl > 0,
(4.2.49)
where 1
to derive the following inequality for E > 0:
J da IN(@,p ,
jR4W(P)I
9
5 Ge
C E'
a, m*
/A
dl IF&,
J R 4 t PI f(P) I IP"'
p)l-r
1
I da I Na,m*(a, P ) I < 00,
(4.2.50)
9
where G, is a positive constant depending on E and we have used the fact that f ( P ) p m '~ 9 ' ( I w ~ " ' ) and that J da N,,,(a, p ) is absolutely convergent.' This completes the proof of the lemma. Inequality (4.2.50) establishes the absolute convergence of T , ( f ) in (4.1.3) for E > 0, and, in particular it implies that to study the E + + O limit of T , ( f )it suffices to study the limit E + + O of the integral
:J
I-Ah) = JR4tP
h(P)N,m,(a,p)F,(a,
(4.2.51)
That j R d m dPl ,/(P)l lpm'l< m follows from the very definition of ,fp"' as a function in Y(R4"). The latter property (see 4.1.1).in particular, states that we may find an arbitrary positive
integer Nandaconstant C z Osuch that If(P)p"'I < C[l + Ip12]-N.Bychoosing Nsufficiently large, the integral J,.,dP [ I + lp12]-N then obviously exists.
4.2 Basic Estimates, f: + +O Limit, and Lorentz Covariance
Fig. 4.1
73
Sketch of the function ~ ( x )
where h(P) E .4"(R4'")sincef(P)p"' E .4"(R4"), as T,(f) is a linear combination of integrals of the form in (4.2.51) with, in particular, coefficients that have well-defined limits for E + +O. To study the E + +O of (4.2.51) we introduce a function 0 I~ ( x I ) 1 defined by
1:
x(x) = I
-
+ 2)'/[1
exp{-(3x
- (3x
+ 2>21)
for x < -4 for -3 I x' < -4 for - 3 1 I x, (4.2.52)
which together with all its derivatives vanishes for x < -5. The function ~ ( x )is sketched in Fig. 4.1. By setting x = pUp/p2, - we may rewrite for (4.2.51), Ieorn*(h)
= I!zrn,(h)
+ I?arn*(hL
(4.2.53)
where (4.2.54)
Because of the x factor multiplying the integrand in (4.2.54) we have, for P contributing to this integral, PUP i.e., p u p
+ M 2 2 PUP + p2 - 2 (-3 + l)p2 = p2/3,
+ M 2 never vanishes for the integral I,&,&). Also I FE(a,P ) I- II F o b , P) I- ' I (3/g2),
(4.2.56) (4.2.57)
where we have used (4.2.56). Accordingly we have
Iconst
J
dPIh(P)I W4m
J da IN,,,,,,.(a, 53
p)l
< a,
(4.2.58)
74
4
E -+
+O Limit of Feynman Integrals and Lorentz Covariance
where we have used the fact that ~ ( x I ) 1 and Lemma 4.2.1(i). Accordingly, from the Lebesgue dominated convergence theorem Dheorem 1.2.2(ii)], lim I:,,@)
=
N,,,.(a, p)FO(a,P)-',
Z;,,,,,(h) =
e++O
(4.2.59) and the latter exists. for E The study of l&,,,,, we may write
-+
+0, is more difficult. To this end we note that
whose validity is easily verified, where (PUP), = p u p - iEp * Up. Substituting the expression (4.2.60) into Z$,,Jh), integrating by parts, and using the vanishing property of h(p). together with all its derivatives, at infinity, we obtain for lim,(h) explicitly
It is easy to see that the expression in the curly brackets may be written in the form of a finite sum of terms each of which is of the form of some nonnegative power of E multiplied by a term of a finite sum of the form
(PUP);'1 hi(P)xi(a,PI,
(4.2.62)
i
where h,(P) E 9'(R4m)and the xka, P) are bounded functions and are vanishing for (a, P) with values outside the set S = {(a, P ) : p U p / p 2 I-31, (4.2.63) (recall the definition of [1 - x(pUp/p2)]). Accordingly, to consider the E +O limit of Z&,,,,(h) it suffices to consider the limit of integrals of the form -+
(4.2.64) For (a, P)ES, Fo(a, P) may vanish. The factor ( p u p ) , , however, does not vanish in (4.2.64) for all E, as
3p2
IPUPl 5 I(PUP),I.
(4.2.65)
4.2 Basic Estimates, E
-+
+O Limit, and Lorentz Covariance
75
Accordingly, the integral (4.2.64) without the In F,(a, P) factor in the integrand, i.e.,
is absolutely convergent, and from (4.2.65) and the Lebesgue dominated convergence theorem exists. Now we consider the expression fie,@) containing the In FE(a,P) factor. To this end we note that for any positive x, y and y o , y o > y > 0, lln(x
+ y)l IIln X I + (In y o [ + 1.
(4.2.67)
The proof of (4.2.67) is elementary. We consider all possible cases. First supposethat x + y > 1. I f x 2 y , then 2x 2 x + y > L a n d ln(x
+ y ) IIn 2x IIln X I + In 2 < Iln x ( + 1 + Iln yol.
(4.2.68)
Ifx
+ y) I(In y o [ + In 2 < (Iny o ( + 1 + (Inx ( . (4.2.69) Now suppose x + y < 1;then (4.2.70) Iln(x + y)J IJln X I < Iln X I + Iln y o ] + 1. If x + y = 1, then (4.2.67) is trivially true. We now use (4.2.67) to obtain I In F,(a, P ) I I+[In I F,(a, P ) I' + 2x1 IIIn IpUp + M21 I + Iln cO(p Up + M2)1 + x + 4, ln(x
*
(4.2.71) where E~ > E > 0. Therefore to establish the limit E +O of T&,3(h), it suffices from (4.2.71) and (4.2.65) to establish the absolute convergence of the following integrals: -+
(4.2.72)
JF,
=
dP
s,
da Na9m'(a' )' hi(P)Xi(a,P) InIpUp
(PUP)'
+ M 2 ( . (4.2.74)
From (4.2.58), J:mm. is obviously absolutely convergent. Also,(p. Up + M 2 )> 0 and U is continuous in 9,and hence J&, is also absolutely convergent. On the other hand, we may, for example, use a very powerful theorem in
4
76
E +
+ O Limit of Feynman Integrals and Lorentz Covariance
distribution theory (the Hironaka-Atiyah-Bernstein-Gel’fand t h e ~ r e m ) , ~ which may be tailored to the problem at hand; i.e., it may applied to the integral J&,, . It states that if the integral
is absolutely convergent, which obviously it is since it coincides with Jf,,,.in
(4.2.72), then so is the integral J:,,,, containing the In I ( p U p + Mz)I factor. This means that the In [ ( p u p Mz)I “singularity” is too mild to break the
+
absolute convergence of Jim,.. Therefore by the Lebesgue dominated convergence theorem we have
-
Iim TLm,(h) = &mp(h) &++O
(4.2.76)
Accordingly, we may state Theorem 4.2.1 : lim,,+o K(f), with defines u Lorentt couuriant distribution.
K(f) as given in (4.1.3), exists and
NOTES
The E -, +O limit of absolutely convergent Feynman integrals, starting in momentum space, is worked out in the elegant papers by Hahn and Zimmermann (1968), where, in particular, the generalized Lagrange’s interpolating formula is derived; Zimmermann (1968; see also Hepp, 1966); and Lowenstein and Speer (1976). In the latter reference one will find how the Hironaka-Atiyah-Bernstein-Gel’fand (HABG) theorem is tailored to the problem at hand. For the proof of the HABG theorem see Hironaka (1964), Atiyah (1970), and Bernstein and Gel’fand (1969). This theorem is quite powerful and goes beyond standard results in distribution theory.
The proof of this theorem is extremelylong and is beyond the scope of the present book, as it requires very special tools.
Chapter 5 / THE SUBTRACTION FORMALISM
This chapter is devoted to the subtraction formalism. In Section 5.1 we present basic definitions and work out examples concerning subgraphs, subdiagrams, canonical variables, Taylor operations, and the associated subtractions. In Section 5.2 we introduce the subtraction scheme, and some examples are then worked out. Here the reader will learn how to write down explicitly the expression for a renormalized Feynman integrand. A recursive formula for the subtraction scheme is also given. The convergence proof of the subtraction scheme is given in Section 5.3. In Section 5.4 we establish the “unifying theorem of renormalization,” which demonstrates essentially of the equivalence of the paths taken in the ingenious approaches of Salam and Bogoliubov. 5.1
BASIC DEFINITIONS AND EXAMPLES
5.1.1
Subgraphs and Subdiagrams
We define a graph G by specifying a set of vertices V = {q,. . ., u,} and a set of lines 9 = {el,.. . ,e,} joining these vertices as obtained from the so-called Feynman rules.’ Given a graph G, a subgraph G‘ of G, which we
’
In an abstract way we may, for our purposes, define a graph G by specifying two universal sets of vertices Y and lines Y with the lines joining the vertices according to given rules. Subgraphs and subdiagrams, as discussed below, are then defined in reference to these universal sets and the given rules. 71
5
78
The Subtraction Formalism
R6 14
9 1 2
14
34
9
4
12
0
9
Fig. 5.1 Parts (b), (c), (d), and (e) represent subgraphs of the graph G defined in (a). Part (f)does not represent a subgraph of G but a subdiagram of G.
write G' c G, is defined by specifying a subset of vertices V ' c V and all those lines in 9 of G that join these vertices in the original graph G. To make this definition more transparent we give some examples. Given a graph G as shown in Fig. 5.la, some examples of subgraphs are given in Figs. 5.lb-e specified, respectively, by the vertices ul, u 7 , u s , v 9 ; u 2 , u s , u 9 , u12, 0 1 3 , 014; u4, u 9 , u l l , u12, 013; and u4, u 9 , u l Z . Note that Fig. 5.le represents a subgraph of G because there are no lines joining the vertices u4, v 9 , and u12 in Fig. 5.la in the original graph G. Clearly, Fig. 5.lf does not represent a subgraph of G, as one of the lines joining the vertices u7 and ug in G is missing in Fig. 5.lf. Quite generally, by specifying a subset of the vertices V ' c V and some, not necessarily all, of the lines in Y of G joining the vertices in Y',we define a subdiagram. Accordingly, a subdiagram is a more general concept than a subgraph and reduces to the latter if it contains all those lines in G that join its vertices in r'.Figure 5.lf, as mentioned, is not a subgraph, but it represents a subdiagram of G of 5.la. In the sequel when we say a subdiagram we may also mean a subgraph. We do not allow a line joining a vertex to itself. A subdiagram is said to be disconnected if it is constructed out of two or more subdiagrams that have no vertices in common. Otherwise a subdiagram is said to be connected. Figure 5.2a represents a subdiagram g of G of Fig. 5.la that is disconnected, as it is constructed out of two subdiagrams g1 and g2 that have no vertices in common. Figure 5.2b provides another example of a disconnected subdiagram. Figure 5.lf represents a connected subdiagram. In this terminology the graph G of Fig. 5.la is itself disconnected,
5.1 Basic Definitions and Examples
79
92
g1
(a)
(b)
Fig. 5.2 (a) A disconnected subdiagram g of G of Fig. 5.la as it is constructed out of g1(a subgraph) and g2 (a subdiagram) which have no vertices in common. (b) Another example
of a.disconnected subdiagram g of G.
as it is constructed out of two subgraphs specified, respectively, by the two disjoint set of vertices {ul, u 2 , . . . , ug} and {ul0, u l l r . . . ,u14}. We say that a subdiagram g has n connected parts if it is constructed out of n subdiagrams each being connected. Trivially, if n = 1, then g is connected. A vertex of a subdiagram g to which is attached just one and only one line or no lines at all will be called an extral vertex of the subdiagram g. Figure 5.3a represents a subdiagram g of G where the vertices ug and u l l are extral vertices of g. The extral vertices of the graph G itself of Fig. 5.la are u l , u 2 , u 3 , ul0, u l l , and u14. By removing the extral vertices from a subdiagram and the lines attached to them, we obtain a new subdiagram, say, g', which may or may not have extral vertices. Figure 5.3b represents the subdiagram g' of g of Fig. 5.3a as obtained from the latter by removing the extral vertices ug and u1 and the lines attached to these vertices, i.e., the lines joining the vertices ug to ug and u1 to u12. The new subdiagram g' of g has an extral vertex ug . By continuing the process of removing the extral vertices (and the lines attached to them) of the resulting new subdiagrams repeatedly, we finally obtain either a nontrivial (i.e., nonempty) subdiagram with no extral vertices or no subdiagram at all (i.e., no lines and no vertices). We call a subdiagram with no extral vertices an amputated subdiagram. Figures 5.3a-c show the process of obtaining the amputated version, say, g" of g. The
Fig. 5.3 (a) A subdiagram g of G of Fig. 5.la where the vertices ug and u I 1 are extral vertices. By removing the extral vertices ug and uI ofg and the lines attached to them, we obtain a new subdiagram g' in (b) having ug as an extral vertex. Finally, (c) represents the amputated version g"of g. Vertices u4, u , , u I 2 , and u I J are external vertices of g" in (c). The vertices us and u6 are internal vertices of all the subdiagramsg,g', and 9".
5
80
The Subtraction Formalism
Fig. 5.4 The amputated version of the graph G of Fig. 5. la.
lines deleted in the process of amputating a subdiagram g are called the external lines of g. That is, the amputated version g" of a subdiagram g is obtained by removing the external lines (and the relevant vertices) of g. The external lines of g will be called external lines to g" (the amputated version of 9). A line that is not an external line of a subdiagram g is called an internal line. Accordingly, an amputated subdiagram may have only internal lines. In Fig. 5.4 we represent the amputated version of the graph G of Fig. 5.la. A subdiagram is proper, or called a proper subdiagram, if it is amputated and its number of connected parts does not increase upon removing any one of its (internal!) lines. Note, by this definition that a proper subdiagram is not necessarily connected. A line in an amputated subdiagram g is called an improper line of g if upon removing it, the number of the connected parts of g is increased. The proper part of a subdiagram g is a subdiagram obtained from g by amputating the latter and removing the improper lines (if any) in the resulting subdiagram. Figure 5.5a represents a subdiagram (with two connected parts) and is not proper, since the line joining the vertex u, to us is an improper line and upon removing it we introduce three, not two, connected parts, as shown in Fig. 5.5b. On the other hand, the subdiagram in 5.52 is proper. The subdiagram g' in Fig. 5.5b is the proper part of g of Fig. 5.5a. The proper part of the graph G of Fig. 5.la was given in Fig. 5.4. In the sequel, the proper part of a graph will be referred to as a proper graph. A subdiagram that is not proper is also called an improper subdiagram. A vertex of the graph G that is not an extral vertex and to which is attached an external line will be called an external vertex of G. A vertex of the graph G that is neither an extral vertex nor an external vertex is called an internal vertex. Now consider a subdiagram g $ G. A vertex ui of g that is not an extral vertex of g, and if ui is an external vertex of the graph G and/or there
(a)
(b)
(C)
Fig. 5.5 The subdiagram g in (a) is not proper, as it contains an improper line joining the vertex u, to cis. The subdiagram g' in (b) is the proper version of g in (a). Part (c) represents a proper subdiagram.
5.1 Basic DeJinitions and Examples
81
is a line t belonging to G, but not to g, attached to ui,then the latter is called an external vertex of g. A vertex of g that is neither an extral vertex nor an external vertex is called an internal vertex of g. Consider the subdiagram g c G in Fig. 5.3a. The vertices u4, u,, u s , u12 , ~ 1 are 3 external vertices of g and u 5 , 06 are its internal vertices. For g’ c G in Fig. 5.3b, u4, u 7 , u 1 2 , ~ 1 are 3 its external vertices and u 5 , u6 are its internal vertices. For the graph G, its external vertices are u4, u 9 , u 1 2 , u13, as they are attached to external lines and they are not extral vertices of G. Its internal vertices are u 5 , V 6 , u 7 , u s . For the proper part of the graph G of 5.la, as given in Fig. 5.4, the external vertices of the latter are also u4, u 9 , u12, u 1 3 and its internal vertices are u 5 , u 6 , u7, u s . In this book we require that each proper and connected graph together with all its proper connected subdiagrams have each at least two external vertices. and Let g1 and g2 be two subdiagrams defined by the pairs (Vl,9,) (V2, p2). If Vl c V2and Y1 c p2, then we write g1 c g 2 . On the other hand, if Vl n V2= 0 (empty), then we write g1 n g2 = 0 and we say that g1 and g2 are disjoint. Several lines may connect any two vertices of a subdiagram. For example, there are three lines connecting the vertices us and u9 of the subdiagram g1 in Fig. 5.2a. We may number these lines. An lth line joining a vertex ui to a vertex uj will carry a momentum Qijldepending on the indices i, j, and 1. The appearance of the indices i and j in the order in which they appear in Qijlis meant, as a convention, to show that the momentum is flowing from ui to u j . Accordingly, we take Qijl = -Qjil when interchanging the indices i and j , i.e., when “reversing” the direction of flow. An object of interest in quantum field theory is a proper and connected graph. Consider a proper and connected graph G. With each line tljoining a vertex ui to a vertex u j of G carrying a momentum Qijland a mass pijl we associate a propogator as a function of Qijland pijl,defined by a polynomial in Q i j r , pijl, and, in general, in ( p U ) - ’ as well: Pijl(Qijl,pijl), multiplied by (see Chapter 2) Accordingly a line will be represented by2 DGLQijl, P i j J
=
PijLQijl, PijJCQ$
+ P$
- i4Q$
+ p&)]-l.
(5.1.1)
We choose phI > 0,3and we do not allow a line connecting a vertex to itself.
’ The polynomial dependence of P,j, on (p,,,)-’, in general, is well known for higher-spin fields. Zero-mass behavior of renormalized Feynman amplitudes will be studied in Chapter 6 .
82
5
The Subtraction Formalism
Let j be fixed and consider {ui(j)}lsisrj, the set of vertices attached by lines to uj in G, and consider the set 6eG(uj)of all the lines joining the vertices uici), 1 I. i Ir j , to the vertex u j . Let {Qiil}lslSsrjbe the set of momenta carried by these lines. Then we assign to the vertex uj a polynomial P j = 9,(Qljl,. . . , Qrjjs,,j). The unrenormalized Feynman integrand associated with the proper and connected graph G, up to an overall multiplicative constant (involving couplings, etc.), is of the form 1, =
r p p D i ; l ,
(5.1.2)
ijl
i<j
consisting the product over all the polynomials q.assigned to the vertices { u j } of G and the propogators DG1 (5.1.1) joining the vertices of G. Let P = { p y , .. .,p:} denote the set of the components of the external independent momenta of the graph G carried by the external lines to G, and let K = {ky, . ..,k:} be the set of the 4n integration variables associated with the graph G. The latter will be called the set of the components of the internal momenta of G. A momentum Qijl flowing in an lth line from a vertex ui to a vertex uj of G is of the form Qijl
= kijl
+ qijl,
(5.1.3)
afjlks,
(5.1.4)
where n
k IJ1 .. = s= 1
(5.1.5) (5.1.6)
That is, kijlis a linear combination of the internal momenta of G and qijl is a linear combination of its external momenta. The sets {kijl}= k and {qijl}= 4' will be called the sets of internal and external variables of G, respectively. Let qj be the total momentum carried by the external lines to G attached to the vertex uj in a direction of flow away4 from the vertex'uj. If a vertex u j of G is an internal vertex, we simply set q j = 0, by definition, as there are no external lines attached to such a vertex. With such a convention we then have at each vertex uj of G qj
=
1"
Qijl,
(5.1.7)
il
The direction of flow of the momentum qjaway from the vertex v j is taken as a convention. Clearly, q j is a linear combination of the external independent momenta in P of G.
5.1 Basic Definitions and Examples
83
by momentum conservation. The sum is over all i corresponding to the vertices ui attached to the vertex u j all belonging to G, and over all the lines d , of G joining the vertices ui to the vertex u j . As mentioned, if uj is an internal vertex, then we simply have q j = 0. Finally, by overall momentum conservation we have, by summing over all j corresponding to all the vertices of G, the expression
c" 4j = 0.
(5.1.8)
j
5.1.2
Canonical Variables
In this subsection we write down the expression for the Qijl in the unrenormalized Feynman integrand I, (5.1.2) in a form suitable for carrying out the so-called subtractions of renormalization. This is important for constructing the renormalized Feynman integrand R. The particular way of writing the expressions for the Qijlin a form suitable for carrying out the subtractions, as will be given, will be called canonical. Once the renormalized Feynman integrand R has been constructed (Section 5.2) one may, of course, freely make a change of variables in R for the purpose of evaluating the renormalized Feynman amplitude s4 = d K R, once its (absolute) convergence (Section 5.3) has been established. We say that the set { q i j l }= ij is a canonical set of external variables of a proper and connected graph G if the qijl are chosen in the following manner: (5.1.9)
at each vertex uj of G, and where these are chosen of the form: 4 t.j .l = uI. - uj ?
(5.1.10)
where u i , u j are four-vectors. Equation (5.1.10) shows, in particular, that the external variables of the lines joining a vertex ui to a vertex uj are all chosen to be equal. If G has # Y cvertices, then (5.1.9) together with the constraint (5.1.8) gives # Y G- 1 independent equations for the # Y G- 1 independent differences ui - u j , and hence the external variables 4ijI of G are then uniquely determined. Equations (5.1.3) and (5.1.7)-(5.1.9) imply
c" ijl
kijl
= 0,
CG kijl = 0
(5.1.11)
il
at all the vertices uj of G for the latter equation. Finally, we note that as we have # Y c- 1 independent external variables 4ijI [see (5.1.10)], we may
5 The Subtraction Formalism
84
write the variables in {qij,} = 4 as functions (linear combinations) of # V G- 1 independent variables in { q j } = q : qij, = qij,(q) [see (5.1.8) and (5.1.9)]. The choice of the Qij, of G such that the qij, are canonical external variables [(5.1.8)-(5.1.10)] and the kij, are any consistent solutions of (5.1.11) will be called a canonical choice of variables. A property of a proper and connected graph G (or for a proper and connected subdiagram, in general) is that if 4n denotes the number of independent integration variables associated with G, then n = #9‘ - # V c + 1, where #9‘ is the number of lines of G and # V c ,already introduced, is its number of vertices. To this end note that the second of the # V Gequations in (5.1.11) with the first equation in the latter as a constraint allows only n of the # Y Gkij, in k to be independent. If (ky, . . . , k;) are the 4n integration variables, then n of the internal variables kij, may be, consistently, chosen to be the internal momenta kl,.. . , k,. More generally, by relabeling the elements in the set { k i j l } = {k(i)}, we may, a i j k j ,i = 1, . . . ,n, consistently, choose n of the k(i) in the form k(i) = with [aij],for 1 I i, j I n, a nonsingular matrix. The remaining k(i) would aijk j , i > n for some a i j .The rank of the matrix be of the form k(i) = [aij] in k(i) = a i j k j ,i = 1, . . .,n, is then equal to n.5 This reduces to the case mentioned earlier if we choose aij = dij for 1 I i, j I n. We work out a few examples for the illustration of the above rules.
cj”=
cj”= cj”=
Example 5.1 : Consider the graph in Fig. 5.6a with its external vertices defined to be u1 and u2 with q 1 + q2 = 0. Let q1 = q ; then q2 = -4. Equation (5.1.9) states q211
+ q212
= 4,
q121
+ q122 = - 4
(5.1.12)
A%
911
2
(4
(d)
(f)
Fig. 5.6 Some proper and connected graphs. The vertices ul, u2 are taken as the external vertices of the graphs in (a) and (b). The external vertices of the graphs in (c) and (d) are taken
to be r1 and P ~ The . external vertices of the graphs in (e) and (f) are taken to be v I and P ~ The . overall momentum conservations are q 1 q2 = 0, q 1 q, = 0. and q , q4 = 0, respectively.
+
+
+
The k(i), for i > n. may be also rewritten as linear combinations of the k(i) with 1 I, i I n.
5.1
Basic Definitions and Examples
85
at the vertices u1 and u 2 , respectively, which are, of course, not independent. Using the definition (5.1.10) and choosing any one of the equations in (5.1.12), we obtain 2(u2 - ul) = q, or
- u1
u2
1
= z4,
i.e., qzll =
1
1 = 192.
24,
(5.1.13)
We also have from (5.1.11)
+ k122 = 0
A121
at the vertex u 2 . Ifwe choose k l Z l as the internal momentum k (the integration variable), we then have (5.1.14) k = -k122. Equations (5.1.13) and (5.1.14) imply that a canonical choice of the variables Q121, Q122 o f G is (5.1.15) Qizi = k + h Q 1 2 2 = - k + &I=
k121
For the subdiagram g in Fig. 5.6b, with external vertices u1 and u2, let q 1 = 4, q2 = -4. At the vertex u l ,
Example 5.2:
4 = q211
+ 4212 + q 2 1 3 ,
(5.1.16)
or u2
- u1
(5.1.17)
= 34, 1
i.e., 1 = 132, 3.
(5.1.18)
+ k 2 1 2 + k 2 1 3 = 0.
(5.1.19)
= $4,
q21/
At the vertex u1 we also have k211
Let k211 = k, and k Z l 2 = k, be the integration variables; then - k, - k2, and a canonical choice of variables would be
=
kz + 34. (5.1.20) with u2 as the only internal Example 5.3: Consider the graph in Fig. 5 . 6 ~ vertex. Let q 1 = -q and q3 = q. At ul, Q211
=
ki
+ 449
Q z i z = kz
+ iq,
k213
- 4 = q211
Q213
=
-k1
-
+ q212,
or u2
- u1
1
= -24.
(5.1.21)
5
86
The Subtraction Formalism
At u2 (with q2 = 0, since u2 is an internal vertex),
+ q 3 2 i + q 3 2 2 = 2(Ui - u2) + 2(U3 - U2),
0 = qi21 -k q122
(5-1.22)
which together (5.1.21) implies u3
1 - U 2 = -rq.
That is, (5.1.23) Let k121 = k l and k231 = k2 be the integration variables. Using the conservation laws at u1 and u 3 , k211
+ k212
= 0,
k231
f
k232 = 0,
(5.1.24)
we obtain k l Z 2 = - k,, k232 = - k2. Therefore acanonical choice of variables is
Example 5.4: For the graph in Fig. 5.6d, u1 and u3 are defined to be its external vertices, u2 its internal vertex, and hence q 2 = 0. At the vertices u l , u 2 , and u 3 , respectively,
(5.1.28) With the constraint q1 + q 3 = 0, we have at our disposal only two independent equations, and we choose (5.1.26) and (5.1.28) with q1 = q and q3 = -q and (ul - u2), (ul - u 3 ) as two independent differences. These lead to
+ 2(Ui - U3), - U2) + 3 ( U 1 - Uj),
-9 = 3(U1 - U2) -4 =
-(U1
(5.1.29) (5.1.30)
5.1 Basic DeJinitions and Examples
87
where in writing (5.1.30) we have used the identity u2 - u3 = (u2 - ul) - u3). Equations (5.1.29) and (5.1.30) lead to the unique solution
+
(ul
u1 - u2 = - n 1q , u1 - u3 = - n4q , or
q23
3
= -nq-
For the internal variables we may choose k121 = k1,
k122 = k2,
k123
=
k3,
k131
(5.1.32)
= kq,
corresponding to the 16 integration variables. For the remaining internal variables, we have at u2 and u 3 , respectively, k121
+ k122 + k123 + k32 = 0,
k131
+
k132
leading to k32 = -k, - k2 - k3 and k132 = -k, A canonical choice of the variables is then Qi2i = ki
+
k23
-
k2 - k3
-i h
Q122
=
k2
-
frq,
Q131
=
k4
- &4,
Ql23
=
k3
Q132
=
-k1
-
- k2 - k 3
- k 4 - A114,
Q32
(5.1.33)
= 0, -
k4.
- k2 - k 3
= -kl
+ Aq. (5.1.34)
Example 5.5: Now we consider the classic example of the graph given in Fig. 5.6e, where the external vertices are u1 and u4. At the vertices ul, u2, u 3 , and u4 we have, respectively, 41
=
q21
= 423
+ q31,
+ q32 + q 4 2 r q 4 = q 3 4 + q24r = 412
+ q43 + q 1 3 ,
(5.1.35)
or 41
=
(u2
-
u1)
+ (u3 - Ul),
0 = (u1 - u2)
0 = (u2 - u 3 ) + (u4 - u 3 )
+ (u1
-
+
(u3
+ ( u 4 - %), q4
= (u3
- u4)
- u2)
+ (u2 - u4).
(5.1.36)
u3),
+
We also have q1 q4 = 0. Let q1 = -4, q4 = q, and choose the first three equations in (5.1.36) with u1 - u 3 , u2 - u 3 , and u4 - u3 as the three independent differences. These equations lead uniquely to UI
- ~3
=
1
Tq,
1.42
- ~3
= 0,
- ~3
= -fq,
fq,
q34
144
(5.1.37)
i.e., q21
= -84,
431 =
-34,
q32
= 0,
q24 =
= fq.
(5.1.38)
The Subtraction Formalism
5
88
For the internal variables we may choose k24 = kl and k 1 3 = k2, corresponding to the eight integration variables. At the vertex u1 we have kll + k31 = 0 or k2, = k2. At the vertex u2 we have k12 k32 k42 = 0 or k32 = kl + k2. Finally at the vertex u3 we have k23 + k13 + k,, = 0 or k43 = k,. Therefore a canonical choice of variables is
+
Q12
=
-k2
Q32 = kl
+
Qi3 =
k2 + 34,
Q34 - -ki + h ?
+ k2,
+
Q24 = kl
+ &?.
(5.1.39)
Since the internal variables kij,may be chosen as any consistent solution of (5.1.1l), another choice of canonical variables Qij is Q12
= &I
+ &2 + iq,
Q32 = ikl -
Q24 =
ik29
Q13
=
Q34
=
- #k2 + iq, -gkl
- $k2
+ 94.
gk1 + 3k2
+
Here we have chosen k12 = &kl 3k2) and k34 = -&3kl coefficient matrix &-; -:) is nonsingular.
+ $4,
(5.1.40)
+ k2), and the
We finally consider another classic example with the graph given in Fig. 5.6f with u1 and u4 as external vertices. At the vertices ul,u2,u 3 , and u4 we have, respectively,
Example 5.6:
0 = q12
+ q322 + q3213
(5.1.41)
Let q1 = q, q4 = -4; then (5.1.41) gives
as only three of the former equations are independent. We choose u1- u2, u1 - u4, u2 - u3 as the three independent differences. From (5.1.42), rewritten in terms of these differences, we obtain
+
- 4 = (u1 - u2) (u1 - u4), - 4 = 2(Ul - u4) - (u1 - U3)Y
0 = (241 - u2) - 2(u2 - UJ),
2 5 giving the unique solution u1 - u3 = -34, u1 - u2 = -74, u1 - u4 = -7q, u2 - u3 = -$q, u4 - u3 = fq. For the internal variables we have
k21 + k41 = 0,
~ 1 :
(5.1.43)
5.1 Basic Dejinitions and Examples
89
A consistent solution of (5.1.43) is k 1 2 = $kl - :k2 = -k14, k231 = $kl - $ k 2 , k232 = -$k1 - $k2, k 3 , = +k, - 9 k 2 , where k l and k 2 denote the eight integration variables. Therefore a canonical choice of the variables Q i j f is Q12
= fk1
- $k, - $4,
Q231 = Qk1 - 4k2 - $4, Q34
5.1.3
=
Qi4 =
Q232
=
-Ski
+ f k 2 - $4,
-qk1 - $k, - 34, (5.1.44)
- $k2 - 34.
Canonical Decomposition of the of Subdiagramsy c G
Qij,
Let G be a proper and connected graph. Its variables Qijr have been defined through (51.3)-(5.1.11), where {ku} = k and {qrjl}= 4" are its sets of the internal and external variables. It is convenient to rewrite Q 1Jf . . = kG. IJf
(5.1.45)
+ qEf,
with a superscript G to emphasize that the kzl and the qE1are the internal and the external variables of the graph G . We may also rewrite { k $ } = k" and {&} = Once the sets k" and 4" have been constructed, we proceed to construct the sets ke and 4 of the internal and external variables of any proper subdiagram g of G. The construction of the sets k" and will be important when carrying out the subtractions of renormalization (Sections 5.1.4 and 5.2). Suppose first that g is a proper and connected subdiagram of G. Let {Qijf}" = Q" and {QijI}O= QQbe the sets of variables of G and g, respectively. If Qijl E QB,then obviously the same QijlE Q". For any Q i j fE QB, we write
4".
4"
Qiji
= kfjf
+
qfjf,
(5.1.46)
and since the kSf and the q:f may be written as linear combinations of independent variables in k" and q", respectively, we expect from (5.1.45) and (5.1.46) that, in general, we may express kfjl and qfjl as functions (linear combinations) of independent variables in k" and 4"; i.e., we may write kfjf = kfjf(k",4") and qfjf = qfjf(kG,4"). We will show below how this may be done. We recall that qG E ( 4 7 ) and k" = {kEl}. The kfjl and qfjl will be defined in the same manner as the kEl and q$, once the Q i j l ,pertaining also to g, are chosen in a canonical way as described in Section 5.1.2. At each external vertex uj of g we have, as before,
c" il
Qijl
= qy(kG, q"),
(5.1.47)
5
90
The Subtraction Formalism
where qy(kG,qG) is the total momentum carried away from the external vertex u j of g and is, in general, a function of the elements in kG and qG.6The sum in (5.1.47) is over all i corresponding to the vertices ui of g attached to the vertex uj and over all the I corresponding to all the lines el of g attaching the vertices ui to the vertex uj in question. If uj is an internal vertex of g, then we define qj = 0, and (5.1.47) then holds for all vertices u j , external and internal, of the proper and connected subdiagram g of G. In the same way as for G, the external variables qyjl of g are chosen in a canonical way as the unique solution of the equations (5.1.48) (5.1.49)
with qfjl = wi - w j .
(5.1.50)
The sum overj, in particular, in (5.1.49) corresponds to all the vertices u j of g. If g has # V Gvertices, then (5.1.48)-(5.1.50) provide # Y e- 1 independent equations for the # Y e- 1 independent differences wi - w j . Once the Qijlare determined in a canonical way, as described in Section 5.1.2, and the qfjl uniquely determined from (5.1.48)-(5.1.50), then the kf,, are also uniquely determined from (5.1.46) by (5.1.51)
We now show the important fact that the kfjl are linear combinations of the kEl only, and hence are independent of the qy (and also of the q$). To this end set the kZ1in (5.1.48)-(5.1.50) equal to zero. We then have
c"q?jXO,qG> = qj(0,
(5.1.52)
il
(5.1.53)
4fj1(0,q G ) = w; - W J ,
(5.1.54)
where w; - wJ = wi - wjIkfi,=,,.Again these equations determine the qfjr(O,q'), i.e., they determine the differences w: - w;, uniquely. However, when the kEl are set equal to zero, we have from (5.1.45) Qijl = q$, and from
' The functions are, actually, in the simple forms as linear combinations of elements from kC and .'4
5 . 1 Basic Definitions and Examples
91
(5.1.47), (5.1.49), and (5.1.10), we have, for i, j , and 1 pertaining to the subdiagram g,
c" 4;r
= qyo, qG),
(5.1.55)
il
(7
4%) = 0,
(5.1.56)
qc1Jl = u!1 - u'. J 3
(5.1.57)
where u; - u; = ui - u ~ ( ~ Rin, =(5.1.10). ~ Therefore q$, with i , j , and 1 pertaining to the subdiagram g, satisfy the same equations as those of qfjl in (5.1S2)-(5.1.54). Since (5.1.52)-(5.1.54) and (5.1.55)-(5.1.57), respectively, determine the # Y e- 1 independent differences w; - w> and u; - u; uniquely, w!1 - w'. = u! - u'. J l J '
(5.1.58)
qfjL0,qG> = q;l'
(5.1.59)
or for i, j , and 1 pertaining to the subdiagram g c G. Finally, we use (5.1.51) and (5.1.59) to obtain kfjl(O,
4')
= q$l - qfjl(O,qG) = 0,
(5.1.60)
kfjl = kfjl(kG).
(5.1.61)
and hence we conclude that Therefore the kfjl depend only on the elements in kG, and are independent of the elements in qG (and also in p). Therefore for any proper and connected subdiagram g c G, we may summarize by saying that the internal kfjl and the external qfjl variables of g are uniquely determined by:
c" qfjl(kG,qG) = c" il
C" (2
(5.1.62)
4')) = 0,
(5.1.63)
4fj1(kG, q G ) = wi - w j ,
(5.1.64)
qfjI(kG9
j
Qijl,
il
il
(5.1.65) kfjl(kG) = Qijl - qfjr(kG,qG), once the Qijlare determined in a canonical way for the graph G as described in Section 5.1.2. The decomposition Qijl = kfjl qfjl in (5.1.46) will be called a canonical decomposition of the Qijrin reference to the subdiagram g c G,
+
92
5
The Subtraction Formalism
with i , j , and 1 pertaining to the subdiagram g. We note that (5.1.62), (5.1.63), and (5.1.65) imply, by analogy with (5.1.11) for G,
c"kg. = 0, IJI
c"kfjf = 0
(5.1.66)
if
ijf
at each vertex uj of g for the latter equation. Again (5.1.66) implies that L(g) = # L P - # V g+ 1 of the kfjf may be independent. Since the k;! are linear combinations of the integration variables [see (5.1.4)], it follows from (5.1.61) that the k7jf also are linear combinations of the integration variables only. Finally, if we set {q?} = q8, where q? is the total momentum carried away from the external vertex uj of g [see (5.1.47) and (5.1.48)], then from (5.1.48)-(5.1.50), we may write the qfjf as linear combinations of the 48, i.e., @jf
=
djl(qg).
In general, if g is a proper and not necessarily connected subdiagram of g, then we apply the above construction for each of the connected parts of g. Similarly, if g' is a proper subdiagram of g (g' c g), then we carry out the = kyj! q f j f , canonical decomposition of the Q i j f in the form Q i j f = kfjf + and repeat the above analysis word for word to determine the internal k:jf and external q$ variables uniquely by equations of the form (5.1.62)(5.1.65) with G and g replaced by g and g', respectively, in the latter, and thus obtaining, in particular, k$ = k$(ke), (5.1.67)
+
q$ = q$(kg, @).
(5.1.68)
Also if we denote by 47' the total momentum carried away from the external vertex uj of g and set {q?'} = qg', then we may also write = q$(qg'). Qijf
We give a few examples to illustrate the canonical decomposition of the in reference to a proper subdiagram g c G.
Example 5.7: Consider the subdiagram g2 in Fig. 5.7a as a subdiagram of the graph of Fig. 5.6b. From (5.1.62), at the vertex u1 qYii
+ qYi2 = Q z i i + Q Z I Z ,
and hence from (5.1.20) and (5.1.64), 2 ( ~ 2- w , ) = k1 or qyij
= $(ki
(5.1.69)
+ kz + $4
+ k2) + $4,
1 = I, 2.
(5.1.70)
From (5.1.65) and (5.1.20), k y i i = ki
+ i q - #k1 + k2) - 34 = #kl
and
- kz) (5.1.7 1)
kY12
= i(k2 -
k1).
5.1 Basic Definitions and Examples
93
92
(cl Fig. 5.7 Parts (a), (b), (c), and (d) represent subdiagrams of the graphs in Figs. 6b, 6c, 6e and 6f. respectively.
As shown in general in (5.1.61), we see that the kQ, are independent of qG, i.e., of q. Similarly, for g 1 in Fig. 5.7a, &'11
=
-ik, + f q ,
ky12
=
k2
+ ik1,
(5.1.73)
k913
=
-ik1 - k 2 .
(5.1.74)
1 = 2, 3,
(5.1.72)
For the subdiagrams g 1 and g2 in Fig. 5.7b as subdiagrams of the graph of Fig. 5.6c, we have at the vertices u3 and u2, respectively,
Example 5.8:
482'31 4Y21
+ q82'32 = Q 2 3 l + Q 2 3 2 + qY22 = Q i 2 1 + Q i 2 2
= q,
(5.1.75)
=
(5.1.76)
from (5.1.25), or (5.1.77) (5.1.78) From (5.1.65) and (5.1.29, we then obtain, (5.1.79) Example 5.9: We consider the subdiagram g , in Fig. 5 . 7 ~ as a subdiagram of the graph in Fig. 5.6e. We may, for example, choose the canonical variables Qijl as given in (5.1.40). At the vertices u2 and u3 we have, respectively, qi'2
+ 4'2
= Q42
+ Q32,
qsqi
+ ~82'3= Q 4 3 + Q239
The Subtraction Formalism
5
94 or ~4
+ + ~2
- ~2
- ~2
~3
= -ik1
- Q k 2 - fq,
(5.1.80) - ~3 - ~3 = ik1 + i k 2 - $4. Choosing w2- w 3and w4 - w 3 as the independent differences, we obtain ~4
w4 - w 3 + 2(w3 - w2)= -ikl - ik2 - )q, wq
- ~3 - (
-3 ~
~
2 = ) $k1
+ g k 2 - $q,
leading to
+
qy4 = k'- 8 1 - i k 2 fq, 482'4= ik1 Finally, for the internal variables of gl, kei 3 4 - Q 3 4 - qy4 =
-kl,
k4k
= Q24
+ ikz + 49.
(5.1.81)
- q94 = kl.
(5.1.82)
- i k 2 + rq, 1
(5.1.8 3)
Similarly, for g2, 4'52
= Ski
+
481j =
$k2,
= -k2,
k'52
k43 =
-k2.
(5.1.84)
The reader may wish to work out the canonical decomposition of the in reference to the subdiagrams g 1 and g2 with the canonical choice in (5.1.39).
Qijr
Example 5.10: Now we consider the classic example of the graph in Fig. 5.6f with proper subdiagrams g1 and g2 as depicted in Fig. 5.7d. At vertex v2 of subdiagram gl, qY21
+ q Y 2 2 = Q321 + Q322,
or from (5.1.44) and (5.1.64), 48321 = - f k 1
+ $k2 + $4,
1 = 1,2.
(5.1.8 5 )
From (5.1.65) and (5.1.44), we then obtain
ky2, = -k 1 -
-kel322.
(5.1.86)
Similarly, for g2, qq24 = - f k 1
+
$k2
- +q, (5.1.87)
and
5.1 5.1.4
Basic Dejnitions and Examples
95
Taylor Operations
Let G be a proper and connected graph. Let us scale its QijIvariables in the expression for the unrenormalized integrand I, in (5.1.2) by a parameter A. If d(PijI)denotes the degree of a polynomial Pijlwith respect to A in (5.1.1), we then define, as in Chapter 2, the degree of D& in (5.1.1) by d(D$ = d ( P i j , ) - 2. Similarly, let d ( q . ) be the degree, with respect to A, of the polynomial Pj associated with the vertex v j of G.We then define the dimensionality d(G) of the graph G by
d(G) =
c“d(D$ ij1 i<j
+ c“ d ( q ) + 4L(G), (5.1.89)
i
L(G) = #YG- # V G+ 1, where L(G) coincides with the number of the independent internal momenta k,, . . .,k, and hence also with the number of independent internal variables k z , associated with G. As before, Y Gand V cdenote the set of lines and vertices of G, and #YC,# V Gdenote the number of elements in them. Similarly, if g is a proper and connected subdiagram of G, then a similar expression to (5.1.89) may be written for the dimensionality d(g) of g , with G replaced by g in the latter, i.e., in particular, with L ( g ) = #Ye - # Y e+ 1. If a proper but disconnected subdiagram g has m proper and connected parts g l , . . . ,g m , we write g = Uy= g i . The dimensionality d ( g ) of g is then given by nt
1d(gi).
d(g) =
(5.1.90)
i= 1
If lei is the unrenormalized Feynman integrand associated with the subdiagram g i , we write I , = I,, . Suppose g’ is a subdiagram of g with unrenormalized integrand I,, ; we then define the expression I,/,, by
ny’
I,
=
(5.1.91)
I,/,J,,.
In other words, I,,,, represents the unrenormalized integrand of g with I,. , corresponding to g’ $ g , replaced by unity. We use the notation I, = 1. By definition, for g’ $ g , a line belonging to g’ does not belong to g/g’. We may interpret g/g‘ diagrammatically as the subdiagram obtained from g by shrinking g’ in it to a point. A few examples of this process are given in Fig. 5.8. Accordingly, the subdiagram g’ is considered as a “vertex” of g/g‘ with which is associated the analytical expression 9,, = 1. If g‘ is a proper subdiagram with s connected parts g ; , . . .,g:, we may then write
n S
I, = I,/,’
i= 1
I,;.
(5.1.92)
5
96
43 9
The Subtraction Formalism
ds‘
(4
Fig. 5.8 The subdiagrams on the right-hand side of each subdiagram ,q is obtained by shrinking the subdiagram g‘g 9 in it to a point.
Finally, we note that if g1 and g2, g, g g2, are proper and connected subdiagrams having a certain number X of vertices in common, then if g2 in g l is shrunk to a point, these X vertices “merge” into a single vertex and we may write # Y e 1 = # V - e * / e 2 -t #Ye2- 1. Also, #Ye’= #Ye2. HenceL(g,) = L(g1/g2) + L(g,)and wenotethatd(g,) = d(gl/g2) d(g2). If g’ is a proper subdiagram with s connected parts, then L(g‘) = L(gj) and we may write for the dimensionality of g in (5.1.92) d(g) = d(g/g’) + EL1 4g;). As before, let {qy} = qG be the set of the total external momenta carried away from the external vertices of G. We define by TG the Taylor operation on I, in the external independent momenta in yc, about the origin, up to the order d(G).If the dimensionality d ( G ) < 0, then we set T G = 0.’ Accordingly, TG I, is of the form
+ +
c:=,
To this end, note that the external variables of G in {q:l} = gG may be written as linear combinations of the external total momenta of G in (4:) = qG:q:, = qc1(qG)[see following Eq. (5.1.1 l)].
’
In general, one may carry out a Taylor operation to any order d’(G) as long as d’(G) 2 d( G). Without loss of generality, we shall take the order associated with TGto coincide with the dimensionality d(G) of G. This will also apply to the Taylor operations in reference to proper subdiagrams g c G to be discussed.
5.1 Basic Definitions and Examples
97
Similarly, if g is proper (not necessarily connected) subdiagram g of G,
TIgis defined with reference to the external independent total momenta in (47) = qg, up to the order d(g). If d(g) c 0, then we set T = 0. Again the external variables of g in {qfjf}= 4 may be written as linear combinations of the external total momenta of g in {q!} = qg [see following Eq. (5.1.66)].
We are particularly interested in the consecutive application of two or more Taylor operations as follows. Suppose g' is a proper subdiagram of g (g' S 9); then we are interested in the operation in G 5.I , . This is precisely defined in the following manner. We write I , in the form (5.1.92) and carry out the Taylor operation q,on I @ ,with reference to the external independent total momenta in { q!'} = 4,.' The expression Isle, q,I,. is of the form Ielee q.I,, = F(Q, kg',@'),
(5.1.94)
where F(Q, kg', q@')is a function of the variables in { Q G } = QB,the variables in {kfjl} = kg',and the variables in { q f } = 4,'. Before carrying out the Taylor operation q on Igleeq . I , . , we express this F as a function of the internal and external variables of g, i.e., of kg and q8.This is easy to do. We use (5.1.46) to write Qijf= kfjf q f j f .Similarly, we use (5.1.67) and (5.1.68) to write kfjl = k$(kg) and qfjf = qfjI(kg,qg),q?' = qfil(kg,q8), with the latter by analogy with (5.1.48), or directly from the expression corresponding to (5.1.47) for g': Qijr= qf, with Qijl = kfjf + qfjf,to write
+
Iff'
El'
(5.1.95)
F(Q, kg', 4,') = F(ke, 4,).
Now we may finally carry out the Taylor operation with respect to the independent components in q8 by applying it to F in (5.1.95). This gives the precise procedure of carrying out the consecutive Taylor operations and may be summarized through the following equation:
q q.1, = T,[I,/,* q.i,,] =
T,[F(Q,k", q")]
=
T,F(ke, q'),
g'
Q 9.
(5.1.96)
The meaning of the consecutive Taylor operations of the form in g2 p . . . p gn are proper subdiagrams of g, is now obvious. We work out a few examples. For the purpose of illustrations we omit the i E factor in the denominators only to simplify the notation.
qIq2... qnIg,where g1 p
Example 5.11 : diagram g1 c g
Consider the graph g of Fig. 5.6b and the proper subin Fig. 5.7a. We take 3
I, =
fl ( Q : ~ I+ p 2 ) - ' ,
I= 1
3
I,, =
n
1=2
(Q?2i
+P~)-',
The Subtraction Formalism
5
98 where d(g) = 2 and d(gl) = 0. Then 1
(5.1.97) where the kt\l are given by (5.1.73) and (5.1.74). Also,
where we write [see (5.1.20)] Q l z l = k t z l + q!21, with k!,, = - k l , qf = -f q . qIIBlis independent of qfjl.We scale 481 in Zelel by 1 and carry out the Taylor operation TB, with respect to 1,up to the order d(g) = 2, and then set 1 = 1, to obtain TBTB,l, =
""11)2
+ p23-l - C2q?21k~21+ (4421)21c(kP21)2+ CL2 1- 2 3
n
+ (2q!21k:21)2c(k421)2 + p 2 1 - 7 1 = 2 C(~!'ZJ~ + p21- l ,
(5.1.99)
and the latter may be expressed as a function of the integration variables k l , k 2 and the external momentum q [see (5.1.72)-(5.1.74) and (5.1.20)]. The expression for TB q,Z, may be similarly obtained. Example 5.12: Consider the graph g of Fig. 5.6~and the proper subdiagrams gl, g2 c g of Fig. 5.7b. For simplicity, we take 2
2
and hence
We have 2
wherethekp31andkqilaregivenin(5.1.79). WemaywriteQ,,, = kg,, + q721, wherek!,, = k l = -kq22andq!21 = $q,l = 1,2 [see(5.1.25)].Accordingly, 2
2
and may be written as a function of the integration variables kl, k 2 , and the external momentum q.
5.1 Basic Definitions and Examples
99
Similarly,
also,
TI, =
2
2
[(k$313)2 1'= 1
+ p'1-l fl [ ( k f 2 1 ) 2+ p 2 ] - l ,
(5.1.105)
I=1
when the kfjl are given by (5.1.25) and the k!$l are given in (5.1.79).
Now we consider a slightly more complicated example. Consider the graph in Fig. 5.9a with external vertices u1 and u2 with q 1 + q2 = 0, q1 = q, and hence q2 = -4. g1 in Fig. 5.9b is a proper subdiagram of g. A canonical choice of the variables Qijlwith respect to the graph g is
Example 5.13:
The canonical decomposition of the Qijlwith respect to the subdiagram g1 is then Qijl= kfjl + qfj1,with qfhl k!hi
= &k1
=
+ k, + +
k3)
&k2
I
- $4,
=
1,2,3,
(5.1.107)
k3),
(5.1.108)
where k , , k , , and k 3 are integration variables. The dimensionalities of g and g1 are, respectively, d(g) = 4 and d(gl) = 2, where 3
4
1, =
fl (Q;n + p 2 ) -
I= 1
'9
I,, =
fl
I=1
9
91
(a)
(b)
(Qf2,
+ ~ ' 1 -',
(5.1.109)
Fig. 5.9 A graph g involving scalar particles for Example 5.13. g, is a proper subdiagram ofg with nonnegative dimensionality.
100
5
The Subtraction Formalism
and the expression for I,, follows from that of I,. A straightforward application of TIon I,, gives
(5.1.110)
where (5.1.111)
+
We also have Ielel = ( Q f 2 4 p2)- with Q124 as given in (5.1.106). In order 1 , have to write the q t i , as to apply the Taylor operation T on 1 , 1 8 1 ~ , 1 ,we functions of the q[!2 1 . We note that 481'21 =
Q121
- 4'21= q421 + ( 4 2 1 - k4'zA
(5.1.112)
with 48121
(5.1.113)
= - 31%
where k!21, are functions of k , , k , , and k 3 as given by (5.1.106) and (5.1.108). Hence we may express Z e i e l ~ , Z , as a function of k , , k,, k , and the external variables qqZ1of g. From (5.1.112) we may express the q81\l in the numerators of (5.1.110) in terms of qfjl and kf2,, k!',,. We may also write Q124 = k424 qq24 in Zelgl = (Qf24 p Z ) - ' , and hence we may finally l to the fourth order in q in carry out the Taylor operation T on I , l e l ~ , I ,up a straightforward, though tedious, manner. Similar examples may be also given when different types of propagators and different masses are involved, and they may be treated in the same manner. Let g and g' be two proper and connected subdiagrams with g' & g. For future reference we show that the actual dependence of the qy and the qfjl on the kfjI is only on the kfjI corresponding to the lines of the subdiagram g'/g, in the notation (5.1.91), i.e., of g' but not of g. To this end we consider the expression (5.1.47) with G in it replaced by g', which gives
+
+
2 i1
Qijr
=
qf(ke', @'I,
(5.1.114)
at each external vertex u j of g. We also write Q111 .. =
k 5. 1.
+
2
(5.1.115)
5.1 Basic Dejinitions and Examples
101
and use (5.1.7), with G replaced by g', to obtain
2'Q . . = qe' J
111
il
or
c" + Cfe"' Qi,I
Qijl
il
il
= q!',
(5.1.116)
where the second term in (5.1.116) is dejined as a sum over all the i and 1 corresponding to all the vertices ui and lines ti,belonging to g' but nor to g, such that the lines 8, join the vertices ui to the external vertex uj of g. From (5.1.114), (5.1.115), and (5.1.116), we then have (5.1.117)
Also, the qfjl may be written as linear combinations of the q! [see following Eqs. (5.1.66) and (5.1.11)]. Accordingly, we see from (5.1.117) that the q! and the qQare linear combinations of the k?jl in g'/g, i.e., q?(ke', @'),qfjr(ke',qe'), with kk, in g'/g. This result will be quite useful later on. The same conclusion may be reached if g' p g, with g' and g proper but not necessarily connected subdiagrams, by replacing g' in (5.1.117) in turn by each of its connected components gi and by replacing g by those connected components of g falling in each g:. Finally let %g' be the set of vertices in Y e 'but not in Ye.Then for uj
E
= 0.
(5.1.1 18)
il
Also, we denote by Ke'the set of vertices in Y esuch that if uj E ce', then we may find at least one line in g'/g that joins the vertex v j . By summing over all the uj E Ke',we have by momentum conservation, (5.1.119)
The
equations in (5.1.118) and (5.1.1 19) together with the constraint' (5.1.120)
* Note that, by definition, the vertices ui and the lines/, joining these vertices to the vertices in +f, do not belong to g. Theconstraint in(5.1.120)simplymeansthat oneofthe d Y '8"gexpressionsontheleft-hand sides of (5.1.1 18)-(5.1.119) is a linear combination of the remaining # V''g - 1 expressions.
102
then imply that may be independent.
The Subtraction Formalism
5
- 1) = L(g'/g) of the #Ye'/s kfi, in g'/g
-
5.2 THE SUBTRACTION SCHEME
Let fG be the unrenormalized Feynman integrand associated with a proper and connected graph G.We recall that a subdiagram is proper if it is amputated and its number of connected parts does not increase upon cutting any one of its lines. We define the renormalized Feynman integrand R, associated with G,involving subtractions as follows: r
1
(5.2.1)
where the sum is over all nonempty sets D such that (i) If g ED, then g is a proper (but not necessarily connected) subdiagram of G with d(g) > 0. If d(G) 2 0, then one of the elements of D may be G itself. (ii) If gl, g2 E D , then either g1 $ g2 or g2 $ gl. If g1 $ g 2 , then the . - ., as defined ordering of the Taylor operations in (5.2.1) is as . . . q2. . in Section 5.1.4. Remarks
1. The Taylor operations are directly applied to the integrand fG (directly in momentum space), and hence no questions of divergences arise in (5.2.1). The sets D in (5.2.1) will be called renormalization sets. 2. In Corollary 5.4.1 we shall prove that in obtaining thefinal expression for R, one may restrict the summation in (5.2.1) over renormalization sets D such that for each connected part gi of a g E D we have d(g,) 2 0, as the other D sets will not contribute to the sum in (5.2.1), and hence to the final expression for R. The more general form for the D sets given will be useful in providing the convergence proof of renormalization in Section 5.3.
The structure in (5.2.1) is very simple, as will be seen in the examples to follow. The simplicity of this structure will also be reflected in the convergence proof (Section 5.3) of the subtraction scheme and in later work in the appendix.
5.2 The Subtraction Scheme
103
The renormalized Feynman amplitude associated with G is then d,(P, P) =
lR4" w, 4, dK
K , P,
(5.2.2)
where K = (@,. . . , k i ) and P = ( p y , . . . , p i ) denote, respectively, the integration variables (components of the internal momenta) and the components of the independent external momenta associated with G. The absolute convergence of (5.2.2) will be given in Section 5.3 for E > 0. The in the sense of distributions then follows existence of the limit E + +O of from the work in Chapter 4. We give a few examples concerning our subtraction scheme (5.2.1) that HseD (- q)]. also illustrate the simplicity of the structure of [l + ID For Example 5.11, the graph g (Fig. 5.6b) and the proper subdiagrams g1 and g2 (Fig. 5.7a) all have nonnegative dimensionalities; hence the renormalization sets D are
Example 5.14:
(5.2.3) Note that { g l , g2} is not a D set since neither g 1 $ g2 nor g2 $ g l . The renormalized integrand is then given by R = (1 - q)[1
-
q, - 7J18.
(5.2.4)
Example 5.15: For Example 5.12, the graph g (Fig. 5 . 6 ~ and ) the proper subdiagrams g and g2 (Fig. 5.7b) all have nonnegative dimensionalities. Therefore the renormalization sets D are as given in (5.2.3) and the renormalized integrand is of the form (5.2.4). Note again that {gl, g2} is not a D set.
For the graph g of Fig. 5.10a, the dotted lines denote scalar particles and the solid lines denote spin-4 particles with, for example, a &,b4 interaction. The degrees of the corresponding propagators are, respectively, -2 and - 1 . g , g l , . . . , g 5 denote all proper subdiagrams (Figs. 5.10a-f) of g with nonnegative dimensionalities. Accordingly, the renormalization D sets are
Example 5.16:
5
104
The Subtraction Formalism
9
91
92
93
94
(a)
(b)
(C)
(d)
(4
95 = 91 u 92 (f)
Fig. 5.10 A graph y in (a) where the dotted lines denote scalar particles and the solid lines denote spin-4 particles. The degree of the corresponding propagators are, respectively, - 2 and - 1. The subdiagrams y, , . . . , ys and g itself denote the proper subdiagrams of g with nonnegative dimensionalities.
The renormalized integrand associated with g is then - q1v - T,C1 - TI - T2Jb
R = c1
-
TI] - T,,c1
-
q21- q s c l - q1- q21 (5.2.6)
It is easy to write down the expression (5.2.6) immediately, by inspection, without writing out first the sets in (5.2.5) in detail. For example, the proper subdiagrams of g, not equal to g, with nonnegative dimensionalities are g 3 ,g4, g5, gl, and g2. Hence in the curly brackets we have
We may now consider each term in (5.2.7). For the q,,the only proper subdiagram of g3, not equal to g3, with the nonnegative dimensionality is gl, accordingly the expression, for the square brackets multiplying T 3 (from the right!) is [l - q,].The 1 factor in [l - TI] occurs, of course, because {g3,gl} as well as {g3} are renormalization sets. Here we note that g1 does not contain a proper subdiagram, not equal to gl, with nonnegative dimensionality; hence we may write [l - T,,[.]] = [1 - TI]. Otherwise, we would have repeated the above procedure for filling in the square brackets multiplying T,, in the same manner as for TJ.1. The square brackets in T,,[.] and T,[.] are filled in the same manner. The square brackets multiplying &,[.I and TJ.1 in (5.2.7) are simply replaced by 1 since g1 and g2 do not contain proper subdiagrams, not equal to g1 and g 2 , respectively, with nonnegative dimensionalities, as just mentioned. The method of writing the expressions for R, as exemplified in (5.2.4) and (5.2.6), is thus straightforward. Example 5.17: We explicitly work out the Taylor operations in (5.2.1) for Example 5.12 corresponding to the graph in Fig. 5 . 6 ~to construct R. According to (5.2.4), corresponding to Example 5.15,
R = (1 - TIC1 -
TI - 7J18'
(5.2.8)
5.2
The Subtraction Scheme
105
L
L
with [see (5.1.25), (5.1.79)]
kY3, = k 2 = -kel2 3 2 , Also, from (5.2.10), kq21
= k1 = -k'
122,
k?',, = k 1 - -ke21 2 2 .
(5.2.1 1)
-ke 2 3 2 .
(5.2.12)
k92-31
= k2 =
Putting (5.2.9)-(5.2.12) in (5.2.8), we obtain, after some simplification,
"
+ P2)1 R = C&q" - ( k 1 d 2 + 2 2 2 + P2)" 1 6 q - (k2q)2 + tq2(k$ (k: + P2)2(k: + P 1 ( k l + + P2)(k:- + P2)(k:+ + P 2 ) ( K + P2)' (5.2.13) where we have defined
It is instructive at this stage for the reader to compare the degrees of R with respect to k,, k , and ( k l , k,) with the corresponding ones of the unrenormalized integrand I, defined in (5.2.9). A recursion formula for determining the renormalized integrand R in (5.2.1) may be also given that will be useful later on. To this end let DG,
106
The Subtraction Formalism
5
denote a renormalization set D such that the largest subdiagram in D G * is G', i.e., if g E D G , ,then necessarily y c G . Let D0 = 0 denote the empty set and define (- T,) = 1. Then (5.2.1) may be rewritten in the form"
n (-q)IG,
1 1
R=
(5.2.15)
0 c G ' c G DG, BEDG,
where CDG, denotes a summation over all renormalization sets D G * with largest subdiagram G' in D G , , and 10cG8cG denotes a summation over all proper (but not necessarily connected) subdiagrams of G with d(G') 2 0 (for G' # 0 )and, by definition, also includes the term with G' = 0.The expression (5.2.15) may be rewritten in the equivalent form
cn
n
1 1
(-q)IG + (-q)IG, (5.2*16) 0cC'gG Dco g € D G , D c BEDG where is the sum over all renormalization sets having G as their largest subdiagram (graph), and the corresponding second term in (5.2.16) reduces to zero if d(G) < 0 since =
xDc
1
n
(-Tg)lG
=
n
1
(-TG)
Do gEDc
(-T)IG,
(5.2.17)
0 c G ' S G Do* g E D c ,
and TG = 0 for the case d(G) < 0, by definition. Using (5.2.16) and (5.2.17), we may rewrite (5.2.15):
[
1+
R=(l-TG)
1 1
n(-q)]IG,
(5.2.18)
0 g G ' g G k ,~ E D G ,
where we have used the definition (- TO) = 1. We define the expression MG,
=
1
n
(5.2.19)
(-q)IG,.
Dc, ~ E D G *
The latter may be rewritten MG'
= (-TG')
1
n (-q)IG'
Do* gEDcs gSG'
[ +1 n [ + Do,
= (-TG,)
~EDc,
02ggG'
1,.
0 S G" !$ G'
1
( - q )IG,
=(-TG') 1
IG*,G**1
n
(-T,)I,s*],
(5.2.20)
Do,* ~ E D c -
where we have used in the process of writing (5.2.20) the relation ZG, = I G , , G * , I G , ,for G" G'. Again using the definition (5.2.19) and noting that G' l o Recall that in this book the symbol c in 0 t C' c G may include equality as well. We use the symbol g to exclude equality. G' = 0 means no lines and no vertices.
5.3 Convergence of the Subtraction Scheme
107
is an arbitrary proper subdiagram of G (with d(G') 2 0), we obtain from the last equality in (5.2.20) (5.2.21) Therefore from (5.2.18) and the definition (5.2.19) we have the following equivalent expression for R :
[
R = (1 - TG) I ,
+
1
IGIG'MG, 9
0$G'$G
(5.2.22)
where TG = 0 if d ( G ) < 0, and M c , is defined recursively by (5.2.21).
5.3
CONVERGENCE OF THE SUBTRACTION SCHEME
The renormalized Feynman amplitude associated with a proper and connected graph G is from (5.2.2) given by d , ( P , p ) = IR4"dKR(P, K , p, E l ,
E
> 0,
(5.3.1)
where R is given by (5.2.1) with the unrenormalized integrand I , having the form (5.1.2). Obviously, R has the very general structure given (2.2.3). Let RE be the Euclidean version ofR by replacing the Minkowski metric grv by the Euclidean metric qcv and setting E = 0 in the latter as defined in (2.2.83). As in (2.2.84) we define dE(p,
p ) = IR4-dKRE(P, K ,
(5.3.2)
In Chapter 2 we have established, in particular, that RE and R , with > 0 in the latter, belong to class B4n+4m+p(I). In this section we prove the absolute convergence of the integral (5.3.2) and the absolute convergence of (5.3.1) for E > 0. The limit E + +O of , d , ( P , p) may be then seen to exist, in the sense of distributions, from Chapter 4. In Section 5.3.1 some basic properties of Taylor operations and the remainder terms are obtained in view of applications to subtracted-out Feynman integrands. In Section 5.3.2 a basic grouping of the Taylor operations is carried out that is indispensable for the convergence proof that is finally completed in Section 5.3.3 by making use of the power-counting theorem established in Chapter 3 coupled to the fact that R Eand R belong to class Bln + 4 m + p ( I ). E
5 The Subtraction Formalism
108 5.3.1
Some Properties of Taylor Operations and Their Remainders
(c)
be a function that is diflerentiable an arbitrary Lemma 5.3.1 I ' : Let f number of times in [ E R'. Define recursively
(5.3.3) (5.3.4) then n- 1
(5.3.5) lor arbitrary positive integer n.
We prove (5.3.5) by induction. For n = 1, f1(5) = [f (0- f(O)]/i, definition, i.e.,
f(0 = f(0) + T f I ( 0 ,
by
(5.3.6)
which establishes(5.3.5)for n = 1. Suppose (5.3.5)is true for some n = k > 1, I.e., k- 1
lkh(0. By definition,f,+ ]([)
(5.3.7)
=
(5.3.8) Substituting (5.3.8)in (5.3.7), we obtain
(5.3.9) i.e., the lemma is also true for n lemma by induction. Lemma 5.3.2 :
k
+ 1. This completes the proof of the
Let L
f(1,C, p ) =
=
n [(Ak, + 5qj)' +
j= 1
+
p; - i & { ( l k j {qj)'
+ pf}]-'.
(5.3.10)
' ' The classic formula in this lemma is known in the literature as Newton'sdivided difference intcrpolating formula (cl Isaacson and Keller. 1966).
5.3
Convergence of the Subtraction Scheme
109
As in Lemma 5.3.1, we dejne recursively
i,PI
fn(A
= Cfn- l(A
i,PI - f n -
l
a
(5.3.1 1)
0, P)I/i*
Dejne a set of coejicients { F,(A p)} by
n [(Akj)' L
+ p;
- iE{(Acj)'
+ pf}]
j= 1
n L
-
[(Lkj
+ iqj)' + P;
- i&{(lkj+ iqj)'
+ pf}]
j= 1
(5.3.12) Then
for
[1 - T i ] f(1, P )
=
1 In < 2L (5.3.13a)
(5.3.14)
id' I & + I(A C, P),
+
where jd+l(,4, (, p ) is given recursively by (5.3.13) with n = d 1, and from the latter, or by inspection, it is the ratio of two polynomials in 1,c, p and is of the form
sd+ l where
a
i,PI
= P&(i4.4, ilk, PIG -
'(h ilk9 P),
(5.3.15)
is u polynomial in its arguments in (5.3.15) and E and L
~ ( i q ak, , P) =
n
[(Akj
j= 1
x [(Akj)'
+ iqj)' + P;
- iE{(lkj
+ Cqj>' + P ~ I I
+ p; - iE{(;lkj)2 + pf}]@',
where pi are some strictly positive integers.
(5.3.16)
110
5
The Subtraction Formalism
To prove (5.3.13a) we proceed by induction. For n
fl(A 5, PI =
=
1, from (5.3.11),
f(4 6PI - f(J,0, PI 5
2L
=
c-3
P)f(4 0, PI
1 r"- 1Fo(4PI,
(5.3.17)
a= 1
where we have used (5.3.12). Equation (5.3.17) coincides with (5.3.13a) for n = 1. Now suppose that (5.3.13a) is true for some n = k with 1 < k < 2L - 1, i.e., k
5, P ) = f(n,0, P )
Ch-a(A
o= 1
C, p ) p a ( A P ) 2L
+ f(A 0, ~ ) f ( n ,5, P ) a = 1 P - k F a ( A P)* k+ 1
+ 1, we obtain L P ) = C f k ( A 5, p) - .MA,0, p)I/L
(5.3.18)
Using the definition (5.3.1 1) with n = k
fk+ l@,
(5.3.19)
and hence from (5.3.18),
or k+ 1
+ !(A
c
2L
0, ~ ) f ( 5,k P )
P - ( k +' ) F a ( A p),
(5.3.21)
a=k+2
+
where we have used definition (5.3.11) again, now with n = k 1 - a, and have absorbed the first term of the second sum of (5.3.20) in the first sum; hence (5.3.21) verifies (5.3.13a) for all n < 2L. Using definition (5.3.11) and the expression (5.3.13a), we readily verify that (5.3.13b) is true for n = 2L, and by induction we easily see that (5.3.13b) is true for all n 2 2L. This completes the proof of (5.3.13).
5.3 Convergence of the Subtraction Scheme
111
It is easy to see from definition (5.3.1 1) that
+ 1, then (5.3.5) in Lemma 5.3.1 implies that
Thus if we choose n = d
d
f (2, 5, PI =
ij
1 J7. f")(4 0, PI + id+'fd+
i(J.9
i,
(5.3.23)
j=o
which in turn implies the result in (5.3.14) since the sum in (5.3.23) is the result of the Taylor operation on f (A, i,p ) up to the order d in about the origin. The final lemma in this subsection is the following: Lemma 5.3.3 :
Consider an expression of the form
where P([q, Ak, p ) is a polynomial in the elements in [q, Ik, p, and, in general, in the ( p j ) - l as well. Forall j = 1, . . . , L we assume k l # 0,for at least one Q E [0, 1,2, 31 (corresponding to each j ) , in the denominators in (5.3.24). As before, let Tf denote the Taylor operation in ( about the origin, u p to an order d 2 0. Then
degr[l - T;]H(A, i,p ) Idegr H(A, <,p ) - d - 1. 1.
1.9
(5.3.25)
5
Also, ifwe scale any subset of the masses by a parameter q, then
degr[l - T f J H ( I ,(, p ) 5 degr H(A, [, p ) - d - 1. 1.. 4
(5.3.26)
1.94.4
To prove (5.3.25) we write P(iq, Ak, P ) =
C italqaPa(Ak,
(5.3.27)
a
where'
' a
=
(aol, ..., a3,,,), q"
la1 = a,,
(qyo'
+
+ a3,,,,
.. . ( 4 P 3 " .
(5.3.28) (5.3.29)
By using the elementary property that for a differentiable function f (0 ~d
'' For 03m
+b
d
(i la1f(C)
=
Pal T ; - l a t f ( o ,
u = (aol.. . . , u3,,,) and b = (bol, . . . , b3,,,), we define a
(5.3.30)
+ b = (aol + bol, ...,
5
112
The Subtraction Formalism
+
for d 1 - la1 2 0. Let h(I, (, p ) be the sum multiplying We use the facts that degrC1
-
T : l H @ , i,
=
r,
in (5.3.33).
degr MI, p ) Idegr MI,C, p )
A
A
=
Cd+l
3 . 2
degr[l
-
5
T : I H ( I , C, p )
-
(d
+ l),
(5.3.35)
1.5
where the last equality follows from the presence of the overall factor multiplying h(A, [,p ) in (5.3.33). But (5.3.36) and hence we conclude from (5.3.35)
+
(5.3.37) degr[l - T t I H ( 1 , C, p ) I degr H ( I , l, p ) - (d l), A 1,c which is (5.3.25) in the lemma. We rewrite P(Cq, I k , p ) in (5.3.24) as (pj)-uj&[q, Ak,p), where the oj are positive integers and p(('q, I k , p ) is a polynomial in [q, Ik, p. By repeating the proof leading to (5.3.35) with p replacing P in (5.3.24) and H and h replaced by the corresponding A and h", we obtain instead of Eq. (5.3.39,
np=
degrC1 -
T:lR@, l, p ) = degr &(A, C, p ) I degr h(I, i,p ) 1.4
194
=
1.4.5
degr[l - T : ] R ( I , C, p ) - (d
+ 1)
.L4.5
I degr A. 4.5
A@,C, p ) - (d
+ 1).
(5.3.38)
5.3 Convergence of the Subtraction Scheme Upon subtractingdegr,
113
n$=
(pj)"jon both sides of this inequality, we obtain
degrCl - T;lH(A, C, p ) Idegr H(A, i,p) 2.
,
- (d
+ l),
(5.3.39)
&4.5
which completes the proof of the lemma.
5.3.2
Basic Grouping of the Taylor Operations in (5.2.1)
Let I , be the unrenormalized Feynman integrand associated with a proper and connected graph G. Let I be a 4n-dimensional subspace of ~4n+4m+p associated with the integration variables in I,. As in (2.2.21), we introduce a vector P in R4n+4m+p such that the integration variables, the components of the external independent momenta, and the masses in G may be written as some linear combinations of the components of P.Let p = LIqlr],
' ' '
qk
+ L,q,
* ' *
qk
+ + Lkqk + c, * '*
(5.3.40)
where k 5 4n and L,, L,, . . . , Lk are any k independent vectors in I . L,, L,, . . . , L,, for all r Ik I4n, span a subspace S, = {L,, L2,. . . ,L,} c I . C is a vector confined to a finite region in R4"+4m+p with pi # 0 for all i = 1, . . . , p . Since, in particular, the integration variables may be written as some linear combinations of the components of P,it follows that for any proper subdiagram g c G, the kfjl of g may be also written as some linear combinak. The latter tions of the components of P. Let r be a fixed integer in 1 I r I property then, in particular, means that a four-vector kfjl may (or may not) depend on the parameter qr in (5.3.40). For a four-vector kfjl, if at least one of its four components depends on a parameter q, in (5.3.40), then we say that the four-vector kfjl depends on q,. Otherwise (i.e., when all the four components of kfjl are independent of q,), we say that kfjl is independent of q,. For convenience, in this subsection we extend the definition of D sets in (5.2.1) to ones including (proper) subdiagrams g with strictly negative dimensionalities as well by simply setting (-T,) = 0 for d(g) < 0. This obviously does not change anything in the expression for R, and will simplify the construction to be given. Choose a set D and consider the set D u { G } obtained from D by adjoining to it the whole graph G.Obviously, if G E D,then D u { G } = D.We arrange the subdiagrams in the set D u { G } in increasing order, i.e., as {. ..,gi, gi + . . .} with gi+ 1 2 gi* As before, let r be a j x e d integer in 1 Ir Ik. Let g 2 g' be any two consecutive (proper) subdiagrams in D u { G } . We denote by c(g/g') the set of all the lines of g/g' that have the internal variables kfjl in g/g' independent
,,
5
114
The Subtraction Formalism
of q r . 1 3 To simplify the notation we write c(g/g’) = g/g‘ if all the k7jl of g/g’ are independent of q,, c(g/g‘) p g/g‘ if not all the kTjl of g/g’ are independent of q,, and c(g/g’) = @ if all the k!j, of g/g‘ are dependent on q,. Finally, by the expression c(g/g‘) 2 0it is meant that at least some (i.e., at least one) of the kfj, of g/g‘ are independent of q,. For any two consecutive subdiagram g, g’ in D u {G} with g 2 g‘, we write g/g’ = 8. From the set D u {G} we induce a set of subdiagrams by deleting, in general, some subdiagrams from D u {G} and introducing, in general, some new subdiagrams not in D u {G}. Let g be a subdiagram in the chosen set D u {G}. (i) Suppose @ $Z c ( g ) $Z 8, and dejne go = g - c ( J )as the subdiagram consisting of the lines in Y e- c(g) and, of course, of the relevant vertices as their end points. We note in particular that g‘ g go g g. By construction, all the kfjl in go/gt, where 8 = g/g‘, are dependent on q,. We shall study later the nature of the subdiagram go. For this case, i.e., with 0g c(8) 9 8, we keep g in D u {G} and we induce a set {go}. (ii) Suppose c(8) = @. Let g’, g, g“ be the consecutive subdiagrams in D u {G} with g‘ g g p g”, 8 = g/g’, jj” = g“/g. If all the k$ in 8”are independent of q,, then we eventually delete the subdiagram g from the set D u {G}, thus inducing the set {g}. Otherwise (i.e., if at least some of the kfi in ij” are dependent on q,) we keep g in D u {G}. (iii) Suppose c(8) = 8; then we keep g in D u {G}. We summarize the above process as follows: (i) @ g c(ij) g 8; keep g in D u {G) and introduce the subdiagram = g - c(8) s! g, thus inducing the set {go}. go if all the k$ in 8’’are independent of q,, then eventually (ii) c(8) = 0; delete g from D u {G} and thus induce the set {g}. c(8) = 125; if at least some of the k$ in 8’’are dependent on q,, then keep g in D u {G} and thus induce no set. (iii) c(g) = S ; keep g in D u {G) and thus induce no set. We carry out this analysis for all g in D u {G}; thus for each g, we either introduce a new subdiagram go and keep g in D u {G}, induce the set {g}, or simply keep g in D u {G} as discussed. We continue the above analysis starting from the smallest subdiagram in D u {G} until we arrive to the graph G itself. If 0g c(G) $Z C, then we introduce a subdiagram Go = G c(C) $ G and thus induce the set {Go}. However, whether c(C) = 0 or c(G) # 0,we delete the graph G from D u {G}. The set of subdiagrams obtained from D u {G}by deleting all those g in D u {G}that were eventually
’’
Recall that g/g’ denotes the subdiagram g with g’ in it shrunk to a point. The k!j, of gig’, then, mean k!j, ofg pertaining to the lines ofg/g’ but not of g’.
5.3 Convergence of the Subtraction Scheme
115
(b)
(a)
Fig. 5.1 1 The overall regions in (a) and (b) denote the proper subdiagram g. In (a) the line joining the vertex r j to the vertex I , , is supposedly an external line ofgo, which is impossible. In (b) the line joining the vertices and v j is supposedly to be an improper line of g o , which is again impossible. [xi
to be deleted as well as deleting the graph G will be called a nuclear set and will be denoted DN.By definition, G # DN.The set of subdiagrams as induced from the set D u {G} containing all those deleted subdiagrams g (including G) and those newly introduced subdiagrams go as discussed in (i) will be denoted D o . It is easy to show that all the subdiagrams in Do are proper (but not necessarily connected) subdiagrams. First, G is in Do and it is proper. On the other hand, all those subdiagrams g deleted from D u {G} and hence included in Do are, by definition, proper. Finally let go = g - c(g) E Do, with 0 $ c(g) $ 8. Consider the subdiagram g depicted by the overall region in Fig. 5.1 la. Suppose that go (p g) has an external line ell4joining some vertex uj to some extral vertex ui of go, as shown in Fig. 5.11a. The shaded region in this figure represents the remaining part of the subdiagram go. By momentum conservation [see (5.1.66)], we have at the vertex ui
kfj, =
-
c'
(5.3.41)
k?j,,t,
j,l'
!,
where the sum is over all those vertices v j , and lines of g/go c g with the lines t,. joining the vertices uj. to the vertex ui. These lines are represented by the dotted lines in Fig. 5.11a. Since the kfjp,,in g are independent of q,, it follows from (5.3.41) that kfj, is also independent of q,. Hence the extral vertex ui and the line L,joining the vertex uj to the vertex ui cannot belong to g - c(g) = go, by definition of c(g). Therefore go has no extral vertices and thus must be amputated. Now suppose that go has an improper line joining some vertex ui to some vertex u j , and a situation as shown in Fig. 5.1 l b arises, with the dotted lines denoting lines belonging to g/go and the
,!
l4
That is, suppose that go is not amputated.
5
116
The Subtraction Formalism
shaded regions representing the remaining part of the subdiagram go, with external vertices uil, ui2, . . . . Again by momentum conservation we have in an obvious notation kg. Ill =
-
c' kfjolo- 1'k9 .
IlJlll
jolo
- ... -
c' k!pjplp)
(5.3.42)
jpb
jdl
corresponding to the dotted lines in Fig. 5.1 lb, which represent some lines belonging to g/go c a. Accordingly, the k!jojo,kfljll,,. .. ,kfpjplpare independent of q,, and so, from (5.3.42), kfjl is independent of q, as well. Therefore the improper line el cannot belong to g - c(g) = go, by definition, and go cannot contain improper lines. Therefore all the subdiagrams in Do are proper subdiagrams. Finally, we show that all the kf; in go/g', where go = g - c(g) E Do [c(g) # 01, with g = g/g', are dependent on q,. From (5.1.67), we know that the kf; = kf;(ke), with go c g, and they are independent of the elements in q8.Thus, for convenience, we set the elements in q8 equal to zero to prove the result stated earlier. From a canonical decomposition as in (5.1.65) and (5.1.46), (5.3.43) kT$(ke) = kfjj - qf$(ke, 0) for i , j , and 1 pertaining to go/g'. Applying the result obtained in (5.1.117) to each of the connected parts of g, we arrive at the conclusion that the qf;(ke, 0) depend only on the kfj, in g/go and are, therefore, independent of q,. On the other hand, we know, by construction, that all the k!jl in go/g' are dependent on q,. Therefore it follows from the (5.3.43) that all the kf; in go/g' are dependent on q, . We have started out with a set D u {G} and induced a set Do. For g E D v {G},with 0g c(g) g g and g = g/g', we denote go/g' by g o . We have thus established that
all the subdiagrams G'E Do - {G} are proper and all the k$i in G' are dependent on qr. (5.3.44) From the sets Do and DN we define a new set N = DNu Do
3
DUD,.
(5.3.45)
We note, in particular, that for c(g) = g or c(g) = 0, all the kfjj in g are independent of q, or dependent on q,, respectively, by definition of c(g). On the other hand, for 0$ c(g) $ 8, all the kfjj in g/go, where go = g - c(g), are independent of q,. Obviously the kEl in C,with G E N, are either all dependent on q, or are all independent of q,. Accordingly, if we arrange the subdiagrams in the set N in an increasing order, i.e., {. . ., g', 9 , . . .} with g 2 g', then we have, from (5.3.44) in particular, the following lemma.
5.3 Convergence of the Subtraction Scheme
117
Lemma 5.3.4: All the subdiagram in the set N , as defined in (5.3.49, are proper, andfor any two consecutive subdiagrams g , g‘ E N , with g P g’, the kfj, in gig‘ = g are either all independent oj’v], or all are dependent on q r .
Starting from a nuclear set DN as obtained from a set D u {G}, we may generate all other possible sets that contain in addition to the subdiagrams in DN any one, or any Two, or . . . or, finally all subdiagram(s) from Do. The collection of all these sets, as just defined, (as obtained from DN and Do), together with the set DN will be denoted 9. Clearly the set 9 has the form 9 = (DN, . . . , (DN u Do)}. The corresponding Taylor operations in (5.2.1) for the subdiagrams in the sets in 59 may be then readily combined, and the corresponding sum over the sets in 9 is then reduced to the expression
(5.3.46) where a set N is defined in (5.3.45) and
s.,
=
1
0
for g ED, for g E D N .
(5.3.47)
The product in (5.3.46) must be taken in the correct order, that is, . . . (S,” - T,) ... (8,“. - q,)... I,, for g 2 9’. Clearly G is the largest subdiagram in N and @ = 1. The sum over all D sets in (5.2.1) then reduces equivalently, for the final expression for R as a sum over all distinct sets N , with the latter as defined in (5.3.45), i.e., R =
1 fl (6,“ - T,)lG.
(5.3.48)
N geN
For any g E N, with d ( g ) < 0, we simply set T, = 0, by definition. Finally we introduce subsets of N : H , ( N ) and H , ( N ) : N = H,W) u H , ( N ) ,
(5.3.49)
where g E H , ( N ) if all the kfj, in are dependent on qr and g E H , ( N ) if all the kfj, in g are dependent of qr (see Lemma 5.3.4). We also introduce subsets F , ( N ) and F , ( N ) by (5.3.50) F A N ) = Hl(N) f l Do and (5.3.51) FIW) = H d N ) - F A N ) . In particular, Hl(N) = FlW) u F A N ) .
(5.3.52)
Also if g E F , ( N ) , then S,“= 1, and if g E F , ( N ) , then 6,” = 0. We also remark that the whole graph G E H , ( N ) u F 2 ( N ) ,8: = 1, by definition.
118
5
The Subtraction Formalism
5.3.3 Completion of t h e Proof of Convergence
Now we apply Lemmas 5.3.2-5.3.4 and the grouping achieved in (5.3.48), with respect to the parameter q,, to prove the following lemma, where r is fixed in 1 Ir Ik I4n, as before. Lemma 5.3.5
degr R E < - dim S,,
degr R <
9r
- dim
S,.
(5.3.53)
Vr
Consider a fixed set N in (5.3.48). T o prove (5.3.53) we introduce the following recursion formula, (5.3.54) F & N ) = (6; - 7JIgFg*(N), corresponding to a set N in (5.3.48), where g, g' are two consecutive subdiagrams in N with g $ 9'. In the notation (5.3.54), we have from (5.3.46) F--(N) E FG(N).
(5.3.55)
We also introduce the notation (5.3.56) and we define L ( @ ) = 0. We note in particular p(G)
+4 1
L(8') = 4L(G).
(5.3.57)
B'EHzW) g'CG
For convenience, for any g E N , we denote byf:(ks, qg, p) any function of the form j-;(kg,
48,
p) =
n
nq' [(ef;,)2
+ p;f
- i&{(qjf)2
+p
; f ) ~ - (5.3.58) ~ ~ ~ ~
g'€Hn(N) i j / g'eg i<j
= The a$ are some for n = 1,2, where Ofj, = Q f j f , and for the g' g, strictly positive integers. Note that if g # H , ( N ) for n = 1 or n = 2, then the product in (5.3.58), for the corresponding n, is over the g' E H , ( N ) with g' $ g, by definition. If g 4 H , ( N ) for n = 1 or n = 2, and there is no g' E H , ( N ) , with g' $Z g, then we definef: = 1 for the corresponding n. Also by definition of H , ( N ) and H , ( N ) , all the kfjf in a' for all g' E H , ( N ) , g' c g, are dependent on qr, and all the kf,f in g' for all g' E H , ( N ) , g' c g, are independent of q, in the denominators in (5.3.58). We also denote by f " ( k g , qg, p ) any function of the form f " ( k g , 4', p) =
fl'
ijf i<j
[(QfjJ'
+
~ $ 1-
+ /L$}]-"~~J'
k{(QfjJ2
(5.3.59)
5.3 Convergence of the Subtraction Scheme
119
for n = 1,2, with the 0 7 ~some ~ strictly positive integers. If the elements in q8 in (5.3.58) and (5.3.59)are set equal to zero, then the corresponding functions will be denoted by f :(ke, p ) andf”(kg, p). Note that all the denominators in f , ’ ( k g ,p ) and f ’ ( k e , p ) are dependent on qr and all the denominators in f i ( k e , p ) andf2(ke, p ) are independent of q r . Finally, we denote by P(ke, q8,p ) [or P,(ks, q8, p ) ] any polynomial in the elements in kg, q8, p and, in general, in the ( p i j l ) - as well. The sum over N in (5.3.48)is over a finite number of sets, and we establish (5.3.53) first for each Y(”, as defined in (5.3.46). As before, we arrange the sets N in an increasing order, i.e., as {. . . ,g’, g, . . .},with g 2 9’. For additional clarity we shall deliberately give more details than is actually necessary. Let g be the first element in N and note that ij = g. Suppose g E F , ( N ) . Then F8(N)is of the form
’
Y&N)=
-
qI,’
(5.3.60)
where I, is of the form I,
me,q8, P ) f ’ ( k 8 , q8, P).
=
(5.3.61 j
We explicitly have for Y , ( N ) , in a convenient notation, (5.3.62) where lal
+ Ibl I 4 g ) .
(5.3.63)
The notation in (5.3.62) is similar to the ones in (5.3.27)-(5.3.29). We note that la1 +degr P“
+ J b l +degr f,’I d(g) - 4L(g).
4r
(5.3.64)
4r
Accordingly, if we scale qr in Pa a n d f i , and q8 as well, by a parameter a, degr F8(N I 4 g ) - P(9)9
(5.3.65)
01
where we have used the fact that 4L(g) = p(g) from the very definition in (5.3.56). We also note that (5.3.66) degr Y , ( N ) I 4 9 ) . 49
Finally, we note from (5.3.62) that Y , ( N ) has the general structure
Y,(W = P,(k? q8,p)f,’(k? PI.
(5.3.67)
Suppose g E F 2 ( N ) .Then F8(N)= (1 - 7p,.
(5.3.68)
5
120
The Subtraction Formalism
If we scale the kf,, in g, which, by definition, depend on qr, by a parameter I, and ifd(g) 2 0, then we may use the result (5.3.25) in Lemma 5.3.3 to conclude that degr Y , ( N ) < -P(g),
(5.3.69)
I
where p(g) = 4L(g). If d(g) < 0, then
5 = 0,
degr Y , ( N ) 5 d ( d - P ( d < -p(g).
(5.3.70)
I
For d(g) 2 0, we see from (5.3.14), (5.3.15), and (5.3.33) that Y , ( N ) in (5.3.68) has the structure =q,(N) = P , ( K qgl P)f#,,
q6,PI.
(5.3.7 1)
If d(g) < 0, then Y , ( N ) = I,, and I, is again of the general form in (5.3.71). Finally, suppose g E H , ( N ) . Then Y , ( N ) = - 51,’
(5.3.72)
where I, is of the form
(5.3.76) Since p(g) = 0 [see (5.3.56)], in this case we may rewrite (5.3.76): degr Y , ( N ) 5 d(g)
-
P(d.
(5.3.77)
a
From (5.3.74) we also note that Y , ( N ) has the structure Y , ( N ) = P,(K q8,P ) . f ; ( e PI.
(5.3.78)
Since all the kf,,, in this case, are independent of q r , we may also trivially write in the notation (5.3.69)-(53.70) degr Y g ( N )= 0. 1
(5.3.79)
5.3 Convergence of the Subtraction Scheme
121
Now as induction hypotheses, suppose that for some subdiagram g E N - { G } ,not necessarily the first element in it, the following are true:
(i) If g E H , ( N ) , then Y,(N)has the structure Y,(N)
=
p,(ke,
q81
(5.3.80)
P)f:W, P)f,2Wl PI.
If we scale all those kyjl in P,, depending on q r , and qr inf: by a parameter A, then” degr q N ) <
(5.3.81)
-P(S)
I
ifp(g) # 0, and degr Y&N) = 0
(5.3.82)
I
if p(g) = 0. Also, (5.3.83)
degr Y , ( N ) I d(g). Yg
Finally, if we scale E. and qg by a parameter
o!
(and set E.
=
l), then
degr Y , ( N ) 5 d ( g ) - P ( d .
(5.3.84)
a
(ii) Ifg E F , ( N ) , i.e., in particular, ijr = 0 [see Eq. (5.3.51) and following it), then Y,(N) has a structure as in (5.3.80). Equations (5.3.83) and (5.3.84) are also true. (iii) If g E F , ( N ) , i.e., in particular, that 6,. = 1 [see (5.3.50)], then Y,(N) has a structure as in
T,(N)
=
P,(kB, qglCL)fglW, q8, Plf,2W9P),
(5.3.85)
andifwescale thek$,inallthe$,g’ E HI(N),g’c g,inff,which, bydefinition, depend on q l , and the kfjl in P,, depending on q r , by a parameter A, then
(5.3.86) Suppose that the next subdiagram to g in N is g’ and g’ E N - {G}; then we prove that (i), (ii), or (iii), as the case may be, is also true for 9’. We finally treat the situation for G itself separately. (i) Suppose that g E H , ( N ) ; then if g’ E H , ( N ) , (g $ g’),
(5.3.87) where la,is of the form
I,. = Pg.(ka’,q8‘,p)f2(kg”,q g ‘ , p). Is
Recall that j ’ i ( k g , p) is independent of qr [see definition (S.3.SS)l.
(5.3.88)
122
5
The Subtraction Formalism
We write ke = k'J(kg'), q g = qg(ke', 4 8 ' ) in (5.3.80). Then with g E H , ( N ) , we explicitly have for Y J N ) in (5.3.87)
Y&v)
=
1 ( q g ' ) P + b + c G . ( k e ' , 0, p)f;(k", p )
-
a. b. c
x G(ke(kg'),qg(kg',01, p)f:(ke(ks'), p)fi(kg(kg"Xp), (5.3.89)
where
+ Ibl + I c l 5 4 s ' ) .
lal
(5.3.90)
If we scale qr infi, and the k$ in P i , P i . , depending on q,, by a parameter 1, we obtain according to the hypotheses in (5.3.81) and (5.3.82)
+ degr f i < - p ( g )
degr
(5.3.91)
A
1
if p(g) # 0, and degr
+ degr fi
1
L
=0
(5.3.92)
if p(g) = 0 (in the latter case, we note thatfi = 1). We note that q8(kg',0) in 5 is independent of qr since the k$, in g'/g, on which q' depends [see (5.1.1 17)], are independent of q,. Also note that G , , f ; a, n d f i are independent of qr, I.e., degr G.= degr 1
ft = degr fi = 0. 1
1
Accordingly, we obtain from (5.3.91) and (5.3.92)
< -dg')
degr
(5.3.93)
1
if p(g') # 0, and degr Y J N ) = 0,
(5.3.94)
1
if p(g') = 0, where p(g') = P ( d .
(5.3.95)
degr F8,(N)Id(g').
(5.3.96)
Also we note from (5.3.90) that 99'
Finally, if we scale 1 and qg' by a parameter a (and set 1 = l), we readily obtain degr Yg,(N)5 4 g ' ) L1
- p(g').
(5.3.97)
5.3 Conver.qence of the Subtraction Scheme
123
We also note from (5.3.89) that Fgr(N)has a structure as in (5.3.80). If g' E F , ( N ) [see (5.3.51) and (5.3.52)], then
qy,,Y g ( N ) ,
q N ) = (-
(5.3.98)
where I,. is of the form I,.
=
P,,(kg', q g ' , p ) f
'(kg',.
q g ' , p).
(5.3.99)
Accordingly, we may write
c (q"y+*+'P;.(kg', 0, p ) f ; ( k e ' ,
Y g , ( N )= -
p)
a.b.c
x
f'@g(kg'), qg(ks',O), p)f #Ykg'), p ) f i ( k g ( k g ' ) p), , (5.3.100)
where 1.e.. q N ) I d(g').
(5.3.102)
If we scale q, infj,f:, the k$ in Pi., 5,depending on qr, and scale as well qg' by a parameter a, then la(
+ Jbl + degr P$+
degr fi I d(3') - 4L(B'),
a
(5.3.103)
a
and the induction hypothesis (5.3.84) for g we obtain
I c J + degr Peg a
+ degr f i I d(g) - p(g).
(5.3.104)
1
Accordingly, degrFg,(N) I&') - P W ) ,
(5.3.105)
a
+
+
where we have used the facts that d(g') = d(8') d(g) and p(g') = 4L(3') p(g). Finally, we note from (5.3.100) that F g , ( N )has a structure as in (5.3.80). If g' E F , ( N ) [see (5.3.50)], then q N ) = (1 - Tg,)I,.Yg(N),
(5.3.106)
and I,. is of the form in (5.3.99). We then have .qI*(N)= f
#g,
W'),P)f ,Z(k"k"'),
p)
x 5 ( k g ( k g ' ) qg(kg', , 0), p)(1
c w')' c
- T:!'"- " ' ) I 9'- '
(5.3.107)
5 The Subtraction Formalism
124
(5.3.108) From Lemma 5.3.3, by scaling the k$, in 8’ by A, which by definition depend on q,, we obtain for d(g’) 2 Ic I degr[l - T ~ ! e ’ ) - ~ c5~d(8’) ] l g s- 4L(g’) - d(g’) I
= - d(g) -
4L(8’)+ I c (
+ IcI - 1 -
1.
(5.3.109)
From the induction hypotheses in (5.3.81) or (5.3.82), we also have that (5.3.91) or (5.3.92) is true. Accordingly, from (5.3.109), degr &09 < I c I - d ( d
- P(S’),
(5.3.1 10)
1
where p(g‘) = 4JY8’)
+
(5.3.1 11)
Finally, from (5.3.108), degr Y J N ) c - p(g’).
(5.3.1 12)
1
If d(g’) < 0, then degr I,. I d(#) - 4L(g’),
(5.3.113)
I
which leads to desrqw)
-
P b ’ ) < - P(S’).
(5.3.1 14)
1.
We also note from (5.3.107), with d(g’) 2 0, together from (5.3.14), (5.3.15), and (5.3.33), that 9 J N ) has a structure as in (5.3.85). The latter conclusion is also reached even if d(g’) < 0. (ii) Suppose g E F , ( N ) . Then if g‘ E F , ( N ) , q ( N ) = (- Tg*)Zg, 9-#(N),
(5.3.1 15)
where I g , is of the form in (5.3.99). We explicitly have
Y g r ( N )=
-
1
( q e ‘ ) ( l + b + c e * ( k e ‘ , 0, p ) f i ( k g ’ ,p )
a.6.c
x
c(kg(ke‘), qe(kg‘,O), p)f;(kg(ke’),p ) f j ( k e ( W ,p), (5.3.1 16)
where
la1 + Ibl +
ICI
5 d(g‘),
(5.3.1 17)
i.e., degr Yg,(N) Id(g’). q9’
(5.3.1 18)
5.3 Convergence of the Subtraction Scheme
125
fi
We scale qr in and f L, and all those k$ in P$ and Pi, depending on qr, and scale qg' as well by a parameter a. By the induction hypotheses in (ii), we have from (5.3.84) as applied to g,
Ic I + degr pEB
+ degrf:
a
Also, by definition of g' la1
Id(g) - p(g).
(5.3.119)
a
E F,(N),
+ Ibl + degr Pi, + degrf,' a
I d($) - 4L(#').
(5.3.120)
a
Accordingly, from (5.3.119) and (5.3.120), degr YgW)I4 s ' ) - p(g').
(5.3.121)
a
We also note from (5.3.116) that Y g , ( N )has a structure as in (5.3.80). If g' E F 2 ( N ) ,then Y g , ( N )= (1 - 7&Yg(N),
(5.3.122)
where I#, is again of the form in (5.3.99). By repeating a similar analysis as the one leading to (5.3.112) and (5.3.114), by using in the process the induction hypotheses in (ii), we obtain
< -p(g'), degr YgW)
(5.3.123)
1
and Y g , ( N )has a structure as the one in (5.3.85). (iii) Suppose g E F 2 ( N ) .Then g' E H 2 ( N ) .In this case Y g * ( N )= (- 7 p , , Y g ( N ) ,
(5.3.124)
where I,. is of the form in (5.3.88). We explicitly obtain g' a + b + c + d p a kg' 0 Ygo9= (4 1 g( , P ) . f 2 ( k g ' *P )
c
3
a. b, c. d
x
pEBE,(kUW'),qYkg',01, PL)fg:d(&W, qg(ke',01,P)f ,2(k"k"), PI, (5.3.125)
where la1 + Ibl
+ ICI + Id1 I&'),
(5.3.126)
i.e., degr Y J N ) Id(g').
(5.3.127)
49'
We make the important observation from (5.1.117), applied to each of the connected parts of g', that the dependence of the qg of g on kg' is only on those in g'/g, i.e., q'(kg', 0) is independent of qr. We scale qr inf;,, [which by
5
126
The Subtraction Formalism
what has just been said about 4fj,(ke’,0) has all its denominators, in (5.3.58) with n = 1, depend on q,] by I. We also scale the k& in 5, P i . , depending on q, by 1.According to the induction hypothesis (5.3.86) in (iii), degr G 1
+ degr f;,d < - A g ) . 1
(5.3. 28)
We also have degrAP;. = 0, which together (5.3.128) implies that degr 1
~ &
(5.3. 29)
Pb‘) = P W
(5.3.1 30)
where If we scale qe’ and I, by a parameter a (and set I = l), we obtain by making use of (5.3.126) degr Ye@) I d@’) - p(g’).
(5.3.131)
a
Finally, we note from (5.3.125) that Y g , ( N ) has a structure as in (5.3.80). Now we apply the above results to the whole graph G itself. If G E F,(N), then (5.3.86) implies degr Y G ( N )<
- p(G).
(5.3.132)
1
Since all the qU ( = (1;) of G are independent of q,, we may simply replace degrl in (5.3.132) by degr,,; i.e., we have degr Y G ( N ) < -p(G).
(5.3.133)
or
If G E H , ( N ) , then (5.3.81) implies, for d(G) 2 0, degr( - TG)Z#&N) <
- p(G),
(5.3.134)
Ir
where g is the largest subdiagram in N contained in G: g G, and we have used the same reasoning as in (5.3.132)-(5.3.133) to replace degr, by degr,,. On the other hand, in this case degr 1~ = 0,
(5.3.135)
9r
or, from (5.3.81) or (5.3.86), degr W q N ) < - P(G),
(5.3.136)
)Ir
since p(G) = p(g), where we have used (5.1.1 17) to conclude that qg(kG,qG) is independent of q,, as the k$ in G are independent of qr. From (5.3.134)
5.3 Convergence of the Subtraction Scheme
127
and (5.3.136), if d(G) 2 0, or just from (5.3.136) if d(G) < 0, we then conclude again that degr Y G ( N )< -p(G).
(5.3.137)
4r
Equations (5.3.133) and (5.3.137) then imply that we always have degr Y ( N <)-p(G),
(5.3.138)
4r
where we have used the identity in (5.3.55). Equation (5.3.56) also implies that p(G)
=
c
4
L(8‘).
(5.3.139)
g’F HI“)
g’tG
From (5.1.4), (5.1.61), and the fact that the integration variables ky, . . . ,k,3 are some linear combinations of the components of P in (5.3.40), we note that for any g’ E Hl(N), we may write for a k$, in g’, r=1
f=r+l
(5.3.140) where the following must be true, from the definition of the set H,(N): r
C (Ar(VOqr
r=1
*
. .q r -
1qr)
+0
(5.3.141)
for at least one of the components of k$,.16 With qr+ . . ,qk fixed, we see from (5.3.140)-(5.3.141) that the components of k$, depending on qr, are functions of the independent parameters q l , q 2 , . . .,qr. The A,($) and the c$ are independent of q l , . . . ,qk. On the other hand, for any g’ E H , ( N ) , the k$, in S’ are of the form k
k$
=
1 A;(gl)qf . . . q k +
C$.
(5.3.142)
r=r+l
The parameters ql, . . , q k (and, in particular, ql, . . . ,qr) are independent. Quite generally, we know from (5.1.118)-(5.1.120) that p(G) of the k$, in the set +
{k$, :g’
E
Hl(N), k7jl in
a’},
(5.3.143)
l6 Recall that, by definition, a four-vector depends on 9, if at least one of its components dependson q,.
5
128
The Subtraction Formalism
may be independent. In particular we note that not all of the four components of the four-vectors in the set in (5.3.143) are necessarily dependent on q,. Accordingly, we may write17 dim S, I p(G), (5.3.144) which, together with (5.3.138) and the fact that, in general, degrqrR I maxNdegrqrF(”,as N runs over the sets in (5.3.48), establishes the statement of Lemma 5.3.5 for R.The same analysis with E set equal to zero and a Euclidean metric establishes the statement of the lemma for R,. The work of Chapter 2 implies, in particular, that we may find a constant br > 1, and the power asymptotic coefficient a($) of R , with q, 2 b,, may be identified with degr,,, R. We also note that (5.3.53) is true for any independent vectors L 1 , .. . , L, in I , as given in (5.3.40), with S, = {Ll, . . . ,Lr}, and for any arbitrary r and k in 1 I r I k I 4n. Therefore for any nonzero subspace S c I , we have from Lemma 5.3.5 that cr(S) < -dim S.
(5.3.145) What has just been said about R is also true for RE.Thecondition in (5.3.145) is nothing but criterion [A], Eq. (3.1.3) of Theorem 3.1.1. Hence we obtain. Theorem 5.3.1 : 7he renormalized Feynman amplitude d E ( P p), , in Euclidean space as given in (5.3.2), is absolutely convergent.
Finally from Theorem 4.2.1 on the c + + 0 limit of absolutely convergent (Feynman) integrals, and from (5.3.145) we obtain. Theorem 5.3.2 : The
renormalized Feynman amplitude d e ( P ,p), in Minkowski space as dejned in (5.3.1), is absolutely convergent for E > 0, and the limit E + +O dejnes a Lorentz covariant distribution.
5.4
THE UNIFYING THEOREM OF RENORMALIZATION AND BASIC IDENTITIES
We define
(5.4.1) and hence from (5.2.19), I ’ Wegiveanelementaryexample where theequalityin (5.3.144)doesnot hold. Suppose that k = 2 in (5.3.40), with L, and L, independent vectors in I . Suppose that G is a graph with only one four-dimensional integral, i.e., K = ( k y , . . . ,k:). We note that dim S, = 1 and dim S, = 2; however, the corresponding p,(G) and p,(G) are simply 4L(G) = 4 as a consequence of the fourdimensional property of space-time.
5.4
The Unijying Theorem of Renormalization a n d Basic Identities
Lemma 5.4.1 :
g l , . . .,g n :g =
129
L e t g be a proper subdiagram w i t h n connected parts
u:=
gi. T h e n
-TBA,I, = (-TBIA,l~,l)...(-TBnA,n~,n)'
(5.4.3)
In particular the identity (5.4.3) says that even if d ( g ) 2 0 and one or more of the connected parts g j are such that d ( g i ) < 0, then - < A , I , = 0, by definition of the corresponding Taylor operations (-?,) = 0 for such (proper and connected) subdiagrams g j . To prove (5.4.3) we note that the left-hand side of it may be written
(5.4.4)
for arbitrary i. For convenience we allow in the sums in (5.4.4) all proper subdiagrams by simply defining, as usual, 5, = 0 if d(gj) < 0. Suppose i in (5.4.4) is chosen large enough so that the g i (for g' # 0)are connected. The lemma is then trivially true for such gi. Suppose as an induction hypothesis, that the statement of the lemma is true for all g 1 si g corresponding to i - 1 square brackets in (5.4.4), i.e., suppose that the statement of the lemma is true for the subdiagrams g l $ g; then we prove that the statement is also true for g itself. According to the induction hypothesis, ( - T,A,)I, = ( -
..,a,!,,,
T,)
1
(-
5;A # ; ) - ( - 5&, A#&&, * *
(5.4.5)
0cg'Sg
where G:,. are the connected parts of a g l , i.e., in particular, g1 = ijf and ml = m1(g'). Clearly, each of the ijf may fall in one and only one of the g i (i = 1,. . . ,n)-the connected parts of g . Also, one or more of the 4; may fall in one of the connected parts of g . Accordingly, for a fixed g 1 in ( 5 . 4 3 , let g : , . . . ,g : , be all proper, though not necessarily connected, parts of g 1 falling in, respectively, g i l r . .. ,g i n , , i.e., g : c g i l , .. . , gf, c gin,,where n, 5 m , , { g i l , .. . , g i n , } is a subset of { g , , . . . , g , } such that g1 = (J:; = g!, and each g! is a union of one or more of the g,!. By using the induction hypothesis once more, we may rewrite (- %;A,:) - . (- ~ L I A , h l ) I ,in (5.4.5) as (- TB:A,t). . (- TBAIABnI)Ig. It should also be noted that since g 1 $ g , it follows that the term [( - TBIABI).. - (- TBnABn)]lB, where g l , . . . ,gn are the connected parts of g , does not occur in the sum in
uyi
0::
5
130
The Subtraction Formalism
(5.4.5). Accordingly, we may add this term to and subtract it from the sum on the right-hand side of (5.4.5) to write equivalently
- C( - Tg,Agl)
*
. (-
TBn-4gn)1
I
(5.4.6)
I,.
For convenience of notation, we now trivially complete the collection (gt,. . . ,gf,) of n , elements into a collection of n elements (g:, . . . ,sf,, 0, . . .,121) = (g:, ...,g;,, .. .,gf) by adjoining n - nl empty subdiagrams to the former collection. In an obvious notation, we may then write (5.4.6) as
I
C( - TB,Ag1) . . (- q"4n)1I,,
-
*
by further relabeling the subdiagrams {gi, . . . ,g;,}. implies
(5.4.7)
Definition (5.4.1) (5.4.8)
Using (5.4.8), we may rewrite (5.4.7) in the following simple form: (-TBAg)Ig
=
(-QW - TBJA,, * * * ( 1- TB")A,"
(5.4.9) - ( - q1 . * .( - 7 p g n ) l l , * We recall that g = gj and I, = I,,. We scale the external variables of g, and hence of gl, .. . ,gn, by a parameter A. If d(gj) 2 0 for all j = 1,. . . , n , then the first expression in the square brackets in (5.4.9) applied to I, is of the form
u;=,
n?=,
where only the dependence on the external momenta of the gj are shown in (5.4.10) and fi are some explicit functions of qe' associated with the g,. By using the fact that d(g) = d(gj), we see that when (- T,) is applied to the expression (5.4.10), it reduces it to zero, by definition of Ti. That is, (- T,) annihilates the first expression in the square brackets in (5.4.9). If for some gj, say, gl,. . .,go we have d(gj) < 0, i.e., Tg,= 0 for j = 1,. . ., s c n, then the product in (5.4.10) is restricted only over j = s + 1,. . . ,n and the resulting expression is simply multiplied by Ih&g').The
n;=
5.4
The Unifying Theorem of Renormalization and Basic identities
13 1
operation ( - T ) then still annihilates the first expression in the square bracket in (5.4.9), since in this case d(y) < ~ ~ = s d(g,). + , On the other hand, = TI . . . qn. Accordingly, we finally obtain T TI. . . -
TAJ,
=
(-TI A,,I,,)
*
* . (- qnABnIgn)'
(5.4.'11)
which is the statement of the lemma. In the definition of the renormalized Feynman integrand R in (5.2.1) one may, in obtaining the final expression for R, restrict the summation over renormalization sets D such that the dimensionality of each of the connected parts gi of a g E D is nonnegative, as the other D sets will not contribute to the sum in (5.2.1). Corollary 5.4.1 :
This useful result follows from (5.2.21), (5.2.22), and (5.4.2), which, in particular, imply that R = (1
-
TG)AGI G ,
(5.4.12)
(-TG*,)AG,,lG*,
(5.4.13)
with AGiG defined recursively by AGSlG,
C
=
0 c G" !$ G'
and from the particular statement of Lemma 5.4.1, which says that - TGet A G , , l G * , is zero if the dimensionality of any one of the connected parts of G" is strictly negative. Lemma 5.4.1, together with (5.4.12) and (5.4.13), gives the following theorem. (The unifying theorem of renormalization) : The renormalized Feynman integrand R for a proper and connected graph G may be equivalently written Theorem 5.4.1
where A, IG is defined recursively by
for each proper and connected subdiagram G'. The sum is over all proper subdiagrams G" (for G" # with proper and connected subdiagrams G ; , . . . ,GL.
a)
The expression (5.4.14), with (5.4.19, provides the definition of the so-called Bogoliubov-Parasiuk-Hepp-Zimmermann subtraction scheme in the Zimmermann form. Since the latter eventually grew out of the approach of Bogoliubov, and our subtraction scheme in (5.2.1) eventually grew out
5
132
The Subtraction Formalism
of the approach of Salam, Theorem 5.4.1 leads essentially to the equivalence of the paths taken in the ingenious approaches of Salam and Bogoliubov (in momentum space). Theorem 5.4.1 allows us to make a transition from one form of R to the other whenever it is convenient. For a proper and connected subdiagram g, we define by Db a D set such that its largest subdiagram is g and any subdiagram in it is a proper and connected subdiagram of g. If {Db} is the set of all such Db sets such that for any Db, and D;, in {Db}, and for any g1 E Db, 9, E Db, ,, we have g1 n 9, = 0, g1 c g, 5 g, or 9, c g, c g, then we denote the union Db of the sets in {Db} by D(g). It is not difficult to see from the definition of the recursion relation (5.4.15) that R may be written as (5.4.14) with
,
u
(5.4.16)
where G’,, . . ., G; are the connected points of G’. The sum xfi(G;Lisover all those sets having their largest subdiagram Gi, and if g l , gz E D(GI), then g1 c g,, or 9, c g l . Finally, (5.4.16) may be rewritten g, n 9, = 0, =
x [
0cG’OG
n
(-Tg)]lG, S I ,..., AG’) B E S I._.. . n(G’)
(5.4.17)
where S l , ,, JG‘) = &G;) u . . . u D(G;), and the product is over all proper and connected subdiagrams g in S, ...,JG’). Note that if gl, g2 E S l , , JG’), then either g1 E GI,g, E G;, for some i # j , or g1,g2E GI for some i, with g1 n g 2 = O , g l = g2,0rg2 c gl.Thatis,inparticular,ifgl,g2 E Sl,...,n(G’)9 then g1 n g , = 0, or g1 c g,, or 9, c gl. Finally, recall that Z Q c G ’ f G in (5.4.16) and (5.4.17) is over all proper subdiagrams G’ (for G’ # 0) with G’ $ G and with G’,, . . . , Gk their proper and connected parts. For G’ = 0, we write as usual ( - T,) = 1. ,
, ,
NOTES
The subtraction scheme was introduced by Manoukian (1976), and we were guided by the classic work of Salam (1951b) in its development. It is important to note that the scheme is carried in momentum space with subtractions, over subdiagrams, applied directly to the Feynman integrand, and hence no questions of ultraviolet cutoffs arise. Canonical momenta were introduced cleverly by Zimmermann (1969), and we followed his definition of the canonical decomposition of the Q i i , quite closely. The
5.4
The Unifying Theorem of Renormalization and Basic Identities
133
convergence proof (Section 5.3) was given in Manoukian (1982a), see also 1977. The “unifying theorem of renormalization” is due to Manoukian (1976). For some other studies of renormalization, apart from the Bogoliubov-Parasiuk (1957)-Hepp (1966)-Zimmermann (1969) one, see Steinmann (1966), Caianiello et ul. (1969), Kuo and Yennie (1969), Speer (1969), Anikin ef a/. (1973), Bergere and Zuber (1974), Anikin and Polivanov (1974). See also Velo and Wightman (1976).
This Page Intentionally Left Blank
Chapter 6 / ASYMPTOTIC BEHAVIOR IN QUANTUM FIELD THEORY
The purpose of this chapter is to study the asymptotic behavior of subtracted-out Feynman amplitudes d E ( P ,p), in Euclidean space as defined in (5.3.2), when some of the elements in P and p “take on” asymptotic values. Throughout this chapter we write d ( P , p ) for d E ( P ,p ) to simplify the notation, and no confusion should arise. Before plunging into a general asymptotic analysis of d ( P , p ) we derive in Section 6.1 some elementary asymptotic estimates f o r d directly from the general definition in (2.2.84) (with 9,in the latter identified with RE) which require no detailed knowledge of the structure of RE.We show in particular that if the dimensionality d(G) of the graph G , with which d is associated, is nonnegative, then if the external momenta are led to approach zero, it follows that d vanishes (Theorem 6.1.1). This is consistent with (and is expected from) the overall subtraction at the origin one carries out in defining R . We also show, whether d(G) 2 0 or d ( G ) 0, if all the masses appearing in d are led to approach infinity, then d again vanishes (Theorem 6.1.2). This is a particular case of the so-called decoupling theorem of field theory, which essentially says that Feynman amplitudes involving “very heavy” masses may be “neglected.” In the remaining sections of this chapter we carry out a general asymptotic analysis for d .In Section 6.2 a general dimensional analysis for RE is given on which the remaining sections are based. In Section 6.3 a study of the high-energy behavior of d is given when all the masses are fixed and nonzero. The study in Section 6.4 generalizes the results in Section 6.3 to deal with
-=
I35
6 Asymptotic Behavior in Quantum Field Theory
136
the situation when not only some (or all) of the external momenta of G become large but some (or all) of its masses as well become large. A study of the zero mass behavior of d is given in Section 6.5. The latter is then applied in Section 6.6 to study the low-energy behavior of d when some of the masses in the theory are led to approach zero. Section 6.7 deals with a very general study of d when some of the external momenta become large, some become small,’ and some of the masses become small. In general we let these various asymptotic components “approach” their asymptotic values at dgerent rates. In Section 6.8 a generalized decoupling theorem is proved which establishes the vanishing property of d when any subset of the masses in the theory become large and, in general, at different rates, and gives sufficiency conditions for the validity of the decoupling theorem when any subset of the remaining nonasymptotic masses are scaled to zero as well. The breakdown of this chapter into the above mentioned sections, as just outlined, will, it is hoped, facilitate the reading of this rather difficult chapter. One thing worth noting in this chapter is that one is able to obtain the asymptotic behavior of d without the need of carrying out explicitly the rather complicated integrals defining .d,as usually no closed expression for d may be obtained when one is involved with complicated Feynman graphs. PRELIMINARY ASY MPTOTlC EST1M A T E S
6.1
We write2 (6.1.1)
n
m
and %P, K, P ) =
c P“%(K, P),
(6.1.4)
a
in a notation similar to the one in (2.2.18), where a = (aol,..., aSm),
dPj2apj20,
la1 = aol
j = 1,..., m,
+ + a3m, p=O,1,2,3.
(6.1.5)
’ Of course, some of the external momenta may remain nonasymptotic. ’ Since in this chapter we consider only a Euclidean metric, we omit the E subscript in Q;c.
6 .I
137
Preliminary Asymptotic Estimates
+
. . . , pydr, denote dPj 1 distinct values for ps. Let P: = (plIo,, 0 . . . , pA13,), where 0 I t P j I dPj. Then the generalized Lagrange interpolating formula (4.1.11) states that we may find a constant C,(P:) depending on a and P: such that Let py,,,
PXK, P ) =
1Ca(P:)P(P:, K ,
(6.1.6)
I
where the sum is over all 0 It P j I d P j ,withj = 1 , . . . ,rn, p = 0, 1,2, 3. We note that with Q = k + p , we may write (6.1.7) (6.1.8) and upon using the elementary inequalities +2kP I 21kl I P I , 2IQIp 5
Q2
-2Qp I 2 I Q I IPI,
+ p2,
2lklp I k2
+ p2,
(6.1.9)
we obtain
+ p 2 ) I (k2 + p 2 ) A , (k2 + p 2 ) I (Q2 + p2)A,
(Q2
(6.1.10) (6.1.11)
or (Q2
+ p2)A-'
I (k2
+ p 2 ) I (Q2 + p 2 ) A ,
(6.1.12)
where A = l + - +IPI+ . P 2 P
Let D(P, K , PI =
n I
P
(Q:
+ p:).
(6.1.13)
(6.1.14)
Then from (6.1.6), the inequalities in (6.1.12) and the definition (6.1.14) we may find a strictly positive constant G depending on P and P:, respectively, such that3
D(0, K , p) means that the elements in P are set equal to zero in D(P, K ,p),
6 Asymptotic Behavior in Quantum Field Theory
138
The inequalities in (6.1.15) imply, in particular, that the absolute convergence of fa.. dK P(PF,K , p)D-'(P:, K , p ) imply the convergence of
and of
Let AP = (Ipy, . . . , I&, I > 0; then
d ( I P , p) =
1 Ilalpa J a
dK
E(K,p ) D - ' ( I P , K , p),
(6.1.16)
R4"
where la1 2 d ( G ) + 1 for d(G) 2 0. Suppose I I 1; then the left-hand side of the inequality in (6.1.15) implies that dK I %(K, p ) ID- '(JP, K , PI IG ( P ) S.4.
IR4" I Eb(K I dK
p)
D -'(0, K , PI, (6.1.17)
and its right-hand side is independent of A.4 Now we consider the limit I -+ 0 and apply the Lebesgue dominated convergence theorem [Theorem 1.2.2(ii)] and conclude from (6.1.17) that we may take the limit I --t 0 inside the integral in (6.1.16), and for d(G) 2 0 we have lim d ( I P , p ) = 140
lim Alalpa a
dK Pa(K,p ) lim D- ' ( I P , K , p ) 1-0
1-0
Accordingly, we may state the following theorem.
0,
Theorem 6.1.1 : For d(G) 2
lim d ( I P , p) = 0,
(6.1.19)
1+0
and from the left-hand side inequality in (6.1.15) we have I d ( I P , p ) I IC 0 I d ' G ' + 1 , Note that G(AP) 5: G ( P ) for L
I; 1
[see (6.1.13)].
0 II I1,
(6.1.20)
6.2 Analysis of Subtracted-Out Feynman Integrands
139
where
and, as we have seen above, the integral in (6.1.21) exists. Finally we write qyc = (r]pl, . . .,upp) to obtain (6.1.22) where P
do(G) = d ( G ) -
1 i=
CT~S
d(G).5
(6.1.23)
1
Accordingly upon identifying q-' in the integral in (6.1.22) with 1 in (6.1.18) we obtain lim c d ( P ,rl, PI 4+m
=
1 lirn yldo(')-la1 a
d K g a ( K ,p) lirn D-'
4-00
4-00
(6.1.24) for all d(G), since if d(G) < 0, then la 1 2 0, and if d(G) 2 0, then la1 2 d(G) + 1. Hence we may state the following theorem. Theorem 6.1.2
(The decoupling theorem) :
For all d(G), i.e., for d(G)
lim d ( P , r]p) = 0
(6.1.25)
< 0 or d ( G ) 2 0, 9-00
and
IJW,?P)l I rl-'co,
r]
2 1,
(6.1.26)
where Co is dejned in (6.1.21). 6.2
GENERAL DIMENSIONAL ANALYSIS OF SU BTRACTED-OUT FEYN M A N INTEG RAN D S
In this section we carry out a general dimensional analysis of subtractedout Feynman integrands. This analysis will then be applied in the remaining part of this chapter to investigate the asymptotic behavior of subtracted-out Feynman amplitudes. See the definition of the positive integers u iin (2.2.17).We assume (without loss ofgenerality) that degr,,, .?p = degrp,K,p.?in (2.2.17) [see also (6.5.4)J
1 40
6 Asymptotic Behavior in Quantum Field Theory
In (5.4.14) and (5.4.17) we have reduced the expression for the renormalized Feynman integrand, associated with a proper and connected graph G, to the form R = (1
-
TG)AGlG,
c
I-I
(6.2.1)
where AGIG=
c 0cG’gG
[
(-T+
(6.2.2)
S I....,n(G‘) B E S I..., , dG’)
with S1,,.,,,(G’)= &G;) u . . . u
b(GL).
(6.2.3)
The sum C Q c G , g G in (6.2.2), for G’# 125, is over all proper subdiagrams G’$L G, with G;, . . . , GL denoting the connected components of G’ and d(G;) 2 0,. .. ,d(G:) 2 0. &G:) is a set with Gf the largest subdiagram in it; if g E D(G:), then g c GY and g is proper and connected with nonnegative dimensionality; if g,, g2 E b(Gi), then either g1 n g2 = 125 or 9, $L g2 or g2 $L 9,. We define S , ,..., = 0 and ( - T O ) = 1. We note that if gl, g2 E S,,...,,(G’),then g,, g2 are proper and connected and if g1 c G : , g2 c GJ, with i # j in [l, . , . , n], then g1 n g2 = 125. Ifg,, g2 c G ; ,for some i in [l, . . . , n], then either g1 n g2 = /a or g, c g2 or g2 c 9,. The proper and connected subdiagrams G;, . . . , C:, are called the maximal elements in S , , ...,,,(G‘) because for any g E S1,...,,,(G’), there is an i E [l, . . . ,n] such that g c G:. Let g E S , , ...,,,(G‘) and suppose g l , . .. ,gmare the maximal elements in S , , ...,,,(G‘), with g,, . . . ,gm$ g. Again the latter means that ifg’ E S , , ...,,,(G’), withg’g g,theng‘c g,forsomeiE[l, ..., m].Wedenoteg/g, u ug,by g. By definition, gl, . . .,gmare proper and connected and pairwise disjoint. As usual, for convenience, we shall remove the restriction on the g in (6.2.2) with d(g) 2 0 by allowing the proper and connected subdiagrams g in (6.2.2) with d(g) < 0 as well by simply setting ( - q)= 0 in the latter case. We may rewrite (6.2.1), (6.2.2) in the form
,,(a)
R
=
X
0cG‘cG
[
C
n
(-q)I1G,
(6.2.4)
S I , ..., dG‘) B E S I..... n(G‘)
where n = 1 for G’= G. We note that a set S , ( G ) = S(G) is of the form S ( G ) = {G}u S1,...,”(G’),where G ; , . . . , GL (the connected components of a proper subdiagram G’)are the maximal elements in S(G) contained in G:G: $L G for i = 1 , . . . ,n. Let I be a 4n-dimensional subspace of R 4 n + 4 m + p associated with the 4n R4n+4m+p integration variables. Let E be a complement of I in R4n+4m+p: = I @ E (see Section 1.3 and Chapter 3). In turn let El be a 4m-dimensional
6.2 Analysis of Subtracted-Out Feynman Integrands
141
subspace of E associated with the components of the independent external momenta. Let E 2 be a complement of El in E ; then we may write R4n+4m+p = I‘ 0 E l = I” 0 E 2 , and we denote by A(E), A(I’), A(I”), A ( I ) the projection operations on the subspaces I , El, E 2 , E along the subspaces E , I‘, I”, I , respectively. We choose E to be the orthogonal complement of I in R4n+4m+p and E 2 to be the orthogonal complement of El in E . The subspace E 2 will be associated with the relevant masses in the theory. such that the elements in K , P , and p may Let P’be a vector in R4n+4m+p be written as some linear combinations of the components of P . Suppose P is of the form
P’ = Liq1q2
“‘qk
+
”‘
+ L k q k + C,
(6.2.5)
where 1 I k I 4n + 4m + p, and Ll, . . . ,L; are k independent vectors in , C is a vector confined to a finite region in R4n+4m+p, such that pi + 0 for all i = 1 . . . , p. Let S: = {Li, . . . , Lk}, where r is a fixed integer in 1 5 r 5 k. Suppose that A(E)S: # {0}, i.e., A(E)S: is not the zero subspace. We carry out an analysis similar to the one in Section 5.3.2 for grouping the Taylor operations in (6.2.4)with respect to the parameter q,. Consider a set { G } u S1,,.,,,,(G’) = S(G). Let g be a (proper and connected) subdiagram in S(G). Let c(g) be the set of all lines in g which have their kfj, independent of v , . ~We may, symbolically, use the convenient notation: 0 = c(B) = 9. [ ~ 4 n 4+m + p
(i) Suppose 0 $ c ( g ) $ g, and introduce the subdiagram g o = g - c(8) as the subdiagram consisting of all the lines L P - c(g) and of the relevant vertices as the end points of these lines. Let g i , . . . ,g h be the maximal elements in S(G) contained in g : gi $ g . Then, by construction, all the kfj, in go = go/g; u . . . u g ; are dependent on q,.’ A similar analysis to that in Section 5.3.2 shows that go is proper and all the k f j in go are dependent on q r . Let g o l , . . . , goN be the connected components of g o . For any subdiagram g E S(G) such that 0 $ c(g) $ we introduce the set O(g) = { g o l , . . . ,gON}. We then define the set M 3 S(G) by (6.2.6) where
u$denotes the union over all sets
O(g) with g E S ( G ) such that
0 $ 4 8 ) $ 3. ‘As usual, we say that a four-vector kf,,depends on q, if at least one of its four components depends on q.. Otherwise we say that the four-vector kfj, is independent of q,. ’ If g is a minimal element in S(G), i.e.,g’, = . . . = gk = 521, then S = g and So = go.
142
6 Asymptotic Behavior in Quantum Field Theory
(ii) Let gES(G) with g $Z G and consider the case where c(g) = 0. Suppose g is a maximal element in S(G) contained in a subdiagram g’: g g’. If all the k$ in S’ are independent of q,, then we introduce the set B(g) = {g}, and otherwise we write p(g) = 0. (iii) Let gES(G) and consider the case where c(g) = g; then we write =
0.
We introduce sets Q X and g o as obtained from the set S ( G ) defined by
(6.2.7) 9 0
=
N - 9”v,
(6.2.8)
where ui p(g) is the union over all sets p(g), with g in category (ii) or (iii), i.e., with c(8) = 0or c(8) = ij. By construction, for any g E JV all the kfjl in ij are either all dependent on qr or are all independent of q,. From the set gXin (6.2.7)we may generate any one, all possible sets which contain in addition to the elements in or any two, or . . . or finally all the elements in go.The collection of all these sets together with the set gXwill be denoted X = { g X.,. . , QX u go}. The corresponding Taylor operations in (6.2.4) for the subdiagrams in the set A’”may then be readily combined, and the corresponding sum over all the sets in X is then reduced to the elementary expression
n
FX(G) =
- q)IG?
(6.2.9)
if g E g 0 if g E g N .
(6.2.10)
geX
where
d:={
1 0
By summing over all distinct sets JV we then obtain an equivalent expression for R in (6.2.4) given by
(6.2.11) with d$ = 1.’ We introduce the following notation: ./v = XlW) u
X2(.w,
(6.2.12)
where g E Xl(J(T) if all the kTj, in g are dependent on q,, and g E S 2 ( N )if all the kfjl in S are independent of q,. We also write Xl(Jlr)= S l ( J ( T ) u
92(JVh
(6.2.13)
Zimmermann (1969) was the first to reduce his Bogoliubov-type subtraction scheme to a form as in (6.2.11) and obtain the estimate of the form in (6.2.44) and (6.2.51).
6.2 Analysis of Subtracted-Out Feynman Integrands
143
) goif G E X l ( N )and R2(N)= g o - {G} if G E X 2 ( N )We . with < F 2 ( N= and 8 ; = 0 ifg E s 1 ( Nu) X 2 ( N ) . note that for g $ G, 8; = 1 ifg E R2(N), We note, in particular, that for g $ G , with g E R2(N),if g is a maximal element in N contained in a subdiagram g’ E N :g $ g’, then all the k$ in a’ are independent of qr. We use the following convenient recursion relation as obtained from (6.2.11) : F 8 ( 4 = (8:
- qYg
n F81(J1T).
(6.2.14)
i
where
R
=
cFG(N),
(6.2.15)
.K ’
and {gi}iin (6.2.14) denotes the set of the maximal elements in N contained in g: gi $ g.9 We also write
(6.2.16) and set L(Qr)G 0. A line L, in G joining a vertex vi to a vertex v j is, in Euclidean space, of the form (see (5.1.1)) DGdQijl, PijJ = PijLQijl, PijJ/CQ$
+ ~$1.
(6.2.17)
We assume that degr D$ I - 1,“’
(6.2.18)
Pill
and degr Pij, I dtgr Piif, QiiisPij1
Qijt
degr D;, I degr D;,. Qzjl. ~ 1 1 1
(6.2.19)
QiJl
The reason for using the expression in (6.2.14). (6.2.15) rather than the one in (5.3.48). (5.3.54). (5.3.55) for R in our asympotic analysis is that it turns out that the latter leads, in general, to an overestimate for thedegr R over the former one. The expression (5.3.48),(5.3.54). (5.3.55) for R is, however, by far much simpler in structure that the one in (6.2.14), (6.2.15). l o For example. for spins 0, I, 2: D& = O(p;;), and for spins 4, : D;l = 0(pGl1),and (6.2.19) is true as well. Equations(6.2.18). (6.2.19) will be used when dealing with the asymptotic behavior of .dwhen some of the underlying masses “take o n ” some asymptotic values, and they will not be needed when all the masses have fixed nonzero finite values.
144
6 Asymptotic Behavior in Quantum Field Theory
Throughout this chapter, since we are interested in the asymptotic behavior of d with respect to the external momenta and/or to the masses of G, we assume that A(1)S; # {O}. Now we prove the following very important lemma for subdiagrams g E N - {G}. The case for the graph G will be treated separately. Lemma 6.2.1 : Let r be a $xed integer in 1 I r 5 k. Suppose that A(E)S; # {O}. Let F,(N)be as dejned in (6.2.14) and suppose that g E N - {G}. For g E Tl(N)u X 2 ( N )ifthere , is
(i) a subdiagram g' $ g with g f E T2(N), andlor (ii) a subdiagram g" t g with g" E Xz(N),such that at least one mass pijrin depends on q,,
a''
then
degr F , W ) < d(g) - 4s).
(6.2.20)
9r
Otherwise, i.e., f(i) and (ii) are not truefor g, then we may replace the sign c in (6.2.20)by <. The latter means that in such a case an equality in (6.2.20) may hold.
The proof is by induction. We suppose that the lemma is true for all those maximal elements g i $ g with g i E Pl(N)u X z ( N ) .In addition to this hypothesis, we suppose, as part of the induction hypotheses, that if we scale the k$ in ij' for all g' E .Xl(N),g' c g i , for all i, and scale as well the masses pijl in the g i , depending on q l , by a parameter I, then
I - 1 - a(gJ (6.2.21) 1
for a g i E X z ( N )and ,
min[d(gi),
- 11 -
a(gi)
(6.2.22)
1
for a gi E .F2(N).11 The condition = O in (6.2.21) and an equality in (6.22) may hold if both conditions (i) and (ii) in the lemma are not true for the corresponding gi with g in these conditions (i), (ii) formally replaced by gi. In addition, suppose that for a gi E P 1 ( Nu) X2(N),an Fgi(N)has a structure as in F,,(N)= Pgi(kBi, q8',p)f:,(ke', p)f:i(kg1, P), It
The notation in (6.2.21) and (6.2.22) will be used whenever convenient,
(6.2.23)
6.2 Analysis of Subtracted-Out Feynman Integrands
145
where P,, is a polynomial in the elements in the sets kei, qgi, and p, and, in general, in the ( p i ) - as well. Here we denote by f : ( k g , q8, p ) any function of the form f i ( k , , 48, p) = [(@$,)2 + p$,]-4J1, (6.2.24)
n
Hi'
g ' ~ J l O ~ ( i t ri j) f g'cg i<j
for n = 1, 2, where @yjl = Qfjf,and for g' $ g, g' E #"(A'-), = k$. The f l y j f are strictly positive integers. We also denote byf"(ke, qg, p ) any function of the form f"(kg, 4'1 = [Qfjf + ~ $ 1 ""li (6.2.25)
ng
ijl i<j
for n = 1, 2. If the qyjfappearing in (6.2.24) and (6.2.25) are set equal to zero, then we denote the corresponding functions f ; ( k e , p ) and f"(ke, p), respec) , we suppose that the corresponding tively. Finally, if a q i e P 2 ( J f / 'then F , i ( N ) has a structure as in F g i ( 4= PJk,', qgi, p)fji(kgi, qsi, p)f,,(2
(6.2.26) We prove the above lemma as well as the results in (6.2.21)-(6.2.26) for the subdiagram g as well. [A] Suppose g E Z 2 ( N ) . Then the g i E X 2 ( N ) u P 2 ( NWe ) . write kei = kei(ke),qsi = qg'(ke,4'). Let 91, . . . , g k l E #2(N) and g k r + 1, . . ,s k i + k z E P2(N).From the induction hypotheses we may then write in a compact notation k9'I,
P).
c (qg)A+BP$(kg,0, P ) f W , n f,z,(k"(k", ki +kz
F,(Jf/') = -
p)
A. B a.6.c
degr a
5;+ degr fji + degr
degr P$ A
a
+ degr a
PI
i= 1
f:j,cj
I - 1- 4gi) ffi
(6.2.29)
7
1
+ degr fi,{ a
5)
min[d(gj),
- 13 -
u(gj). (6.2.30)
146
6 Asymptotic Behavior in Quantum Field Theory
Accordingly if degr, 1 ; I- 1 and/or at least one of the conditions = O (for some i ) in (6.2.29) or an equality in (6.2.30) (for somej) does not hold, then we obtain degr F , ( N ) I.
-1
(6.2.31)
- o(g).12
1
If k , = 0, (i) and (ii) in the lemma are not true for the g i , i = 1, . . . , k , , i.e., the conditions = O in (6.2.29) hold for the gi,and degr, f i = 0, then degr F,(N)= 0,
(6.2.32)
a
and a(g)
=
0. Finally, from (6.2.27)-(6.2.31) we have
{I
degr F,W) or d(g) Ir
-
ds),
(6.2.33)
where an equality in (6.2.33) may hold if the conditions (i) and (ii) in the lemma are not true for g. We also note from (6.2.27)that F , ( N ) has a structure as in (6.2.23). [B] Suppose g E .Fl(N).Then the g iE 9 , ( N )u X 2 ( N )According . to the induction hypotheses we may write
where
+
1-41 IBI
lai/
+ degr P: + degrf; 1
I d(8) - 4L(g),
+ degr Pf;+ degr fii + degr ff, a 1
(6.2.35)
1
d(gi)- o(gi). (6.2.36)
1
Accordingly we have
For degr, F , ( N ) # 0, the expression on the right-hand side of the inequality in (6.2.31) gives an upper bound value for degr, F,(.&").
147
6.2 Analysis of Subtracted-Out Feynman Integrands
where, again, an equality in (6.2.37)may hold if both (i) and (ii) in the lemma are not true for g. From (6.2.34)we also note that F , ( N ) has a structure as in (6.2.23). [C] Finally, suppose that g E S2(N).Then the gi E X 2 ( N u ) Sl(N), and by the induction hypotheses we may write F,(N) = a
n
(qeY'~~(kei(ks), q W e , O), p)f
#YW, p )
i
x f ,2,(k"(k", p)[1 - T p ' - q I g .
(6.2.38)
For d(g) 2 la 1, Eq. (5.3.26) in Lemma 5.3.3 implies that degr [l - T:'8)-Iat]IgI d(8) - 4L(ij) - d(g)
+ J a l - 1,
(6.2.39)
I
where we also have degr I , I d ( 8 ) - 4L(g).
(6.2.40)
A
From (6.2.38),(6.2.39), and (6.2.36) we then obtain for d(g) 2 0
I:[
degr F , ( N ) or I
-1
-
a(g).
(6.2.41)
If d(g) < 0, then (6.2.38),(6.2.40),and (6.2.36) imply that (6.2.42) Equations (6.2.41) and (6.2.42) may be combined to yield min[d(g),
- 13
- a(g),
(6.2.43)
I
in the notation in (6.2.22). O n the other hand, (6.2.38) shows that F & N ) has a structure as given in (6.2.26). This completes the proof of the lemma together with the results in (6.2.21)-(6.2.26) for the subdiagram g itself. Now we apply the above lemma to the graph G in question. [I] Suppose G E F2(N). Then (6.2.43) implies that degr F , ( N ) c 1
in the notation of (6.2.22).
- a(C),
(6.2.44)
148
6 Asymptotic Behavior in Quantum Field Theory
Let G , , . . . ,Gm be the maximal elements in N contained in G: Gi $ C . We may then directly apply the estimate in (6.2.37) to obtain for d(G) 2 0 m
degr( - TG)Zafl FG,(N)
(6.2.45)
i= 1
Ir
Also, (6.2.20) and Lemma 6.2.1 imply that
n
m
m
degr I a
[d(Gi) - o(Gi)].
FG,(N)
(6.2.46)
i=l
P J ~
To simplify the notation, let G' be that subdiagram of G with c' = G ' / u y = Gi corresponding to all those lines in G depending on qr. Then from (6.2.45) and (6.2.46) we may write
c o(Gi), m
- 4L(Q -
d(G')
i= 1
'Ir
m
- 4L(G') -
1o(Gi)]
(6.2.47)
i= 1
for d(G) 2 0, and (6.2.46) implies that
c o(Gi) m
d(G') - 4L(G') -
(6.2.48)
i= 1
'Ir
for d(G) < 0. An equality in (6.2.47), (6.2.48) may hold if there is no subdiagram g' $ G such that g' E S2(N),and there is no subdiagram g" c G in 3Ep2(N)such that at least one of the masses pijl in depends on q r . [11] Suppose that G E 3Ep2(N). Then (6.2.31) implies that for d(G) 2 0
a''
degr(-
TG)lG
n
FG,(N)
< -o(G),
(6.2.49)
i
1
where the Gi denote the maximal elements in N contained in G: Gi $ G.i Since degr, 1,- = 0, (6.2.31) and (6.2.43) imply, when applied to the G i , that degr Ic
ci
1
fli F G I ( ~ <) - O ( G ) ,
(6.2.50)
Hi
where o(G) = o(G!). If d(G) < 0, F G ( N ) coincides with I E FG,(N); accordingly from (6.2.49) and (6.2.50) we always have, i.e., for d(G) < 0 or for d(G) 2 0, degr FG(N)< -o(G). 1
(6.2.51)
6.2 Analysis of Subtracted-Out Feynman Integrands
149
Suppose that the Gi are such that G,, . . ., Gkl, G k l + l , .. ., G k l + k z ~ 9 z ( J V ) with d(Gi) 2 0 for i = 1 , . . ., k,, and d(Gi) < 0 for i = k, + 1 , . . . , k, + k , ; and G k ,+kz+ . . . , G , E sEc;(JV). Let { G i j } be the set of maximal elements in JV contained in G i : Gij $ G i . Let GI be that subdiagram of Gi with Gi = G f / U jGij corresponding to all those lines in Gi depending on q r . Let Gfj be that subdiagram of Gij with GIj corresponding to all those lines in Gij depending on q,. We may use (6.2.33), (6.2.47), and (6.2.48) to write
,,
d(G‘) - 4L(G‘)
+
9.
-
c a(Gij)] + i
i= 1
max[d(Gi) - 4L(Gi), d(GI)
m
1
(6.2.52)
[d(Gi) - o(Gi)]
i=kl+kz+l
for d(G) < 0, and
max[d(G), d(G‘) - 4L(G‘)]
-kk1ik2{max[d(Gi)
i=l
4L(Gi),d(G:) - 4L(G;)] -
c a(Gij) i
m
(6.2.53) for d ( G ) 2 0. An equality in (6.2.52) may hold if the conditions (i) and (ii) in Lemma 6.2.1 are not true for all the Gi. An equality in (6.2.53) may hold if (i) and (ii) in Lemma 6.2.1 are not true for all the Gi and, in addition to these constraints, we have
Now consider the situation when A(E)S: = (0). In this case the integration variables are independent of qr. Then we may simply replace Sf by 0 for g $ G in (6.2.14) and, as usual, replace :8 by 1, and JV “becomes” simply Z Z ( J V )= S(G). We may then use directly the estimates in (6.2.52)
6 Asymptotic Behavior in Quantum Field Theory
150
and (6.2.53) by replacing k, and k , in them by zero, as well as setting o(Gi) = 0, and obtain m
d(G') - 4L(G')
+ 1 d(Gi)
(6.2.54)
i= 1
vr
for d(G) < 0, and
f<1
rn
max[d(G'), d(G') - 4L(G')]
+ 1 d(Gi)
(6.2.55)
i= 1
vr
for d(G) 2 0. An equality in (6.2.54) and (6.2.55) may hold if the Gi have no masses depending on q, . This completes our dimensional analysis of subtracted-out Feynman integrands R and will be applied in the remaining sections of this chapter. We note in particular that we may write a(G)
+4
L($) = 4L(G),
(6.2.56)
0' E H A X ) g'cG
and quite generally we have dim A(E)S: Ia(G).
(6.2.57)
In particular, to have an equality in (6.2.57) the subspaces S: must be such that dim A(E)S: is a multiple of 4, and all the four components of the kfr,, in a with g E X,(N),depend on q,. Let P be a vector in E, and L,, . . . ,Lk be k independent vectors in E such that
P
=
Llql
* * '
qk
+ + Lrqk + * * *
* * '
+ Lkqk + c,
(6.2.58)
where C is confined to a finite region in E, with the pi # 0 for i = 1, . . . , p, and the external momenta and the masses of the graph G in question may be written as some linear combinations of the components of P. We may then write d ( P , p)
d(Llq1
'*
qk
+ + Lkqk + c). * ' *
(6.2.59)
According to Theorem 3.1.1, the power asymptotic coefficient a,(S,) of d ( P , p ) associated with a subspace S, = { L l , . . . , L,} is given by a,@,) = max [a@) A(I)S = S,
+ dim S - dim S,]
(6.2.60)
where a(S) is the power asymptotic coefficient of the with S c R4n+4m+p, integrand R, and, according to the analysis in Chapter 2, may be identified
6.3 High-Energy Behavior
151
with the degree of R . In the subsequent sections we shall construct the class of the maximizing subspaces A (see Chapter 3) for the various situations at hand directly from the dimensional analysis carried out in this section. This will lead to both the power and logarithmic behavior of d . We recall that if S E A, then a,(S,) = a(S)
+ dim S - dim S,.
Note that for any S c R4n+4m+p ,dim S Idim A(I)S dim S - dim A(I)S 5 dim A(E)S.
6.3
(6.2.61 )
+ dim A(E)S
or
HIGH-ENERGY BEHAVIOR
In this section we are interested in the high-energy behavior of renormalized Feynman amplitudes with all the masses involved in the graph G in question fixed and nonzero. Technically we are interested in the behavior of ed(Llq1 " ' q k
+
' * '
+ Lrqr
* ' '
qk
+ + Lkqk + c), * ' '
(6.3.1)
for q l , q,, . . . , q k + 00 independently, where L,, . . . , Lk are k independent vectors in E , and 1 I k 5 4m. C is a vector confined to a finite region in E such that pi # 0 for all i = 1 , . . . , p. As usual we write S, = { L l , . . ., L,} with 1 5 r Ik. (Recall that R4"+4m+"= I @ E , E = El @ E,, where E , is associated with the external momenta.) We may specialize Lemma 6.2.1 to the problem at hand through the following corollary [see also (6.2.5)]. Let r be a fixed integer in 1 5 r Ik . Suppose that A(E)S: # (0). L e t F , ( N ) beasdefinedin(6.2.14)andsupposethatgEN - { G } . For g E Y l ( N u ) X2(M),if there is a subdiagram g' $ g with g' E Y 2 ( N ) , then
Corollary 6.3.1:
degr F,(J-) < d(g) - 4s). 9r
Otherwise, if there is no subdiagram g' $ g with i g' E P2(N), then the < sign may be replaced by I, which means that an equality may hold for the latter case.
By treating the situation for the graph G in the light of Corollary 6.3.1 we arrive to the estimates for degr, FG(N)in (6.2.47) and (6.2.48) for G E P2(N),and to the estimates (6.2.52) and (6.2.53) for G E JEL2(N)by completely deleting, in the process, condition (ii) in Lemma 6.2.1, as all the pi are fixed (nonzero), i.e., are independent of q,. Similarly, for A(E)S: = (0)
152
6 Asymptotic Behavior in Quantum Field Theory
we have the estimate in (6.2.54) and (6.2.55) with a possible equality holding, as, again, all the masses are kept fixed. For each line el joining a vertex ui to a vertex uj of the graph G , we introsuch that, with P as given in duce vectors V0(iJ), . . . , V,(ijl) in R4n+4m+p (6.2.5). Vo(ijl) P = Q$l, (6.3.2) V,(ijl) * P’ = Qil.
-
We denote by So($) the subspace generated by the vectors Vo(ijl),. . . , V3(ijl). We also introduce vectors Vb(ijf), . . . , V;(ij/) such that Vb(ijl) * P’ = /&
-
(6.3.3)
V;(ijf) P‘ = k;,.
A close examination of Corollary 6.3.1, together with the estimates (6.2.47),(6.2.48),(6.2.52)-(6.2.55) for the present situation with fixed masses, as discussed above, after summing over JV in (6.2.15), suggests defining the following class d oof subspaces, which turns out to form, in general, a subset of the maximizing subspaces for the I integration of R relative to S, = {Ll, . . . , L,}, with L 1 , .. . , L, as given in (6.3.1). Definition of class M o : We define a class Jt0 = { S ’ , . . .} of subspaces ~4n+4m+p and a set of subdiagrams zo = {G‘, . . .} in such a way that
s
the following are consistent: (i) A(Z)S = S,. (ii) Let G’ be the subdiagram of G , corresponding to all those lines in G (and, of course, corresponding to the vertices as the end points of these lines), such that all the subspaces So(ijf)of G in G’ are not orthogonal to S’.13 (iii) The proper part Gb of G‘ corresponds to all those lines in G with their Vb(ijI), . . . , V;(ijl)not orthogonal to S’.’“ (iv) If S” c R 4 n + 4 m + p is such that (i)-(iii) above are consistently true, with G”,in particular, corresponding to all the lines in G with the subspaces So(ijr) of G in G” not orthogonal to S”, then d(G”) Id(G’).If d(G”) = d(G‘), then S” E d oand G” E T ~ . In light of this definition, we say that the subdiagram G’ is associated with the subspace S’. l 3 The subspaces S O ( i j 1 ) are defined following Eq. (6.3.2).By all subspaces SO(ijl)ofG in G’ we mean the subspaces So($) for all i, j , and I pertaining to the lines and vertices of G in G’. l4 The condition that all the Vb(ij/),. . . V;(ijl) in the lines of G appearing in G’ be not orthogonal to S’ will be necessary in order to have dim A(E)S’ = 4L(Gb).
.
6.3 High-Energy Behavior
153
We note that G’/G’,(if not empty), with G’ E z0 associated with a subspace S’ E d oand Gb being the proper part of G’, corresponds to the external and improper (if any) lines of G’. An analysis very similar to the one given in Section 5.3.2 in reference to Fig. 5.11 shows that all the vectors Vb(ijl), . . . , V;(ijl) of the external and improper (if any) lines of G in G are orthogonal to S’. This is consistent with condition (iii) in the definition. Note, however, that condition (iii) requires that the whole proper part of G , not just a proper subdiagram of G’,have the Vb(ijl),. . . , V;(ijl) of G in its lines not orthogonal to S’. We also note that for any proper subdiagram g 2 Gb, we may write k$ - (qYjJ0(kC,0) = (kYjJo, or conveniently as Vb(ijr) - (qfjI)O(Vb(ijr),. . . , V;(ijl)) = VS,(ijl)’and V$(ijr)‘ P = ( k f j I ) O . Here(qfj,)’ is a linear combination of the Vb(ijl),. . . , V;(ijl) with i, j , and I pertaining to the lines and vertices in G/g [see (5.1.117)]. Since these latter vectors are orthogonal to S’, and Vb(ijf),with i, j , and I pertaining to G/Gb, is orthogonal to S’ as well, it follows that the V$(ijl)’in g/Gb are orthogonal to S’. Repeating the same analysis for the V!(ijl)’,. . . ,Vq(ijr)’, similarly defined, as well, we arrive to the conclusion that for any proper y $ Gb, all the VS,(ijf)’, . . . ,Vg(ijr)’in g/Gb are orthogonal to S’ since the Vb(ijl),. . . ,V;(ijl) in G/Gb are orthogonal to S‘. If the whole graph GET,, and is associated with a subspace S E ~ ~ , then (6.2.47), (6.2.48), applied to the problem at hand, imply for the set A” = { G } that we may take for an estimated degree of F c ( N ) , with respect to q r , the expression d(G) - 4L(G), i.e., degr F G ( N ) = d ( G ) - 4L(G),
(6.3.4)
,r
with 4L(G) = dim A(E)S. More generally, if G‘ E t o ,with G‘ $ G, and is associated with a subspace S’ E d othen , (6.2.52), (6.2.53), applied to the problem at hand, imply for the set A” = { G , Gb,, . . ., Gb,}, where Gb,, . . . , Gb, are the connected parts of the proper part Gb of G’, that we may take
degr F G ( N ) = d(G’) - dim A(E)S’,
(6.3.5)
,r
,
where dim A(E)S’ = 4 L(G;,) = 4L(Gb). On the other hand, if Gb, , . . . , Gb, are contained (as maximal elements) in some maximal elements G,,.. . , rn I n, with the latter contained in G : Gi $ G , in some set N, then according to Corollary 6.3.1 and the definition of (do, z0) [in particular, condition (iv)], degr,, Fc(J)T) for such a set A” will not be greater than the one in (6.3.5). A moment’s reflection then shows that the power asymptotic coefficient of R itself [see (6.2.15)] for a subspace S’ E d omay be taken to be
c,,,,
ct(S’) = d(G’) - dim A(E)S’,
(6.3.6)
154
6 Asymptotic Behavior in Quantum Field Theory
and that all the subspaces S’€Ao are maximizing subspaces for the I integration relation to S, = {Ll,. . . , L,}. The latter, in particular, follows from conditions (i)-(iv) in the definition of Ao,the power counting conditions (6.2.44), (6.2.51) [see criterion [A], (3.1.3) in Theorem 3.1.11, and (3.1.4), which imply that the power asymptotic coefficient a,(&) of d in reference to the parameter qr in (6.3.1) may be taken for the bound of Id1 [see (6.2.60),(6.2.61),(6.3.6)]:
(6.3.7)
a,@,) = d(G’).
Accordingly, we may state the following theorem. Theorem 6.3.1 : The power asymptotic coe@cient
u,(S,) of the renormalized
amplitude
Cd(L141 ’ ” V k
+ + L,qr***)lk + + Lkqk + c) ”’
’ * *
is simply given b y ~ , ( S R= ) d(G’),
Sr
c
El,
(6.3.8)
where G’ is any subdiagram in 7 0 , respectively in r, with 1 Ir Ik I4m. We note that when some (or all) of the external independent momenta of the graph G becomes large, specified by a parameter, say, q, -+ 00 in (6.3.1), then in reference to this parameter a subdiagram G’ E zo cannot have an extral vertex at which all the momenta carried by all the external lines t o G, at this vertex, are nonasymptotic.15 As a matter of fact, if G’ has an extral vertex, then the total external momentum at that vertex must be asymptotic. This follows from the fact that an extral vertex u i of G’ is necessarily an external vertex of G and an external line t , of G‘ attached to this vertex has its k$ independent of q,, and the corresponding q:; is dependent on q,, by definition of G’. Momentum conservation then requires that qy must depend on q, (q, -+ a).Conversely, G’ contains all those vertices of G at which the total momentum carried by the external lines to G, at each of these vertices, is asymptotic. Note also (for G’ G), G‘ cannot have a subdiagram, say, G i ,as one of its connected components with all the external momenta of Gi being nonasymptotic. Because, whether d(Gi) < 0 or d(Gi) 2 0, this can only “decrease the value” of a($) below d(G’), since then < - a(Gi) [see (6.2.43)]. Therefore the determination of the degrqrFG,(N) subdiagrams G’ E zo is not difficult. Finally, we recall that if any other subdiagram G” similarly defined as G is such that d(G”) 5 d(G’), then G” E 70 only if d(G”) = d(G’). Is
Such a point was also emphasized by Weinberg (1960, p. 847).
6.3 High-Energy Behavior
I55
The subspaces in A. do not, in general, constitute all of the maximizing subspaces for the I integration relative to S , = { L l , . . . , L,} due to the simple fact that if there is a proper and connected subdiagram g’ in a set Jf, with d(g‘) 2 0 and g‘ $ g , g’ E X2(Jf), in Corollary 6.3.1, then degr,, F , ( J f ) may still coincide with d(g) - a(g). Accordingly we may readily extend the definition of the class A. as follows. Consider a subdiagram G’ in t oassociated with a subspace S’ in Ao. Let J,. = {g’,, g i , . . . ,g;.} be the set of all proper, but not necessarily connected, subdiagrams of the proper part Gb of G’ such that the connected part of each of the subdiagrams in .Icp has a nonnegative dimensionality. In particular, we note that if each of the connected parts Gb has a nonnegative dimensionality, then Gb E J , , , by definition of J , , . Let g ; E J,. . We define a generalized subdiagram (G‘lg;) obtained from G’ by shrinking gIl in it to a point and replacing the analytical expression I,,,, in the unrenormalized integrand I,. for G’, by a polynomial in the external uariables of g’, of degree I d(g’,).Therefore (G’ 18;) is nothing but the subdiagram G’ with g’, in it replaced by a vertex, which we call a generalized vertex, with the corresponding analytical expression for the latter as a polynomial in the external varibles of g’,. In this respect, we also note from (5.1.1 17), that the external variables of g‘, may be expressed as linear combinations of the Qijrin G’/g’,,with the vi being vertices in G’/g‘,,but not in g’,, and the uj being external vertices of g’,, and also, in general, as linear combinations of external variables of G’.Accordingly we may formally define an integrand in the same way as we defined I , . . By considering all the elements in J,. , we may generate the following generalized subdiagrams: (G’lg’,), . . . ,(G’ Ig’,.). Finally, we repeat the above construction by considering all of the remaining subdiagrams in t oand generate: (G”I g i ) , . . . , (G” I &-), . . .; G”,. . . E to. We extend the definition of the class d oto the class A by simultaneously enlarging the set t o to include as well all the generalized subdiagrams (G’lg’,),. . . , (G’Ig;.); (G”Ig;‘),. . . ,(G”)g;;.,);. . . for all G‘, G”, . . .E t o ,as follows. To this end we define (G’ 10)= G’. Definition of class M :
We define the class
.I= {S,s;, . . ., s;,; S“,s;, . . . , She,;. . .},
where d o= { S ’ , S”, . . .), and the set of subdiagrams t = {G’,(G’lg‘l),. . . , (G’lg;), G”,(G”Ig‘;),. . . ,(G”(g;;..); . . .} such that the following are consistent: For any 3 E A (i) A(I)s = S,. (ii) Let be the subdiagram of G, corresponding to all the lines in G , such that all the subspaces So($) of G in G are not orthogonal to 3.
e
156
6 Asymptotic Behavior in Quantum Field Theory
e e,
(iii) The set of all the lines of G in having all their Vb(ijl), .. . , V;(ijl) not orthogonal to $ coincide with the set of all the lines in the proper part of the generalized subdiagram (GI& = where 4 is either empty, or otherwise 4 is a proper subdiagram of such that each of the connected components of 4 has nonnegative dimensionality. (iv) G belongs to zo (and obviously to z) and G belongs to z.16 As before, we also say that the generalized subdiagrams (G’lg;),. . . , (G’Igb.); (G”lg:), . . . ,(G”Ig&);.. . are associated with the subspaces S’,, . . . , Sb,; S‘;, . . . , S’h,,; . . . , re~pective1y.l~ The basic difference between a subspace S’ and S;, say, is that
e
a(S’) = d(G’) - dim A(E)S’
(6.3.9)
a@‘,) = d(G’) - dim A(E)S;
(6.3.10)
and where 4L(Gb) = dim A(E)S’, 4L(Gb/g;) = dim A(E)S;, and hence dim A(E)S; < dim A(E)S’. Both subspaces S’ abd Y1are obviously, however, maximizing subspaces for the 1 integration relative to S,, and from the definition of the set t otogether with (6.2.60), (6.2.61), we may take
a,(S,) = 4 G ’ )
(6.3.11)
when considering both subspaces S’ and S;. Now with the class A of the maximizing subspaces for the 1 integration of R relative to the subspace S , = { L , , . . ., L,}, with L,, . . . , L, as given in (6.3. l), we determine the logarithmic asymptotic coefficients p,(S,) of d . We decompose 1 as a direct sum of 4n one-dimensional subspaces: 1 = I , @ l , @ . . . @., l Lemma 6.3.1 : All the maximizing subspaces for I, integration relative to S,, afer performing the l 2 @ . . . @ 14,,,a w given in the set
A’ = { A(12 @
. @ 14,,)S: all S E A}.
(6.3.12)
All the muximizing subspaces for the 1, integration relative to any one of the subspaces in A’,say, S’ E A’,a f e r performing the I 3 @ @ I,,,, are given in the set A2= {A(l, @ . . . @ 14”)S;S E A and A(1, @
0 14,,)S = S ’ } . (6.3.13)
Of course the classes d oand .M depend on the integer r. Note that with the connected components g‘liE%2(.N) of a g;, all the F&,(.h.) are some polynomials of degree
”
6.3 High-Energy Behavior
157
More generally we have recursively that all the maximizing subspaces for the I i integration relative to any one of the maximizing subspaces in M i - ' ,say, Si- E Ai- after performing the I i + Q - QI 14, integration are given in the set
'
',
.Ai= { A ( I i + Q . . . @ 14,)S : S E A and A(Ii Q
. . Q 14,)S = S i - ' } , (6.3.14)
'
'
for i < 4n. W e set Ao= {S,} and we define for some S4"- E A4,A4"= {S : S E A and A(14,)S = S4"- '}.
(6.3.15)
Suppose that the lemma is true for all i in 1 I i I k < 4n. Let Sk be any given subspace in Ak.The latter means, in particular, that there is some subspace S' E A such that A(Ik+ 1 @
* *
0 14,)s
(6.3.16)
Sk.
We now show that for any subspace S E A such that (6.3.17)
A(Ik+ 1 6 . .. @ 14,)s= Sk
-
it follows that A(Ik + Q . Q 14,,)S is a maximizing subspace for the I k + integration relative to Sk after performing the I k + @ . . @ I,, integration. This follows from the following chain of inequalities: 'Ik+
-
I @ "'@14n(Sk)
-
max
[aIk+2e...e14n($ + dim
s - dim SkJ
A(lk+I)S=Sk
2
'Ik+
z@
...@14n(A(lk+
+ dim A(Ik -
+2
2
Q * * * Q I4n)V
Q . . . Q 14,,)S- dim Sk
max
[a(S")
+ dim S
- dim SkJ
A ( I k + z@ . ' . @ I 4 n ) S " = h ( l k + l @ '"@/4,)s
+ dim S - dim Sk = a,(S,) + dim S, - dim Sk
2 a(S)
-
max
[alk
+
@.
+
.. @ 14m(g) dim
- dim Sk]
A(ll@."@Ik)S=s,
2
-
[email protected],(Sk)
+ 1 @ "'@ 14"(Sk),
+ dim Sk - dim Sk (6.3.18)
where we have used the facts that S' and S in (6.3.16) and (6.3.17) are in A and the fact that (6.3.19)
6 Asymptotic Behavior in Quantum Field Theory
158
Since the extreme left-hand side and the extreme right-hand side of the inequalities in (6.3.18) are identical, we may replace all the inequality signs in it by equalities. Therefore we obtain, in particular, that 'II, + I
@
..' @14.(Sk)
=
'1,'
+
z@
..' @14n(A('k
+2 @
' *
0 I4t1)~)
+ dim A(fk+2 @ . - .@ I4,)S - dim S k ; (6.3.20)
i.e., I \ ( I k + @ . . . @ 14")Sis a maximizing subspace for the I k + integration relative to Sk after performing the I k + Z @ . . . Q 14, integration. Now we prove that any maximizing subspace for the I,+ integration relative to Sk after performing the l k + @ . . . Q 14, integration is in the set A k + l
= { ~ ( ~ k + 2 $ ' . ' @ ~ 4 , ) S : S E ~ a n d A (@ I k* + . *,@ I , , , ) S =
sk},
(6.3.21)
thus completing the proof of the lemma by induction.'* integration Suppose that So is a maximizing subspace for the relative to Sk, after performing the 1, + @ . Q 14,, integration, and that So 9 A k + l .We shall then reach a contradiction. By hypothesis I @'" @14n(Sk)
= 'Ik+ z@
"'
@14.(So)
+ dim '0
- dim S k ,
A(lk+ 1)So = sk.
Let
(6.3.22)
s be a maximizing subspace for the @ . @ 14,,relative to S o , i.e., aIkt2@...@f4,(So) = a(S) + dim S - dim S o , Ik+2
A(I~+ 2 0 ...
(6.3.23)
14,)s = SO.
From (6.3.22) and (6.3.23) we have = a(s) aIk+ ... e14m(Sk) A ( I k + 1 @ * * @ 14,)s = Sk.
+ dim s - dim Sk, (6.3.24)
Equation (6.3.18),however, implies that [email protected](Sk) = a,&)
+ dim S , - dim Sk,
(6.3.25)
which upon comparison with (6.3.24) shows that E A and hence, by definition of Ak+', A ( ] k + 2 @ . - .@ 14,,)S = So EA'", thus leading to a contradiction of the initial hypothesis that So $ Ak+'. Note that the proof does not depend on the fact that the I i are one dimensional, and the same Wenotethat theset.1'" (6.3.16). belongs to .XktI .
isnot emptysinceA(lkt2 @ . . . @ I,,)S',withS'introducedin
159
6.3 High-Energy Behavior
analysis as above shows that all the maximizing subspaces for the I , integration relative to S, after performing the I 2 Q . . . Q I,, integration are in A’.This completes the proof of the lemma by induction. Since the dimension numbers p j , j = 1, . . . , 4 n , in the expression for the logarithmic asymptotic coefficients fi,(S,) of A in (3.1.6) may be computed relative to any one of the maximizing subspaces for the Ij-l integration, after performing the I j Q - . Q I,, integration, we may use the results in Lemma 6.3.1 and (3.1.6) to state +
Theorem 6.3.2: The logarithmic asymptotic coeficients
fi,(Sr) of the re-
normalized amplitude .d are given b y 4n
PdSr) =
CPj,
(6.3.26)
j= 1
where p j , j = 1 , . . . ,4n, is equal to zero same dimension, and p j = 1 otherwise.
if all the subspaces in Aj have the
We shall simplify Theorem 6.3.2 further to a form more suitable for applications. Choose some subspace in Ao;call it So. Let A2in (6.3.13) be the set of all the maximizing subspaces for the I 2 integration relative t o A(12 0 . . Q I,,)SO after performing the I 3 Q . . Q I,, integration. By definition, we observe that A(I3 0 . . . 0 14,)S0 is necessarily in A2 [see (6.3.13)]. Recursively, then, let Aj be the set of all the maximizing subspaces for the I j integration relative to A(Ij Q . . . @ 14,)S0 after performing the I j , , 0 . . . 0 14” integration. Obviously A ( I j + Q 0 I,,)SO belongs to 4’ since A ( I j ) A ( j + Q . . . Q 14,)S0 = A(Ij Q . . . Q 14,)S0. Accordingly we may replace Theorem 6.3.2 by the following simpler version : Theorem 6.3.3: Choose So to be any subspace in d o Let . {S,S”,
set of all those subspaces in A such that A(Ij 0 . . * Q 14,)s’ = A(Ij Q *
.
*
. . .} be the
l9
Q 14,)s” =
-
* *
= A(Ij Q *
.
*
@ 14,)S0,
(6.3.27) then
(6.3.28) where p j = 0 ifall the elements in
{dim A(Ij+ 0 . . Q 14,,)S - dim S,, dim A ( I j + Q . . . @ I,,,)S” - dim S,, . . . ,dim A ( I j + Q . . . Q 14,)S0- dim S r } (6.3.29) l9
Obviously this set is not empty sincc it contains the subspace So itself.
6 Asymptotic Behavior in Quantum Field Theory
160
are equal, and p j = 1 otherwise. The subspaces S', S", . . . are given by (6.3.27). The dimensions of the A(Ij+ Q . . Q 14,,)S',. . . have been measured relative to dim S,. The chosen subspace So in d owill be called a reference subspace. In many applications, d oconsists of only one element d o= { S O } and Theorem 6.3.3 may be readily applied. From (2,1.1), the work in Chapter 2, Theorem 6.3.1, and Theorem 6.3.3 we may then state Theorem 6.3.4 d(L1ql
"'qk
-
+ + L r V r " ' q r + + Lkqk + c) . . qid{Lls...sLk)) (In qnl))'I - (In q n k ) y k } , " *
" '
(6.3.30) ....Y k where { L l , ..., Lr} = S , c E l , 1 I r I k I 4m, C is confined to a finite region in E , with the masses pi # 0 for all i = 1, . . . , p. T h e sum in (6.3.30) is over all nonnegative integers y l , . . . , y k such that o { q a 1I ( ( L I ) ) .
?I.
I
(6.3.3 1) for all 1 5 t Ik , and the logarithmic coeficients
P
have been arranged in
increasing order (6.3.32) P({L,, . . . , Ln,H I - - * 5 P((L1, - L n k h where {?rl,. . . , nk} is a permutation of the integers in { l , . . . , k } . The power al(Sr) and the logarithmic Pl(Sr) asymptotic coeficients are, respectively, given in Theorem 6.3.1 and Theorem 6.3.3. 7
Consider the self-energy graph G of a fermion shown in Fig. 6.1. Suppose we let the external momentum q become large. Consider the amplitude d ( q 4 , m, p ) and let q + co. Here m denotes the mass of the fermion and p denotes the mass of the boson, with both masses assumed to be nonzero.2o Let 1, = I , 0 ... Q I4 be associated with integration variables k,, and 1, = I, @ . . @ I s be associated with integration variables k , . I . . . , I , denote one-dimensional subspaces. We note that in Fig. 6.1, degr, D:3 = degro D14 = - 2 for the spin 0 propagation, and degr, D:2 = degrv D t 3 = degrQD14 = - 1 for the spin 4 particle. Also the dimensionalities of G, g , , and 9 , are as follows: d(G) = 1, d ( g l ) = d(g2) = 0. Obviously zo = (G}. As a matter of fact any other subdiagram g $C G has d(g) c 1. Canonical decompositions of the Q i j ,for G in Fig. 6.1 are given in (5.1.40), and the corresponding expressions for k$,, q$, k $ , qf$ are given in (5.1.81)-(5.1.84). Example 6.1 :
,,
2"
Further generalizations to this will be given in subsequent sections,
6.3 High-Energy Behavior
161
Fig. 6.1 A self-energy graph G of a fermion with a $+#I coupling contributing to i t with the dashed lines representing a spin 0 and the fermion is of spin f.
The graph G E T , , is associated with the subspace So = S ( q ) 0 1, 0 I,, where S ( q ) is a subspace associated with the momentum q. We readily infer from Theorem 6.3.1 with no further work that (6.3.33)
a,(S(q)) = d(G) = 1.
From (5.1.82) we note that 1, is associated with k$, and 1, is associated with k f j . From the definition of (A,T ) we may write
.x = {S', s;, s;,s;} 7 =
(6.3.34)
{ G , ( G I g A (Gig,), (GIG)},
(6.3.35)
where
s = so= S(q) 0 i, 0 I;, (6.3.36)
s; = S(q). The subdiagrams G , ( G l g J , ( G i g , ) , (GIG) are associated, respectively, with the subspaces S', S ; , S,, S ; . Hence in the notation of Theorem 6.3.2 and Lemma 6.3.1 we have .N' = (A(I2 0 .. @ 1 6 ) s : s = so, s;, s 2 , &}, (6.3.37) j=2,3,4,6,7,8,
J T J = { A ( l j + l 0 . . . 0 I 6 ) S ",)
JT5 = {A(Z6 @ 17 0 1 6 ) s : S = So, S;}.
(6.3.38) (6.3.39)
Accordingly by the application of Theorem 6.3.3, using {dim A(Z2 0 ... 0 1,)s - dim S ( q ) : S
=
So, S ; , S2,S3} = {l,O, l,O}, (6.3.40)
we obtain p ! = 1, and from (6.3.38) we obtain p j = 0 f o r j = 2, 3,4,6, 7, 8 since the A',for suchj, constitute only one element. Finally, from {dim A(16 0 Z7 0 1 6 ) s - dim S(q) : S
=
So, S2}= {5,4},
(6.3.41)
we have p s = 1. From (6.3.28) we then obtain (6.3.42)
162
6 Asymptotic Behavior in Quantum Field Theory
Therefore we may write (6.3.43)
6.4
GENERAL ASYMPTOTIC BEHAVIOR I
In this section we generalize the results given in Section 6.3 and consider the asymptotic behavior of d when not only some of the momenta become large but also some of the masses in the theory become large as well. The analysis here is similar to the one carried out in Section 6.3, and we shall be brief. Applications of this analysis will be given in the remaining part of this chapter. As in (6.3.2), we introduce for each line e, joining a vertex ui to a vertex v j of G, vectors V0(ijr), .. .,V3(ijr) and now an additional vector V,(ijr) such that, with P as given in (6.2.5),
Vo(ijr) P = QP,,, (6.4.1)
P V,(ijr) P V3(ijr)
= Q&, = pt,,.
We denote by S(ijr) the subspace generated by the vectors Vo(ijr), . . ., V3(ij0, V,(ijr). We also introduce vectors Vo(ijr), .. , ,V;(ijr) as in Section 6.3 satisfying (6.3.3). Technically we are interested in the behavior of d(LlV1 ‘ * *
tlk
+ + Lrqr ’ * *
*.*
vk
+ + Lkqk + c) ‘ * *
(6.4.2)
for ql, q 2 , . . .,tlk + 00, independently, where L 1 , .. . , Lk are k independent vectors in E = El 6 E 2, 1 5 k I 4m + p, C is a vector confined to a finite region in E, such that pi # 0 for all i = 1, . . . , p.
a class A. = {S’, . . .} of subspaces and a set of subdiagrams r0 = {G’,. . .} in such a way that the following are consistent: Definition of class .MAY,: We define
s
~4n+4m+p
(i) A(Z)S’ = S,. (ii) Let G’ be the subdiagram of G, corresponding to all the lines in G (and, of course, corresponding to the vertices as their end points) such that all the subspaces S(ijr) of G in G‘ are not orthogonal to S’. (iii) The proper part Go of G’ corresponds to all those lines in G with all their Vb(ijl),. ..,V;(ijl) [see (6.3.3)] not orthogonal to S’.
6.4
General Asymptotic Behavior I
163
(iv) If S“ c 5!4n+4m+p,with which a subdiagram G” is associated, is such that (i)-(iii) are true, then d ( G ) I d(G’). If d(G”) = d(G’), then S” E A. and G E to. By the same analysis leading to Theorem 6.3.1 we have Theorem 6.4.1 : The power asymptotic coeficients a,(Sr) of the renormalized
amplitude d(LIV1 with 1 I r I k I4m
”’
qk
+ .” + L r q r
* * *
Vk
+ + L k V k + c), ’ * *
(6.4.3)
+ p are given, respectively in r, by = d(G’),
(6.4.4)
where G‘ is any subdiagram in to. The subdiagrams G’ E to are determined as in the case for the high-energy behavior given in Section 6.3. There are, however, some differences in the nature of the diagrams in this case. For example, a subdiagram may contain an extra1vertex at which the total external momentum carried by the external line to G’ at that vertex is nonasymptotic as long as the external line of G’ attached to this vertex carries an asymptotic mass.” Definition of class A :
We define the class
A = {S’,s;, . . . ,SN,; S”,s;, . . . , s;,.;. . .},
where A. = {S’,S”.. .}, and the set of subdiagrams z = {G’, (G’lg’l),. . . , (G’lgh,);G”, (G”Ig);),. . . , (G”IgK,,);. . .} with to = { G’, G”, . . .}, such that the following are consistent: For any 5 E A (i) A(1)S = S,. (ii) Let be the subdiagram of G, corresponding to all the lines in G, such that all the subspaces S(ijl) of G in are not orthogonal to 5. (iii) The set of all the lines of G in G having all their Vo(ijr), . . .,V3(ijr) not orthogonal to coincide with the set of all the lines in the proper part of the generalized subdiagram 14) = G,where 4 is either empty or otherwise 8 is a proper subdiagram of such that each of the connected components of 4 has a nonnegative dimensionality, and all the masses in the lines in 4 are independent of q,. In the notation in (6.4.1) the latter means that the V4(ijl) of the lines in 4 are orthogonal to 5. (iv) G E to c t and G E T for all nonempty 0 as defined above.
e
e
(e e
I ’ In such a case. of course, the momentum-dependent part of this external line is nonasymptotic by momentum conservation.
164
6 Asymptotic Behavior in Quantum Field Theory
Theorem 6.4.2
L , ~ , ” ‘ ~ k + ‘ *+’L k q k
d(Llq1”’qk+“’+
..
- O { ~ 1~ I ( { L I.) ) ,,,;I((LI..-.L~H 71.
1 (In vJ1 ...
+ c, . . . (In q*,)YkI,
(6.4.5)
.Yk
where { L l , . . . ,L,} = S, c E, 1 Ir I k I 4m + p , C is conjned to a j n i t e region in E, with the masses pi # 0 for all i = 1, . . . , p. The sum in (6.4.5) is over all nonnegative integers y l , . . . , Yk such that I
1
Yi
i= 1
5
P({L,,
* * 3
Lzt}),
1 I t I k,
(6.4.6)
and the logarithmic coeficients P have been ordered in increasing order
P(L
* * * 9
LIDI
*
*. I P((L1, . . . I Ln&
(6.4.7)
where {n,,. . . , nk} is a permutation of the integers in { 1, . . . ,A}. a,(S,) is given in Theorem 6.4.1, and the P,(S,) may be obtained from Theorem 6.3.2 or dejned above.22 Theorem 6.3.3from the classes A,and 6.5
ZERO-MASS BEHAVIOR
The purpose of this section is to study the zero-mass behavior of renormalized Feynman amplitudes (Section 6.5.1) and finally give sufficient conditions to guarantee the existence of the zero-mass limit of renormalized Feynman amplitudes (Section 6.5.2).The study is general enough to deal with the most general cases when some (not necessarily all) of the masses of a Feynman graph G become small and, in general, at different rates. That we have to consider (i) the most general cases when some of the masses as well and not necessarily all the masses become small and (ii) the approach of such masses to zero at different rates as well are clearly physical requirements. In quantum electrodynamics, for example, one would be interested in the behavior of a renormalized Feynman amplitude for p -+ 0, m + 0, and (p/m) -+ 0, where p is a photon “mass” and m is the mass of the electron. For a propagator D&, carrying a mass pijrthat we wish to scale to zero, we write (in Euclidean space) D$I = P i j L Q i j r , P i j J / ( Q & + P&), (6.5.1) where Bijl(Qij,,p i j r )is a polynomial in Q i j r and pijl but not in (l/pij,) such that DGdQijls
0) = B i j L Q i j r ,
o)/Q?j~
(6.5.2)
’’ Of course. Theorems 6.3.2 and 6.3.3, as they stand, apply to the present situation as well.
Zero-Mass Behavior
6.5
165
denotes the mass p i p = 0 p r ~ p a g a t o r With . ~ ~ the propagator in (6.5.1) we then carry out the subtractions of renormalization as usual to obtain the final expression for R . All those propagators carrying masses which we do not wish to approach zero will be written as in (5.1.1). We note that in general we may rewrite (51.1) as (in Euclidean space with E = 0) (6.5.3) where d i j , is some nonnegative integer, and Pij, is a polynomial in Qijrand p i j r ,but not in l/pij,. Also quite generally we suppose that degr J‘ijdQijr?
PijJ
=
Qiir
degr
PijLQijl, PijJ
Qlif. Ptil
=
degr P i j / ( Q i j / ,
(6.5.4)
PijA
Qijr
and with D;, = ( p i j l ) - ” i i l D & we , have degr fi;, = degr Qtjfv@#Jl
DGI.
(6.5.5)
QtJf
The expressions (6.5.4) and (6.5.5) will be assumed explicitly. By working with the propagators 06,in (6.5.1) and in (6.5.5) we may introduce an unrenormalized integrand T, given by
n P
I,
=
-
(6.5.6)
(pJ)-OJlc
j= 1
in a form as in (2.2.17), where the oj are some positive integers. 6.5.1
Zero-Mass Behavior of d
The structure of a Feynman amplitude associated with a proper and connected graph G is, from (2.2.3), of the form
”
For example, in quantum electrodynamics we write for the spin f propagator (up
-
m)/
+ m2)and for the photon propagator (in the Feynman gauge)g,,/(Q’ + p2).Quitegenerally, wemayalsoallow higher powersofthedenominator in(6,5,I):(Q$ + @“,with integersn 2 I .
(p’
.
The latter does not necessarily mean that one isallowinga double,triple, . . . etc.. pole term in the propagator. as this depends very much on the structure of the polynomial P i j f .In any case we always have to take the correct dimensionality degru,,, D;f(Qijr, pijJ of D;r(Qijf, p i j f )when carrying out thc subtractions of renormalization.
6 Asymptotic Behavior in Quantum Field Theory
166
where [see (2.2.14), (2.2.17), (2.2.18)-(2.2.20), and (6.5.6)] P
R
=
fl
(6.5.8)
(p')-"jR,
j= 1
RPY,.
3. kO ., prn, 1 3 . . , k:; P ' , - - .
9
L
pP) =
1 A f , , , t , i P ' ~ ' i ~ "CQ:i l n+ d 1 9 i
1= 1
(6.5.9)
= pf' =
P'
( k y l
* . . (k+,
SLj
2 0,
(~!)'61
. . . (p:)f$m,
tij
2 0,
. . . (/p)4,,
u; 2 0,
p"'
(6.5.10)
xi
with Q,= afki + C j bjp,. The Af,,,,, are some suitable coefficients. Suppose that { p ' , . . . ,ps}, with s I p, denotes the subset of the masses that we wish to approach zero. We scale the masses in the set { p ' , . . . ,,us}as follows:
(6.5.11) ps + 1112
* * .
AspS.
The masses in the subset { p ' , .. . , p'} have been arbitrarily labeled from 1 to s for convenience of notation. Without loss of generality, we assume that all those masses that we wish to approach zero at the same rate have been identified with pl, or p2, or . . . , ,usdepending on the rate we wish them to approach zero, etc. By the definition in (6.5.2), the factor (pj)-"J in (6.5.8) is invariant under the scaling in (6.5.1 1); in particular, o1 = o2 = . - . = os = 0. Quite generally we have under the scaling (6.5.11).
mZl
k'fp''p"'
+
( I , . . . A,)d(N)(k')"'(P')'i(p')"',
(6.5.12)
where
(6.5.13)
6.5 Zero-Mass Behavior
I67
and d ( N ) is the dimensionality of the expression on the left-hand side of (6.5.12),i.e., d ( N ) = (sb1 + . . . + s i n )
+ ( t i , , + . . + ti,) + (u; + + u;), * * *
(6.5.14)
and is a fixed number of all i, for which A:,,,,, # 0, and coincides with the dimensionality of the numerator in (6.5.9).Similarly the denominator in (6.5.9)is transformed as L
n CQ:
1=1
L
+ d1
+
(A1
*
n CQ;’
I= 1
+ ~;’l,
(6.5.15)
where Q; = Q,/Al . ..A,. d(D) is the dimensionality of the denominator on the left-hand side of (6.5.15),i.e., d(D) = 2L.
(6.5.16)
Accordingly i? in (6.5.9)is transformed to
( A , . . . A,)d(R)d(p;o, . . . ,p z ; k;’, . . . , kk3; p’l, . ..,P ’ ~ ) ,
(6.5.17)
d(R) = d ( N ) - d(D).
(6.5.18)
where
Hence finally the renormalized amplitude d formed to
E
npZ1
( p j ) - u J g is trans-
(6.5.19) where d(G) = d ( N ) - 2L
+ 4n,
(6.5.20)
and we infer from (6.5.5)that d(G) coincides with the dimensionality of the graph G in question. We choose the external (independent) momenta pl, . ..,pm such that no partial sums of these momenta vanish; i.e., p i , + . + pi, # 0 for all subsets {il, . . . ,i t ) c ( 1 , . . ., This, in particular, 24 Such external Euclidean momenta have been called nonexceptional momenta (cf. Symanzik, 1971). We recall that these external momenta are momenta carried by the external lines to the graph G taken, by convention, in a direction away from the external vertices. In general, at each external vertex, the total external momentum carried away from that vertex may be written in the form f ( p i , . . . + pi,).
+
168
6 Asymptotic Behavior in Quantum Field Theory
means that the total external momentum carried away from each exernal vertex is nonzero. As the elements in p are also chosen to be fixed and nonzero, the factor L(pLi)-uJ is independent of the parameters A,, . . .,As. The amplitude J$ in (6.5.19) is of a particular form of the amplitude d analized in Section 6.4, where in the former the propagators are of the form in (6.5.1) and b$, introduced below Eq. (6.5.4). We decompose E 2 as E2 = E$ @ E i , where E i is an (s - 1)-dimensional subspace and E: is its orthogonal complement in E 2 ,We introduce orthogonal vectors L2, . . . , L, in E i with nonvanishing components p l , .. . ,ps- I , respectively. We may also introduce a vector L, in El @ E i with nonvanishing components py, . . ., p i , ps' ', .. . ,pp and a vector C E E: orthogonal to L1 with nonvanishing component ps. Finally, we may write .d in (6.5.19) in the form s?(P), where
m=
P=(A1*..AS)-'L,+(A2-**3Ls)-1L2 +
*
*
a
+
&'L,+C.
(6.5.21)
The conditions I,, . . . ,A,+ 0 in (6.5.21) mean, in particular, that all the external momenta become asymptotic and the total external momenta at all the external vertices are asymptotic. Obviously the vectors L1,. .., L, are s independent vectors in IW4n'4m'p. We may now apply Theorem 6.4.2 (see Section 6.4 for details) and infer from (6.5.19),together with (6.5.21), that
for Al, A,,
...,As -+
0, and where
(6.5.23)
d i = d(G) - a,(&)
for i = 1 , . . . ,s, Si= {Ll,...,Li}, with adsi) being the power asymptotic coefficients of 2 ( P ) with respect to the parameters l/&. The sum in (6.5.22) is over all nonnegative integers y,, . . . , ys such that (6.5.24)
for all t in 1 I t Is, and the B have been arranged in an increasing order:
B(IL1, - - *
9
L,})I I B({L,, * * *
* * 3
LZJ.
(6.5.25)
From the definition of the classes 4, and A in Section 6.4 and (6.5.22) we may then readily give sufficient conditions for the existence, with nonexceptional external momenta, of the zero-mass limit of renormalized Feynman amplitudes d for A,, .. . ,A, + 0.
6.5 Zero-Mass Behavior 6.5.2
I69
Rules (Sufficiency Conditions) for the Existence of lim .a?
Consider the amplitude d,as given in (6.5.19), associated with a proper and connected graph G. Let i be fixed in 1 I i I s. Let IT;. be the set of all the subdiagrams of G such that the following are true. If Gi E IT;., then Gi ( cG) contains all of the external vertices of G but not necessarily all of its lines. The lines in G/Gi (if not empty) do not carry any external momenta, and all their masses are from the set { p i ,pi+ . . . , p”}.z5Any external line of Gi depends on the elements from the set P and/or { p l , . . . ,p i - 1 , p S + l , .. . ,P P } . * ~ We repeat the definition of IT;. for all i = 1 , . . . , s, thus generating the sets T,, . . . , T,. If the following are true, for all i = 1, . . . , s, then limyl,...,As-,o d exists.” (i) for any Gi E T , d(Gi) I d(G). (ii) If d(Gi) = d(G), then Gi does not have a proper subdiagram gi c Gi in it such that the masses in the lines in gi are from the set { p i , pi+ . . . ,p”} (if not empty), and the dimensionality of each of the connected components of gi is nonnegative. The above rules are very simple to apply and one may, in general, infer the existence of limA,,...,A,+o d by a mere examination of the graph G from these rules with almost no extra work. In particular, the rules state as one of the sufficiency conditions for the existence of the latter limit that the graph G itself is not to contain a proper subdiagram g such that all the masses in g are from the set { p ’ , . . . , p”}, and that the dimensionality of each of the connected components of g is to be nonnegative. Before giving some examples applying these rules we wish to note the following. This will save us time in applications. The above conditions if satisfied imply that di 2 0 for all i = 1, . . ., s and for the corresponding i for which di = 0 we have fl,(Si) = 0. If fl,(Sj) = 0 for some j, then no In(l/Aj) terms will appear in (6.5.22). The reason is that with the ordering as in (6.5.25) with n1 = j, i.e., 0 = B(S,) I . . . I B({L,,. . . ,Lz,}), then from (6.5.24), we have 0 I y1 I fl(S,) = 0, and hence we obtain y1 = 0.
’,
’,
Example 6.2 : Consider the graph in Fig. 6.2 representing the lowest-order contribution to the self-energy of the electron in quantum electrodynamics. We write the photon propagator (in the Feynman gauge) as D,,.(Q) = g,,,/(Q2 + p2). We consider the behavior of the amplitude d ( q , m,Ap) for A -, 0, where m is the electron mass and q # 0. Any subdiagram G’ c G is such that d(G’) I d(G). We also have d(G‘) = d(G) only if G’ is the graph G zs This simply means that the lines in GIGi,in reference to the amplitude .d,are independen1
0r,ir1.
’’This simply means that the external lines of Gi depend on A:’. 27
More precisely, we should say that .dremains bounded in this limit.
170
6 Asymptotic Behavior in Quantum Field Theory
Fig. 6.2 Lowest-order electron self-energy graph in quantum electrodynamics. The wavy line represents a photon, and the straight line represents the spin particle.
itself. Also G does not contain a proper subdiagram having all its masses from the set { p } . Accordingly the limA+od ( q , m, Ap) exists. This example demonstrates the simplicity of the application of the rules for the existence of lim d given before. For the convenience of the reader we give the explicit expression for the renormalized amplitude corresponding to Fig. 6.2 with subtractions performed at the origin and with p = 0: d ( q , m, 0) = -
211
yq Jolx dx In( 1
+
2
x)
(6.5.26)
where a is the fine-structure constant a = e2/4a. The expression d(q,m, 0) in (6.5.26)obviously exists for q2 > 0 (also for q = 0) and m # 0. Example 6.3: Consider the photon self-energy graphs in quantum electrodynamics. Some of these graphs are shown in Fig. 6.3. For the photon propagator we write DPv= g,,,/(QZ + p2). Let G be any graph in Fig. 6.3. We note that for any g c G,we have d@) 5 d(G). Also, G does not contain any proper subdiagram having all its masses equal to p. Accordingly the limit 1 + 0 of the renormalized amplitudes d(q,m, Ap), corresponding to all the graphs in Fig. 6.3. exist in the limit A 0. Finally, we give an example where the behavior of d may be studied as of one of the masses of the graph in question becomes small and another one goes to zero.
Fig. 6.3 Some low- and high-order photon self-energy graphs in quantum electrodynamics.
6.6 Low-Energy Behavior
171
Fig. 6.4 A Fourth-order electron self-energy graph in quantum electrodynamics.
Consider the behavior of the renormalized amplitude d(q,A,m, AIA,p) associated with a fourth-order self-energy graph of the electron in quantum electrodynamics, shown in Fig. 6.4., for A,, A, + 0. Consider the amplitude d ( q / A l ,m, A,p). We refer to Example 6.1 to infer, in the notation in (6.3.33) and (6.3.42), that a,(S(q)) = 1, b,(S(q)) = 2. On the other hand, repeating the same analysis as the one given in the previous two examples shows that the limit A, + 0 of d ( q / A , , m,A2p) exists. Accordingly we have for A,, A, + 0
Example 6.4:
(6.5.27) where we have used the fact that d(G) = 1. 6.6
LOW-ENERGY BEHAVIOR
In Theorem 6.1.1 we have seen that d ( A P , p ) + 0 (A + 0) for d ( G ) 2 0, with fixed nonzero masses, as expected. The same analysis leading to this theorem shows that d ( A P , p ) remains bounded for A + 0 when d ( G ) < 0 since Ia 1, in (6.1.18), is positive." In this section we generalize these results to the cases when some of the underlying masses approach zero as well, and some, not necessarily all, of the external momenta become small. On physical grounds we let these masses approach zero at a rate faster than the corresponding external momenta become For generality we let these vanishing external momentum components and those vanishing masses become small at different rates. We write P=PuP, (6.6.1) Note also that (a1 is bounded above. Applications may be also given when some of the external momenta become small at a faster rate than some of the vanishing masses; however, these are of less interest than the general cases given here and will not be discussed. 29
172
6 Asymptotic Behavior in Quantum Field Theory
where P" is that subset of P containing those elements we wish to scale to zero, and P' constitutes the remaining elements in P. We decompose El as (6.6.2)
El = E ; 8 E:,
where E: is, say, a k-dimensional subspace of El associated with the vanishing momenta, and E i is the orthogonal complement of E: in El. We also decompose E 2 as E2
=
(6.6.3)
E: 8 E ; ,
where E: is, say, of s dimensions and will be associated with the masses that we wish to become small, and E t is the orthogonal complement of E: in E 2 . Let P1be a vector in E : such that the elements in P' may be written as some linear combinations of the components of P,.Let P2 be a vector in E: of the form
+ +
P2 =
' '
11
' * *
AkLk+ 1,
(6.6.4)
,
where L 2 , . . . ,Lk+ are k independent vectors in E:, and suppose that the elements in P" may be written as some linear combinations of the components of P2 such that every element pfl in P" may be conveniently written as pfl = I, Aj(il)jfl for some j(il) Ik, and where jf1 is independent of 11, 1,. Let P, be a vector in E, of the form 9
p3
= I1 *"1k(1k+lLk+2
+
" *
+ Ik+l
"'Ak+sLk+s+l)+ L
= P, + L,
(6.6.5)
,
with P3E E : , L E EZ, where Lk+2,. .. ,Lk+s+ are s independent vectors in E: such that the masses that we wish to become small may be written as some linear combinations of the components of P, in such a way that every mass pi' we wish to approach zero may be written in the form p i t = 1, ... & A k + l Ak+j(il)pil for some 1 I j(il) Is, and where pi' is independent of I , , . . .,& + s , and the remaining nonasymptotic masses may be written as some linear combinations of the components of L. Accordingly the subtracted-out amplitude d ( P , p) may be written as d ( P ) , with
P = P, + P, + P,. Introducing the vector P by
(6.6.6)
6.6 Low-Energy Behavior
173
defining
c= L k + s + L1
=
P1
(6.6.8)
1,
+ L,
(6.6.9)
and hence writing
(6.6.10) we obtain the following expression for d ( P ) : P
d ( P )=
fl ( p J ) - ' J ( I l
' ' '
&+s)dcG)d(p')
(6.6.11)
j= 1
[see (6.5.19)], and for all those masses that we wish to become small the corresponding aj = 0 (see Section 6.5.1). From (6.6.9) and (6.6.10) we note are proportional to (&+ . . . Ik+s)- '. that all the external momenta in J(P) For I , = . . = I k + s = 1, we choose the external momenta p l , . . . ,pm such + pi,# 0 for that no partial sums of these momenta vanish, i.e., pil + all {i,,. . . , i t } c (1,. . . , m } . For 11,. . . , I , + , + 0 independently, the total external momenta carried away from each external vertex is asymptotic. We may then apply Theorem 6.4.2 to investigate the behavior of d ( P ) from (6.6.11). We may also apply the rules for the existence of lim d for I k + l , . . . , I k + s + 0, given in Section 6.5.2, to infer the behavior of d at low energies, in the zero-mass limit. Example 6.5 : Consider the elementary graph G in Fig. 6.2. We are interested in the behavior of d ( I , q , m, I,I,,u) for I,, I, + 0, where ,u is the mass of the
photon and m is the mass of the electron. We write for the photon propagator D J Q ) = y,,,/(Q2 p'). From Example 6.2 we know that lim d for I, + 0 exists. We may write
+
d ( I , q , m, I I I 2 , u ) = I , d ( q , m / 4 , &PO
=
I , & d ( q / I , , m / 4 f b 9PI.
In reference to the parameter l/I,, it is easy to show that for the amplitude d ( q / I , , m / I I I z ,p), we have 7 0 = { G } = t, a,(S(q, m)) = d(G), and flr(S(q, m)) = 0. In reference to the parameter l/I,, we note that for d ( q / I , , m / I I I , , p), as in Example 6.2, to = { G } and hence a,(S(m)) = d(G) = 1. From the definition of class A in Section 6.4 we also note that G has no proper subdiagram g [with d(g) 2 01 dependent only on the mass p [i.e., independent of 1/11,in reference to the amplitude d ( q / 1 2 ,rn/IIIz,p ) ] . Accordingly fl,(S(m)) = 0. Hence we obtain from (6.6.11) and (6.4.5) lim d(l,q,m,AlA,,u) = 0 11,12+0
(6.6.12)
174
6 Asymptotic Behavior in Quantum Field Theory
It is instructive to study the expression explicitly for d ( q , m, p), corresponding to Fig. 6.2, with p = 0 and to determine its behavior for q + 0. The expression for d ( q , m,0) is given in (6.5.26).We use the identity (x 2 0)
(
5)
In 1 + - x
= -q2 x - - x q4 m2
m4
’s’
Y dY [ I + (q2/m2)xyl’
(6.6.13)
and hence the bound (6.6.14) to bound the expressions
(5)_ - - 2*, (5)
a -
-
= -
5 +5
Jolx dx ln( 1
Jol dx In( 1
+
x)
x),
(6.6.15) (6.6.16)
as (6.6.17) (6.6.18) Accordingly we have
and for A, --* 0, we have from (6.6.17)-(6.6.19) that d remains bounded (actually, it vanishes) consistent with the result in (6.6.12), as expected. Example 6.6:
Consider the behavior of d ( A l q , A1A2m, A l A Z A 3 p ) for
A1,A,, A3 -+ 0 corresponding to the graph in Fig. 6.4. We may write d ( A , q , A1A2m, A1A2A3p)= A l A z A 3 d ( q / A 2 A 3 , m / A 3 , p). In reference to the latter amplitude we have from Example 6.1, a I ( S ( q ) ) = 1, flI(S(q)) = 2. From Example 6.4.,we may infer that aXS(q, m)) = 1, fl,(S(q, m)) = 0. Accordingly we have from (6.6.11) and (6.4.5)
2
(6.6.20)
6.7 General Asymptotic Behavior 11
- IP1
+
P2 + P 3
I
175 1P3
I
I
I
II
I
I
I
I
I
4Fig. 6.5
p2
A fermion-fermion (spin f ) scattering graph with a
$@ coupling
Example 6 . 7 : Consider the process depicted in Fig. 6.5, with a &,h@ coupling, where the dashed lines denote scalar bosons, and for simplicity the fermion and the scalar bosons are assumed to have the same mass p. Here d ( G ) = -2. We are interested in the behavior of &(Alpl, Alp,, p 3 , A l l z p ) for 11,A, + 0. As in Example 6.4, the limit A, --t 0 may be taken. In reference to the parameter 1/11, we readily obtain for the amplitude J4PlIAZ
9
P2/&
9
P 3 I A l f b1
that a,(S(p3)) = - 1. Also, G does not contain any proper subdiagram g [with d(g) 2 01, and hence P,(S(p,)) = 0. Accordingly we have
for nonexceptional momenta. 6.7
GENERAL ASYMPTOTIC BEHAVIOR II
In this section we generalize our applications to cases when, in general, some (or all) of the external momenta become large, some (or all) become small, and some of the masses are led to approach zero. We repeat the definition of P in (6.6.1), with the subset P' now consisting of those elements in P becoming either large or remaining nonasymptotic. Let El and E , be written as in (6.6.2) and (6.6.3), respectively. Let PI be a vector in E : of the form P1
=
L1q1 . * . q,
+ + LJ, + Ll+ * * *
1,
(6.7.1)
where L 1 , .. ., L,, with a + 1 I4m - k, are a + 1 independent vectors in E : . As in (6.6.4) and in a notation similar to it, let P, E E: be such that
Pz = AlL,+Z
+ + ' * '
11
."&La+k+l-
(6.7.2)
As in (6.6.5), let P3E E , be such that p3 =
= P;
"*Ak(Ak+lLp+k+Z + "'
+ Lb',,.
+
&+I
"'Lk+sLa+k+l+s)+ G+l, (6.7.3)
176
6 Asymptotic Behavior in Quantum Field Theory
We may then write the amplitude d ( P , p ) as d ( P ) , with
+ P,,
(6.7.4)
lk+s)P,
(6.7.5)
P = P1 + P2 or
P = (A1
' ' '
with (6.7.6)
l/Ai = v . + ~ ,
i = 1,. . ., k
+s
(6.7.7)
The amplitude d ( P ) may be then written as in (6.6.11), and the limits ql, . . . ,u a + k + s + co may be then studied directly from (6.4.5) for the amplitude d(P') with P now defined in (6.7.6). We note that L,, . . .,La+,+, are independent vectors. Again, with p i , pi, # 0 for all { i l , . . . ,i,} c { 1, . . . ,m},all the external momenta at the external vertices then become asymptotic for ql, . . .,)l,+k+s -,co for the amplitude d(P).
+
+
Example 6.8 : Consider the renormalized amplitude d ( q , p, m,p ) associated with the graph G in Fig. 6.6, where a dashed line represents a scalar particle of mass p, and a solid line represents a spin-4 particle of mass m. Let (k;, .. .,k:) be integration variables, with ke = k z , associated with the subdiagram g, and (ky, . . . , k:) be the integration variables associated with the graph G. We note that d(g) = 2, d(G) = 0. We write R'* = I @ E. We also write I = f l $ ~ 2 , f l = l l @ ~ ~ ~ @ I , , ~ z = Z , $ ~ ~ ~ ~ I ~ ( d1,..., imIj=l,j= 8), and E = El $ E 2 . The subspace 1, is associated with the integration variable kZ. We make the further decomposition El = E : @ E: and
Fig. 6.6 A vertex correction with a @ $, coupling, with the dashed line representing a scalar boson with mass p, and a solid line representing a spin 4 particle of mass m.
6.7 General Asymptotic Behavior II
177
E2 = E: Q E : , with E : , E:, E : , E: associated, respectively, with q, p , m,p. We define the subspaces
(6.7.8)
We are interested in the behavior of the amplitude
We consider the parameter q l first. In reference to this parameter, it is readily seen, as in Example 6.1, that aI(S(q))= d(G) = 0, as 70 = {G}. From , in Section 6.3, we have 7 = {G, (Gig), (GI G)} with the the definition of ( M 7) latter subdiagrams associated with the subspaces in A = {So, S', S" = S ( q ) } . By noting that A(I)So = A(Z)S' = A(I)S" = S(q), A(Ij Q . . * Q 18)s' = A(Ij Q *
*
Q 18)s' = I1 Q *
*
Q Ij- 1 @ Ei, j = 2 , . . . , 5,
(6.7.9)
we obtain from Theorem 6 . 3 . 2 ~ '= p s = 1, and p j = 0 forj # 1,5, and hence
B,(S(d) = 2. We note that d ( v ] l q , Alp,
J1A2m,p) = d ( q l A ;
'A; ' 4 , A; ' p , m, A; '&'p).
Consider the parameter A;'. We note that 70 = {G}, i.e., a,(S(q, p)) = 0, and 7 = {G, (G Is)}. Note that (GIG)4 7 since G (trivially) contains the mass p. The subdiagrams in 7 are associated with the subspaces in A = {Sl, S , } . By repeating an analysis similar to the one given before, we obtain p s = 1 and pi = 0 for i # 5, thus leading to B,(S(q, p)) = 1. Finally, we note that the subdiagram g does not depend on the mass p, and as before we obtain a,(S(q, p, p ) ) = 0 and P,(S(q, p, p ) ) = 1 in reference to the parameter I ; .
'
178
6 Asymptotic Behavior in Quantum Field Theory
Accordingly we have from Theorem 6.4.2, for nonexceptional momenta,
for ql + 00, A, + 0, A2 Y1, Y 2 9 Y 3 such that
-,0, and the sum is over all nonnegative integers 71 I 1,
+ Y2 71 + Y2 + Y 3
1,
Y1
(6.7.1 1)
2-
Other examples may be also carried out where, for example, only some (or all) of the external momenta, of the graph in question, become large, and some (or all) of the masses become small. 6.8
GENERALIZED DECOUPLING THEOREM
In this section we generalize the decoupling theorem given in Section 6.1. In Section 6.8.1 we consider the behavior of d when any subset of the masses in d become large, and in general, at different rates. In section 6.8.2 this theorem is generalized further to cases when some of the remaining masses in the theory are led to become small as well, corresponding to theories which, on experimental grounds, may contain zero-mass particles. 6.8.1
Generalized Decoupling Theorem I
Consider the renormalized integrand R with argument P' as defined in (6.2.5). In this section all the external momenta are kept fixed and hence we require that A(I @ E2)Si = (0). For the problem at hand we suppose that N r 0 E1)Si # (0). First suppose that # (0). Then we may apply (6.2.21) and (6.2.22) to the graph G itself. If G E F2(N)[see (6.2.13)], then (6.2.22) applied to G implies that min[d(G),
- 13
- o(G).jo
(6.8.1)
1r
30 Since the external momenta of G are independent of q,, we took the liberty of replacing degr, by degrq, in (63.1).
6.8 Generalized Decoupling Theorem
179
An equality in (6.8.1) may hold if there is no subdiagram g c G in H 2 ( N ) such that at least one of the masses in the lines in S is dependent on q,, and Finally suppose that G E H2(N). there is no subdiagram g' $Z G in .F2(N). Then we may apply (6.2.21), (6.2.22) [and (6.2.18), in general, applied to l ~to] conclude that degr 1,9r
fl FG,(.N)I
k
min[d(Gi),
- 13
- a(G),
(6.8.2)
i= 1
i
where G , , . . . , Gklare those maximal elements in N contained in G : Gi $ G such that Gi E .F2(N), with i = 1, . . . , k,. We denote the remaining maximal elements in N contained in G by Gkl+ . . . , G,. We may also apply (6.2.21) directly with gi in it simply replaced by G, for d(G) 2 0, and use the estimate in (6.8.2) to conclude for d ( G ) 2 0 or d ( G ) < 0 that
,,
degr FG(N)I - 1 - a(G),
(6.8.3)
since k , 2 1, for A(E)S: # (0). From (6.8.1) or (6.8.3) and the fact that dim A(E)S: 5 a(G),
(6.8.4)
we obtain upon summing over the sets N in (6.2.11) that degr R I
-1 -
dim A(E)Si.
(6.8.5)
9r
If A(E)S: = {0},then directly from (6.2.4) and (6.2.18) we conclude that degr R I - 1.
(6.8.6)
4r
Now consider the renormalized amplitude -d(f', 41 . . . qSp1, ~2 . . * qsp2,. . . , q,pS, pS+', . . . , p'),
(6.8.7)
with s I p. We conveniently decompose the subspace E 2 into s one-dimensional orthogonal subspaces Ei and introduce s vectors L1 E El, . . . , L, E Es with nonvanishing components p ' , . . . , ps, respectively. We then rewrite (6.8.7) as d ( L l q , ...qs
+ ... + L,qs + C),
(6.8.8)
where C is confined to a finite region in E , with the pi # 0. We are mainly interested in the vanishing property of .d for q,, . . . , qs --+ CQ, and we shall not carry out an analysis regarding the logarithmic asymptotic coefficients fi,(S,), as the latter is quite cumbersome.
180
6 Asymptotic Behavior in Quantum Field Theory
From (6.2.44), (6.2.51), (6.2.60), (6.8.5), and (6.8.6) we may then state Theorem 0.8.1 : For I ] , , .. . , I],+ oo, d in (6.8.8) vanishes, as the power
asymptotic coeficients al(S,) are bounded above as a,(S,) I- 1.
(6.8.9)
In particular, since the pl(S,) are finite, we can always find positive integers N , , . . . , N , and a real constant C, > 0 such that
6.8.2
Generalized Decoupling Theorem II
Here we are interested in generalizing Theorem 6.8.1 to the cases when some of the remaining masses p S + l ,. . . , pp vanish; i.e., we are interested in the behavior of (6.8.11) where s + k Ip, for I ] ~ .,. . , I],+ co and A,, . . . , 2, + 0. We have already , + 0 of d established sufficiency conditions for taking the limits A,, . . . ,1 in Section 6.5.2. Accordingly Theorem 6.8.1 remains true if the sufficiency conditions stated in Section 6.5.2 are satisfied with respect to the parameters 11,.
..
9
Aka
The renormalized amplitudes d(q,m, p ) corresponding to the graphs in Figs. 6.2 and 6.3, for example, all’vanish form + 00, p =fixed and m + co, p + 0. We note, in particular, from the estimates in (6.6.17) and (6.6.18) that for the graph G in Fig. 6.2 we have for p = 0 (6.8.12) with (6.8.13) and (6.8.14) which show the vanishing property of d ( q , qm, 0) for q consistent with our conclusions.
+
oo-a result that is
Notes
181
NOTES
Section 6.1 is based on Manoukian (1981a) and on some analysis, in particular, estimate (6.1.12), due to Hahn and Zimmermann (1968). Section 6.2 is based on Manoukian (1980a, 1981b), Section 6.3 on Manoukian (1978, see also 1980c), Section 6.4 on Manoukian (198Oc, 1981c), Section 6.5 on Manoukian (1980b, 1981c), Section 6.6 on Manoukian (198Od, see also 1979a), Section 6.7 on Manoukian (1981c), and Section 6.8 on Manoukian (1981d). Asymptotics were also studied by completely different methods (in the so-called a-parameter representation) by, e.g., Bergere et al. (1978). The decoupling theorem with only one mass scale becoming large and with no zero mass particles was proved by Ambjgfrn (1979) (again in the a-parameter representation). A proof of the general case with several mass scales becoming large and with zero-mass-particle limit was given in Manoukian (1981d). Interesting applications of the decoupling theorem have been carried out [see Appelquist and Carrazzonne (1975), Poggio et al. (1977), Collins et al. (1978), Toussaint (1978), Kazama and Yao (1979), Ovrut and Schnitzer (1981), and Hagiwara and Nakazawa (1981)], and many other papers on the subject are still appearing.
This Page Intentionally Left Blank
Appendix / SUBTRACTIONS VERSUS COUNTERTERMS
In this appendix we show by a direct and simple method that the subtractions introduced in Section 5.2, as carried out directly in momentum space, are equivalent to formally adding counterterms to the (unrenormalized) interaction Lagrangian density. This equivalence theorem had a very important role in the history of quantum field theory. A.l
THE FORMAL UNRENORMALIZED THEORY
Let K = {mi, Q2, . . .} be the set of all basic (free) fields and their adjoints of interest in the theory, added to which are all their derivatives of arbitrary orders, suppressing, for simplicity of notation, all tensor and spinor indices. We may define the interaction Lagrangian density without counterterms (in the interaction picture, i.e., in terms of the elements in the set K) in the form
where U,(x) is formally hermitian and the cG are constant matrices (couplings) depending, in general, on tensor and spinor indices and summation over them, when the cG are multiplied by the fields, is understood. The double dots : : in (A.l) denote the so-called Wick ordering.' Y(1)
' Wick (1950).or see Schweber (1961). I83
Appendix Subtractions versus Counterterms
184
corresponds to the set of all the terms in the interaction specified with the combinations {G,: a l l(x). - ale(&):) and such an element is denoted by G.This notation will be indispensable when we come to counterterms. In momentum space variables we may write P A X ) as uI(x) =
CG
E€9(1) X
fdQGexp{XQ11 + * * *
+ Qle(~)Ix}:all(Ql~)...a~e(~)(Qle(~)):, ('4.2)
using the same symbols for the Fourier transform of the fields, for simplicity of notation, and (A.3) (Q?I, . . Q&G)) Consider the chronological product2 of n(n 2 2) Wick-ordered terms each of the form in (A.2): QG
= ~ ~ G , ( c G* * ,
=
C G , ~ G , ) ( ~ G~, ( Q . . - Q .
,J:@:G~(Q~G~) * * @ ~ G ~ ( Q ~ G ~ ) : , 1
(A.4)
where the superscripts c,p denote connected and proper parts; we shall come back to their precise meaning. We have introduced the superscripts f on the fields on the right-hand side of (A.4) to remind us that they resulted in the final expression after carrying out the (covariant) time ordering product. 05, is the so-called parity associated with the commutation of the fields within a chronological product, on the left-hand side of (A.4), so that a pair of fields which are to be contracted together are brought next to each other. The chronological pairing of two fields has the general structure
~ d Q d ~ A =QS(Q1 d + Q2) AldQd, and fGoin (A.4) is defined by f ~ = o
n A(Qi),
64.5) (A4
cont.
as a product of the contracted pairs of fields. We shall not go into the details in defining explicit forms of the A(Qi), as we are working at a very general level. The factor SG0)in (A.4) occurs as the product of &functions appearing in pairings such as in (AS). The sum over & is over all possible contractions among the fields on the left-hand side of (A.4) not occurring
(n
We assume covariant chronological products. We shall not go into details of such definitions as these details are not necessary in understanding the content of the appendix. For such details cf. Nishijima (1969).
A .1
185
The Formal Unrenormalized Theory
within the same Wick-ordering signs : : such that when we integrate over the momentum components Q I G , , . . ., we obtain for (A.4), after having it multiplied by nl=l(S(z‘ Qij)) =conly i’ proper and connected subgraphs Go of order n, i.e., an expression of the form
where Go is a proper and connected subgraph with precisely n vertices. The expression
I,
= CG, ’. ’ CG,lbo
(A.8)
defines the unrenormalized Feynman integrand associated with Go. The expression ic0in (A.6), by definition, contains no vertices to be distinguished from I,, corresponding to a genuine subdiagram of order n. The introduction of the factor I;, in (A.8) will be useful later on. For simplicity of notation we have absorbed the oGofactor in I;,. The one overall Q i G , ) occurs in (A.7) as the result of a restriction over connected parts only and momentum conservation. K,, denotes the internal momenta of Go and Q G o its external momenta. Expression (A.7) prepares us to define the concept of a Wick polynomial of order n.
S(c:=
Definition of a Wick polynomial of order n : We denote the range of possible expressions Go in (A.7) by B(n) and we call the expression
I,,
:@:Go
* *
. @:Go:,
(A-9)
a Wick polynomial of order n, symbolized by Go.The latter may be defined by the combination {Go, - @Go:}r where Gois a proper and connected graph of order n. A Wick polynomial of order n is specified by both Go and a@ ,:,:
:@{Go
* * *
@Go:.
We may then summarize by saying that 9 (n) is the set of all possible Wick polynomials of order n as arising in the expansion (A.4) leading to (A.7). This definition will be quite useful later on. Definition of counterterms: Let Go be a graph with a set of vertices Y and a set of lines 9. Consider n disjoint subsets of Y : f 1..., , Yl associated with subgraphs G;,. . . ,Gi of G. By definition the GI contain all those lines in Y joining the vertices in Y;.Let 9’denote ; the set of lines of G:. Consider the sets Y ’ = Y’,u - . u Vi and 9’ = Y’, u - . u Yn.The
-
latter obviously define a proper subdiagram G’. Note, however, that G’ is not necessarily a subgraph of Gosince the set 9’ does not necessarily contain all those lines in Y which join the vertices in Y‘. We call such a proper
Appendix Subtractions versus Counterterms
186
subdiagram G' as a union of vertex-disjoint subgraphs. We may then repeat the proof of Lemma 5.4.1 to provefor such a G' that (A.lO) where the
are defined recursively by
/TG'IG' =
c
-
(A. 11)
(- T,.)A,..I,.,
0 c G" !s G'
r0
G p , G , corresponds to a sum over all proper where the notation subdiagrams $G' (not necessarily connected) with the latter as unions of vertex-disjoint subgraphs. We then define an interaction Lagrangian density including counterterms as m
m= 1
G E 3(m)
where
Z!(QG, K G )=(-
TG) &I G( QG, KG) : @ m l
(Qml)
*.
@me(G)(Qme(G)):
>
(A.13)
with & defined by (A.11). For m = 1, Y(l) corresponds to the terms of the interaction Lagrangian density U , ( x ) of the unrenormalized theory as discussed above with (-TG)&IG, formally, replaced by cG. For rn # l,Y(m), as also defined above, is the set of all Wick polynomials of order m arising in expressions (A.4) and (A.7) with n replaced by m. Accordingly the Wick polynomial in the sum and in (A.12) is 1,
:@ml
'
. .@me(G):
3
(A.14)
where G is proper and connected graph of order m. By definition (- TG)= 0 if d(G) < 0. In the study of renormalized Feynman amplitudes J R we do not have to introduce any cutoff in actual computations, since we perform the subtractions in momentum space directly on I G to define R. To make all the subsequent analysis rigorous when working with counterterms, however, we have to introduce cutoffs. Evidently then with such an assumption the 9:will be cutoff dependent. Throughout this appendix we shall assume that the cutoffs are strong enough, if necessary, so that all the subsequent integrations are meaningful. We shall not, however, go into the nature of such cutoffs.
A . 2 Equivalence
187
The proper and connected parts of the formal Dyson-Feynman perturbation expansion of the theory with interaction Lagrangian density -YC(x)is3 1+
(-iy c7 j(dx n.
1
* * *
(dxn)(pc(x1 1*
*
yc(xn)y; p
(A.15)
where we have m 1 +
1 . .
+ m, = N and
(A.16) I G 1 .* .I G " . The S(c;?i) Qmij) in (A.15) arises upon integration over x l , . . ..x,. Recall that G1,...,, is a subdiagram as a union of vertex-disjoint (connected) subgraphs. I G , , . . ., n
A.2
=
EQUIVALENCE
Using the expansion (A.4) now applied to (A.15) we obtain (n 2 2)
=
Cjc0(HSc0(Q..
-
Q.
:@;
co(Q1 Go)
. @L,,(Qwo) :.
(A. 17)
GO
Upon replacing this expression in (A.15) and carrying out intermediate integrations to make use of the delta-functions in (A.17) we obtain for each fixed N 2 2, n 2 2: N 2 n.
x (-~
G J &. . .~(-
~ G , ) & , I G ~ ,. .,I~,:@;G(QIG) - . . @ ~ G < Q ~ ~ >(A.18) :, , ,
It is, ofcourse, expected that the redder is familiar with the left-hand side ofthis expression (cf. Bjorken and Drell, 1965). The conditions N 2 2, n 2 2 corrcspond to contractions between different Wick-ordered products.
Appendix Subtractions versus Counterterms
188
where the overall momentum conserving delta-function arises as a consequence of the fact that the (proper) subdiagram G, defined by (A.19)
I G = IGl.....”~GO~
must be connected, by definition. Alternatively, Ibo may be defined by ‘Lo
=
IWG~,.,.,~
(A.20)
i.e., by replacing each of the IG1,. . . , ZGn in the expression I, by unity. We recall that the G l , . . . , G, are connected, having no vertices in common. K G in (A.18) includes, in addition to the components in K G r ,.. . ,K , , the internal momentum components carried by the lines in IGo.Also, QG
=
. ., QZG).
(A.21)
[see (A.17)] We note that, as contractions between fields occur in pairs, lGo “contains” no vertices, i.e., n
number of vertices of G =
C (number of vertices of Gi) i= 1 n
mi = N ,
=
(A.22)
i= 1
and hence G has precisely N vertices. We examine the expression (A.18). For n = 2, ml
+ m2 = N, we have
k X
( - TGi)JGI(-
JG~ 1 ~
I&:
@L(QiG):
1 2
-
(A.23)
i= 1
By using the symmetry over the summations &lEg(ml)&2E9(m2), the fact that m1 + m2 = N, and using the identity in (A.10), we may rewrite (A.23) as
The summation z i c G , g Gis over all subdiagrams G’ $ Gi such that, for G’ # 0, each G is a union of at most two (i.e., 1 or 2) vertex-disjoint (connected) subgraphs and ZG,Go corresponds to a subdiagram with exactly two vertices. The situation G’ = 0 occurs when N = 2. In this latter case m1 = m2 = 1 and the -TG,&,IG, are simply replaced by cG,, respectively; IG then has also associated with it exactly two vertices. In general, if, say, ml = 1, then m2 = N - 1 and - TG1&lIG1 is replaced by cG1.
A.2 Equivalence For n = 3, m,
189
+ m2 + m3 = N, we have
The summation C & c G , g G , for G' # 0, is over all subdiagrams G' $ G such that each G' is a union of at most three (i.e., 1 or 2 or 3) vertex-disjoint (connected) subgraphs. Again the situation G' = 0 arises when N = 3. l G , G , corresponds to a subdiagram with exactly three vertices. In general, we may then write for (A.18) n!
C3 ( N ) P Q c dKG a(
EE
il
Qic)
k
1" ( - TG,)JG*ZG : fl %dQiG)
:
3
(A.27)
i= 1
0cG'SG
where rOcG.gG, for G' # 0,denotes a sum over all G' $ G such that each G' is a union of at most n (i.e., 1 or 2 or . . . or n) vertex-disjoint subgraphs. The situation G' = 0 occurs when N = n. ZGIGt corresponds to a subdiagram with precisely n vertices.' For G' = @, i.e., when N = n, rn, = ... = m, = 1, ZG, again, corresponds to a subdiagram with N = n vertices. Upon summing over n and N we obtain for (A.15), from (A.27), the expression 1
f P Q G ~ K G ~ ( ~1' ~ (-TG,)JG*ZG :n@G(QiG):. (A.28) (-i)N
GE9(N)
QiG)
0cG'cG
N= 1
k
x
i= 1
For N = 1, x 0 c G ' c G is to be replaced by one. For G' # 0 and G' # G, the summation CbgGegG is over all subdiagrams G' such that each G' is a union of at most N (i.e., 2 or 3 or . . . or N) vertex-disjoint (connected) subgraphs and l G / G * corresponds to a subdiagram with at most N (i.e., 2 or 3 or . . . or N) vertices. The analytical expression for these latter vertices in ZG/G, is unity. By definition - TG,iG. is to be replaced by one if G' = 0 ; and for G' # 0, ( - TG,)= 0 if d(G') 0. The G E % ( N ) have exactly N vertices, by definition.
-=
' Note that the analytical expression for these n vertices in IG,G, is unity, by definition.
Appendix Subtractions versus Counterterms
190
Fig. A.l A subdiagram G' such that G " g G ' with d(G") 2 0. The subdiagram G" is depicted by the shaded area in the figure. G" has the same number of vertices and the same number of connected parts as G', but the latter has one or more lines each of which joining two vertices within the same connected parts of G'. The subdiagram on the right-hand-side depicts G'/G",showing, in particular, that the number of vertices of G'/G'' coincides with the number of connected parts of G' (or of G").
Consider a subdiagram G' in (A.28). Suppose that G' contains a subdiagram G": G" $ G such that G" has the same number of vertices as G' and the latter has one or more lines each of which joining two vertices within the same connected parts of G' (see Fig. A.l) and d ( G ) 2 0. Note that G" has the same number of connected parts as G'. Consider the following expression: [( - TG*)( - TG")
+ (- T&)]
* *
1G*
= (1
- T&)(- T&)
*
I,, . (A.29)
If G' is a union of, say, n vertex-disjoint subgraphs, then G" is not a union of vertex-disjoint subgraphs, and hence both terms in (A.29) do not occur, in the summation in c' in (A.28). Integrating (A.29) over internal momenta of G" we obtain an expression (A.30) (1 - TG,)IG,,WPG,* CQG,*I where PG,[Qc,,] is a polynomial of degree < d ( G ) in the extremal momenta of G" by the definition of the (- TG")operation. The external momenta of G" are now to be written as linear combinations of the external variables of G' and the internal variables in GIG". Integrating (A.30) over the internal variables of GI/"' we obtain an expression of the form, 9
(1
- TG*)PG"QG,I,
(A.31)
where PG,[QG,] is a polynomial of degree
This integral representation is easily proved by induction in d(C').
' The vanishing property ofexpressions like in (A.3l)arewell known (cf. Hepp, 1971. p. 483).
A.2 Equivalence
191
Conversely suppose that G” is any proper subdiagram, with d(G”) 2 0. We add to it some lines, each of which joins two vertices within the same connected parts of G”,and we form a subdiagram G’with the same number of vertices as G”.Then, by definition of the dimensionality of a subdiagram, we note that d(G’) 2 d ( G ) . Accordingly for any proper subdiagram G” with d(G”) 2 0, with n connected parts, but which is not a union of vertexdisjoint (n connected) subgraphs we can always find a proper subdiagram G’ with d(G’) 2 0, with n connected parts, having the same vertices as G” but with one or more lines, and a similar expression as the one in (A.29) involving a pair of terms may be written down which cancel out, as seen in (A.32). [Note that such a subdiagram G’ may or may not itself be the union of vertex-disjoint ( n connected) subgraphs.] As a result of cancellation between such pairs, we may then rigorously include subtractions over proper subdiagrams in the sum c G , c G not necessarily being vertex-disjoint subgraphs. Hence we may simply replace the sum ~ ~ c G , in c c (A.28) by a sum over all proper subdiagrams G’ with d(G’) 2 0, and we may replace the JG,, defined through (A.1 l), by the Act introduced in Chapter 5 involving general proper subdiagrams. The additional sets of subdiagrams thus introduced involving one or more subdiagrams which are not the union of vertex-disjoint subgraphs will not contribute to the final expression for 1 0 c G ‘ B G (-TG’)AG’lG* Accordingly we may rewrite (A.28)
z0
k
x
:fl @iG(QiG):,
(A.33)
i= 1
Z0
B G , B G represents a sum over all proper, but not necessarily connected, subdiagrams of G with (-TGt) = 0 if d(G’) < 0. Upon using the basic property [see (5.2.22) and (5.4.2)].
we obtain for (A.33) m
I-
/ k
\
k
where for N = 1, RG is to be replaced by cG. Expression (A.35) is the proper and connected part of the renormalized perturbation expansion with R ,
192
Appendix
Subtractions versus Counterterms
(the renormalized Feynman integrands) introduced in Chapter 5 replacing ZG (the unrenormalized Feynman integrands).* This completes the demonstration of the equivalence of the subtractions and the counterterms formalisms. We shall not, however, go into the philosophical implications of counterterms. The formal local character of the interaction Lagrangian density 9, including the counterterms is readily seen. We may rewrite (A.12) and (A.13) as m
U,(x) =
C1(-
m=
i)m- 1
1 BE
9(m)
e(G)
fdQ~exPCiC
j=1
Q~*~~IPGCQGI (A.36)
where for rn > 1, PG[QG] is a polynomial [of degree Id(G)] arising as a result of the Taylor operations in (-TG)AG. The terms in the expression (A.36) are well defined with cutoffs. Finally, upon integrating over Q G we obtain for U , ( x ) (A.37)
exhibiting the formal local character of U , ( x ) .
NOTES
There is a long history associated with the equivalence of subtractions and counterterms. For a partial list of references see Schwinger (1958), Gupta (1951), Takeda (1952), Matthews and Salam (1954), Bogoliubov and Shirkov (1959), Bjorken and Drell (1969, Jauch and Rohrlich (1976), Speer (1969), Zav'yalov and Anikin (1976), and Manoukian (1979b). The appendix is based on Manoukian (1979b).
* We note that with cutoffs adopted, as mentioned earlier, to make all the intermediate steps leading to (A.35) justifiable, R , will depend on the cutoffs. More precisely, then, the renormalized Feynman integrand will coincide with the cutoff-independent part of R G . In any case, if mass regulators are suitably introduced as cutoffs, for example, then we may apply our generalized decoupling theorem (Theorem 6.8.1) to conclude formally that the part of the amplitude, dependingon these mass regulators, will vanish when the limit of infinite mass regulators is taken.
REFERENCES
Abers, E. S., and Lee, B. W. (1973). Gauge theories. f h y s . Rep. 9C,I . Adler, S. (1969). Axial-vector in spinor electrodynamics. f h y s . Re[>.177, 2426. Akulov, V. P., Volkov, D. V., and Soroka, V. A. (1975). Gauge fields on superspaces with different holonomy groups. JETP Left. 22. 187. Ambjnrn, J. (1979). On the decoupling of massive particle$ in field theory. Comm. Math. f h y s . 67, 109. Anderson, P. W . (1963). Plasmons, gauge invariance and mass. Phys. Reii. 130,439. Anikin, S. A., and Polivanov, M. K. (1974). Proof of the Bogoliubov-Parasiuk theorem for nonscalar case. Theorer. Math. and f h y s . 21, 1058. Anikin, S. A., Zav’yalov, 0. I., and Polivanov, M. K. (1973). Simple proof of the BogoliubovParasiuk theorem. Theoret. Math. and f h y s . 17, 1082. Appelquist, T., and Carrazzonne, J. (1975). Infrared singularities and massive fields. f h y s . Rev. D 11, 2856. Arnowitt. R., Nath, P., and Zumino, B. (1975). Superfield densities and action principle in curved superspace. Phys. Lett B 56, 8 I . Atiyah, M. (1970). Resolution of singularities and division of distributions. Comm. Pure Appl. Math. 23. 145. Becchi, C.. Rouet, A,, and Stora, R . (1975). Renormalization of the Abelian Higgs-Kibble model. Comm. Math. f h y s . 42, 127. Becchi, C., Rouet, A., and Stora, R. (1976). Renormalization of gauge theories. Ann. Phys. 98, 287. Beg, M. A,, and Sirlin, A. (1974). Gauge theories of weak interactions. Annual Rev. Nuclear sci. 24, 379. Bell, J. S . . and Jackiw, R . (1969). A PCAC puzzle: no + 77 in the a-model. Nuoro Cimento A 60A, 47. Berberian, S . K . (1965). “Measure and Integration.” Macmillan, New York. Bergere, M. C., and Zuber, J. B. (1974). Renormalization of Feynman amplitudes and parametric integral representations. Comm. Math. f h y s . 35, 113. Bergere, M. C.. de Calan, C., and Malbouisson. A. P. C. (1978). A theorem on asymptotic expansion of Feynman amplitudes. Comm. Murh. Phys. 62, 137. 193
194
References
Bernard, C., and Duncan. A. (1975).Lorentzcovariance and Matthew’s theorem for derivativecoupled field theories. Phys. Rev. D 11,848. Bernstein, I. N., and Gel’fand, S. I. (1969). Meromorphic property of the functions PA. Functional Anal. Appl. 3, 68. Bernstein, J. (1974). Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that. Rer. Modern Phys. 46,7. Bethe, H. A. (1947). The electromagnetic shift of energy levels. Phys. Rev. 72, 339. Bialynicki-Birula, 1. (1962). On the gauge covariance of quantum electrodynamics. J . Math. Phys. 3, 1094. Bjorken, J. D. (1972). In “The Proceedings of the XVI International Conference on HighEnergy Physics at Chicago-Batavia” (J. D. Jackson and A. Roberts, eds.), Vol. 2, p. 304. Natl. Accelerator Lab., Batavia, Illinois. Bjorken, J. D., and Drell, S . (1965). “Relativistic Quantum Fields.” McGraw-Hill, New York. Bogoliubov, N. N., and Parasiuk, 0. S. (1957). Uber die Multiplikation der Kansalfunktionen in der Qudntentheorie der Felder. Actu Math. 97, 227. Bogoliubov, N. N., and Shirkov, D. V. (1959). “Introduction to the Theory of Quantized Fields.” Wiley (Interscience), New York. Born, M., Heisenberg, W., and Jordan, P. (1926). Zur Quantenmechanik 11. Z. Phys. 35, 557. Bouchiat, C., Iliopoulos, J., and Mayer, P. (1972). An anomaly-free version of Weinberg’s model. Phys. Leu. B B B ,519. Brandt, R., and Preparata, G. (1971). Operator product expansions near the light cone. Nuclear Phys. B 21, 541. Brink, L., Gell-Mann, M., Ramond, P., and Schwarz, J. H. (1978). Supergravity as geometry of superspace. Phys. Lett. B 74, 336. Brodsky, S. J., and Drell, S. D. (1970). The present status of quantum electrodynamics. Annual Rep. Nuclear Sci. 20. 147. Buras, A. J. (1980). Abymptotic freedom in deep inelastic processes in the leading order and beyond. Rea. Modern Phys. 52, No. I , 199. Buras, J., Ellis, J., Gaillard, M. K., and Nanopoulos, D. V. (1978). Aspects of the grand unification of strong, weak and electromagnetic interactions. Nuclear P h p . B 135.66. Caianiello, L. R., Guerra, R. F., and Marinaro, M. (1969). Form-invariant renormalization. NUOLIO Cimento A 60A, 713. Callan. G. C. (1970). Broken scale invariance in scalar field theory. Phys. Rev. D 2, 1541. Collins, E. G. J., Wilczek, F., and Zee, A. (1978). Low-energy manifestations of heavy particles: Application to the neutral current. Phys. Rev. D 18.242. Costa, G., and Tonin, M. (1975). Renormalization of non-Abelian gauge theories. Rh?.Nuovo Cimenfo 5, No. 1, 29. Deser, S . , and Zumino, B. (1976). Consistent supergravity. Phys. Lett. E62.335. deWitt, B. (1964). Theory of radiative corrections for non-Abelian gauge fields. Phys. Rev. Lei!. 12, 742. deWitt, B. (1967). Quantum theory of gravity. 11. The manifestly covariant gauge. P h p . Rev. 162, 1195. Dirac, P. A. M. (1927). The quantum theory of the emission and absorption of radiation. Proc. Roy. Soc. London Ser. A 114,243. Dyson, F. J. (1949a). The radiation theories of Tomonaga, Schwinger and Feynman. Phys. Rer. 75, 486. Dyson, F. J. (1949b). The S-matrix in quantum electrodynamics. Phys. Reo. 75, 1736. Englert, F.,and Brout, R. (1964). Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Letf. 13, 321.
References
195
Fadeev, L. D., and Popov, V. N. (1967). Feynman diagrams for the Yang-Mills field. Phys. Lett. B 25B, 29. Feynman, R. P. (1949a). The theory of positrons. Phys. Reo. 76, 749. Feynman, R. P. (1949b). Space-time approach to quantum electrodynamics. Phys. Rev. 76, 769. Feynman, R. P. (1950). Mathematical formulation of the quantum theory of electromagnetic interaction. Phys. Reo. 80, 440. Feynman, R. P. (1963). Quantum theory of gravitation. Acta Phys. Polon. 24,697. Feynman, R. P. (1972). “The Development of the Space-Time View of Quantum Electrodynamics,” Nobel Lect. Phys. 1965, p. 155. Elsevier, Amsterdam. Fink, J. P. (1967). Asymptotic behavior of Feynman integrals, Doctoral Dissertation, Stanford University, Stanford, California. Fink, J. P. (1968). Asymptotic estimates of Feynman integrals. J . Math. Phys. 9, 1389. Fradkin, E. S., and Tyutin, I. V. (1969). Feynman rules for the massless Yang-Mills field, renormalizability of the theory of the massive Yang-Mills field. Phys. Lett. B 30B, 562. Fradkin, E. S., and Tyutin, 1. V. (1970). S-matrix for Yang-Mills and gravitational fields. Phys. Reo. D 2, 2841. Fradkin, E. S., and Tyutin, I. V. (1974). Renormalizable theory of massive vector particles. Rill. Nuoao Cimento 4, No. I , 1. Freedman, D. Z., van Nieuwenhuizen, P., and Ferrara, S. (1976). Progress toward a theory of supergravity. Phys. Rev. D 13, 3214. Frishman, Y. (1970). Scale invariance and current commutators near the light cone. Phys. Rev. Lett. 25. 966. Gell-Mann, M., and Low, F. E. (1954). Quantum electrodynamics at small distances. Phys. Rer. 95, 1300. Georgi, H., and Glashow, S . L. (1974). Unity of all elementary particle forces. Phys. Rea. Lett. 32. 438. Georgi, H., Quinn, H. R., and Weinberg, S. (1974). Hierarchy of interactions in unified gauge theories. Phys. Rev. Lert. 33, 451. Glashow, S. L. (1959). The vector meson in elementary particle decays. Ph.D. Thesis, Harvard University, Cambridge, Massachusetts. Glashow, S. L. (1961). Partial-symmetries of weak interactions. Nuclear Phys. 22, 579. Glashow, S. L. (1980). Toward a unified theory: Threads in a tapestry, Novel Lectures in Physics 1979. Rev. Mod. Phys. 52, No. 3,539. Goldstone, J. (1961). Field theories with “superconductor” solutions. Nuoco Cimento 19, 154. Goldstone, J., Salam, A., and Weinberg, S. (1962). Broken symmetries. Phys. Rev. 127, 965. Gross, D. J., and Jackiw, R . (1972). Effect of anomalies on quasi-renormalizable theories. Phys. Rer. D 6,477. Gross, D. J., and Wilczek, F. (1973). Ultraviolet behavior of non-Abelian gauge theories. Phys. Rev. Lett. 30, 1343. Gupta, N. (1951). On the elimination of divergencies from quantum electrodynamics. Proc. Phjs. SOC.London Ser. A 64, 426. Guranlik, G. S . , Hagen, C. R., and Kibble, T. W. (1964). Broken symmetry and the mass of gauge vector mesons. Phvs. Rew. Lert. 13,32I . Hansch, T. W., Schawlow, A. L., and Series, G. W. (1979). The spectrum of atomic hydrogen. Sci. Am. 240, No. 3, 94. Hagiwara, T., and Nakazawa, N. (1981). Perturbative effect of heavy particles in an effectiveLagrangian approach. Phys. Rev. D 23,959. Hahn, Y., and Zimmermann, W. (1968). An elementary proof of Dyson’s power counting theorem. Comm. Math. Phys. 10, 330.
196
References
Halmos, P. R. (1974a). “Measure Theory.” Springer-Verlag, Berlin and New York. Halmos, P. R. (l947b). “Finite Dimensional Vector Spaces.” Springer-Verlag, Berlin and New York. Heisenberg, W. (1938). h e r die in der Theorie der Elementariteilchen auftretende Universelle Lange. Ann. Phys. (5) 32, 20. Heisenberg, W., and Pauli, W. (1929). Zur Quantendynamik der Wellenfelder. Z. Phys. 56, 1. Heisenberg, W., and Pauli, W. (1930). Zur Quantentheorieder Wallenfelder. 11. Z. Phys. 59, 168. Hepp, K. (1966). Proof of the Bogoliubov-Parasiuk theorem of renormalization. Comm. Murh. Phys. 2, 301. Hepp, K. (1971). In “Statistical Mechanics and Quantum Field Theory, 1970 Les Houches Lectures” (C. deWitt and R. Stora, eds.), p. 429. Gordon & Breach, New York. Higgs, P. W. (1964). Broken symmetries, massless particles and gauge fields. Phys. Lett. 12, 132. Higgs, P. W. (1966). Spontaneous symmetry breakdown without massless bosons. Phys. Reo. 145, 1156. Hironaka, H. (1964). Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Murh. 79, 109. Hoffman, K. (1975). “Analysis in Euclidean Space.” Prentice-Hall, Englewood Cliffs. New Jersey. Isaacson, E., and Keller, H. B. (1966). “Analysis of Numerical Methods.” Wiley, New York. Jauch, J. M., and Rohrlich, F. (1976). “Theory of Photons and Electrons.” Springer-Verlag. Berlin and New York. Kanazawa, S., and Tomonaga, S. (1948). Corrections due to the reaction of “Cohesive force field” to the elastic scattering of an electron. I. Progr. Theoret. Phys. 3,276. Kazama, Y., and Yao, Y.-P. (1979). Effects of heavy particles through factorization and renormalization group. Phys. Rer. Len. 43, 1562. Kibble, T. W. B. (1967). Symmetry breaking in non-Abelian gauge theories. Phys. Reo. 155, 1554. Koba, Z., and Tomonaga, S. (1948). On the reaction of collision processes. I. Progr. Theoret. Phys. 3, 290. Koba, Z . , Tati, T., and Tomonaga, S. (1947). On a relativistically invariant formulation of the quantum theory of wave fields. 11. Prog. Theorei. Phys. 2,101 (1947). Kuo, P. K. and Yennie, D. R. (1969). Renormalization theory. Ann. Physics 51, No. 3,496. Lam, C. S. (1965). Feynman rules and Feynman integrals for system with higher spin fields. Nuoi.0 Cimento 38, 1755. Lamb, W. E., andaetherford, R. C. (1947). Fine structure of the hydrogen atom by a microwave method. Phys. Rev. 72, 241. Lautrup, B. E., Peterman, A., and deRafael, E. (1972). Recent developments in the comparison between theory and experiments in quantum electrodynamics. Phys. Rep. 3C,193. Lee, B. W. (1972). Renormalizable massive vector-meson theory, perturbation theory of the Higgs phenomenon. Phys. Ret.. D 5,823. Lee, B. W. (1973). Transformation properties of proper vertices in gauge theories. Phys. Let!. B 46B,214. Lee, B. W., and Zinn-Justin, J. (1972a). Spontaneously broken gauge symmetries. I. Preliminaries. Phys. Reit. D 5, 3121. Lee, B. W., and Zinn-Justin, J. (l972b). Spontaneously broken gauge symmetries. 11. Perturbation theory and renormalization. Phys. Reit D 5,3137. Lee, B. W., and Zinn-Justin, J. (1972~).Spontaneously broken gauge symmetries. Ill. Equivalence. Phys. Rev. D 5, 3 155. Lee, T. D., and Yang, C. N. (1962). Theory of charged vector mesons interacting with the electromagnetic field. Phys. Rer. 128. 885.
References
197
Lowenstein, J. H., and Speer, E. (1976). Distributional limits of renormalized Feynman integrals with zero-mass denominators. Comm. Math. Phys. 47,43. Lukierski, J. (1963). Gauge transformations in quantum field theory. Nuouo Cimento 24, 561. Mandelstam, S. (l968a). Feynman rules for electromagnetic and Yang-Mills fields from the gauge-independent field theoretic formalism. Phys. Reu. 175, 1580. Mandelstam. S. (l968b). Feynman rules for the gravitational field from the coordinate-independent field-theoretic formalism. Phys. Reu. 175, 1604. Manoukian, E. B. (1976). Generalization and improvement of the Dyson-Salam renormalization scheme and equivalence with other schemes. Phys. Reu. D 14,966,2202 (E). Manoukian, E. B. (1977). Convergence of the generalized and improved Dyson-Salam renormalization scheme. Phys. Rev. D 15, 535; 25, I157(E) (1982). Manoukian, E. B. (1978). High-energy behavior of renormalized Feynman amplitudes. J. Math. Phys. 19. 917. Manoukian, E. B. (1979a). Low-energy behavior of massive field theories. Acta Phys. Austriaca 51, 31 I . Manoukian, E. B. (1979b). Subtractions vs. counterterms. Nuouo Cimento A 53A,345. Manoukian, E. B. (1980a). Dimensional analysis of subtracted out Feynman integrands. J. Phys. A 13,2903. Manoukian, E. B. (1980b). Zero-mass behavior of Feynman amplitudes. J. Math. Phys. 21, 1218. Manoukian, E. B. (1980~).High-energy behavior of renormalized Feynman amplitudes. 11. J. Muth. Phys. 21, 1662. Manoukian, E. B. (1980d). Low-energy analysis of subtracted Feynman amplitudes. Nuouo Cimento A 57A,377. Manoukian. E. B. (1981a). On the decoupling theorem in quantum field theory. J. Math. Phys. 22, 572. Manoukian, E. B. (198 Ib). Dimensional analysis of subtracted-out Feynman integrands. 11. J. Phys. G 7, 1149. Manoukian, E. B. (1981~).General asymptotic behavior in quantum field theory. J. Phys. G 7. 1159. Manoukian. E. B. (1981d). Generalized decoupling theorem in quantum field theory. J. Math. Phys. 22, 2258. Manoukian, E. B. (1982a). Class B,-functions: Convergence of subtractions. Nuouo Cimento A 67A,101. Manoukian, E. B. (1982b). Class B,-function property of Feynman integrals. J. Phys. G 8, 599. Marciano, W., and Pagels, H. (1978). Quantum chromodynamics. Phys. Rep. 36C,No. 3, 136. Matthews, P. T. (1949). The application of Dyson’s methods to meson interactions. Phys. Rer. 76, 684. Matthews, P. T., and Salam, A. (1954). Renormalization. Phys. Reu. 94, 185. Munroe, M. E. (1959). “Introduction to Measure and Integration.” Addison-Wesley, Reading, Massachusetts. Nishijima, K . (1969). Fields and Particles: Field Theory and Dispersion Relations.” Benjamin, New York. Oppenheimer, J. R. (1930). Note on the theory of the interaction of field and matter. Phys. Ref.. 35, 461. Ovrut, B. A,, and Schnitzer, H. J. (1981). Gauge theories with minimal subtractions and the decoupling theorem. Nuclear Phys. B 179,381. Parasiuk, 0. S. (1960). On Bogoliubov’s theory of R-operation. Ukrainian Math. J . 12,287. Pati, J. C., and Salam. A. (1973). Unified Lepton-Hadron symmetry and a gauge theory of the basic interactions. Phys. Rer. D 8. 1240.
I98
References
Peierls. R. E. ( I 973). The derelopmenr oJquan/umjeld theory. In “The Physicist’s Conception of Nature” (J. Mehra, ed.), p. 370. Reidel Publ., Dordrecht, Netherlands. Poggio. E. C., Quinn, H. R., and Zuber, J. B. (1977). Renormalization group and the infrared behavior of gauge theories. Phys. Rev. D 15, 1630. Politzer, H.D. (1973). Reliable perturbation results for strong interactions. Phys. Rev. Lerf. 30, 1346. Politzer, H. D. (1974). Asymptotic freedom: An approach to strong interactions. Phys. Rep. 14C, 130. Royden, H. L. (1963). “Real Analysis.” Macmillan, New York. Rudin, W. (1964). “Principles of Mathematical Analysis.” McGraw-Hill, New York. Rudin, W. (1966). “Real and Complex Analysis.” McGraw-Hill, New York. Salam, A. (1951a). Overlapping divergences and the S-matrix. Phys. Reo. 82, 217. Salam, A. (1951b). Divergent integrals in renormalizable field theories. Phys. Rev. 84,426. Salam, A. (1952). Renormalized S-matrix for scalar electrodynamics. Phys. Rev. 86,731. Salam, A. (1968). Weak and electronra,qne/ic in/erac/ions. In “Elementary Particle Physics” (N. Svartholm, ed.). Nobel Symp. 8. p. 367. Wiley, New York. Salam, A. (1980). Gauge unification of fundamental forces, Nobel Lectures in Physics 1979. Rer. Modern Phys. 52, No. 3, 525. Salam, A,, and Strathdee, J. (1972). A renormalizable gauge model of Lepton interactions. Nuovo Cimenro A 11A, 397. Salam, A,, and Strathdee, J. (1978). Supersymmetry and superfields. Forrschr. Phys. 26.57. Schwartz, L. (1978). “Theorie des distributions.” Hermann, Paris. Schweber. S. (1961). “An Introduction to Relativistic Quantum Field Theory.” Harper & Row, New York. Schwinger, J. (1948a). On quantum electrodynamics and the magnetic moment of the electron. Phys. Re[-.73. 416. Schwinger, J. (1948b). Quantum electrodynamics. 1. A covariant formulation. Phys. Rev. 74, 1439. Schwinger, J. (1949a). Quantum electrodynamics. 11. Vacuum polarization and self-energy. Phys. Rev. 75, 65 I. Schwinger, J. (1949b). Quantum electrodynamics. 111. The electromagnetic properties of the electron-radiative corrections to scattering. Phys. Rei’. 76. 790. Schwinger, J. (1951a). On gauge invariance and vacuum polarization. Phys. Rev. 82.664. Schwinger, J. (1951b). The theory of quantized fields. I. Phys. Rev. 82,914. Schwinger. J. (1957). A theory of the fundamental interactions. Ann. Physics. 2,407. Schwinger, J. (1958). *’Quantum Electrodynamics.” Dover, New York. Schwinger, J. (1972). “Relativistic Quantum Field Theory.” Nobel Lect. Phys. 1965, p. 140. Elsevier, Amsterdam. Shaw, R. (1955). The problem of particle types and other contributions to the theory ofelementary particles, Ph.D. Thesis, Cambridge University. Slavnov, A. A. (1972). Ward identities in gauge theories. Theoref.and Math. Phys. 10,99. Slavnov. A. A. (1974). Generalized Pauli-Villars regularization in the presence of zero-mass particles. Theoref.and Math. Phys. 19, 3. Slavnov, A. A. (1975). Renormalization of gauge invariant theories. Sorier. J. Particles und Nuclei 5. No. 3. 303. Speer, E. (1969). “Generalized Feynman Amplitudes.” Princeton Univ. Press, Princeton, New Jersey. Steinmann, 0. (1966). Renormalizability in LSZ perturbation theory. Ann. Physics. 36, 267. Stuckelberg. E. C. G., and Petermann. A. (1953). Normalization of constants in quantum theory. Hell:. Phys. A r i a 26, 499.
References
199
Symanzik. K . (1970). Small distance behavior in field theory and power counting. Cornrn. Math. Phys. 18, 227. Symanzik. K . (1971). Small distance behavior in field theory. Sprin,qer Tracts Modern P h p . 57, 222. Takeda, G . (1952). On the renormalization theory of the interaction of electrons and photons. Pro.qr. Theorer. Ph)Is. 7. 359. Taylor, J. C. (1971). Ward identities and charge renormalization of the Yang-Mills field. N u c h r Phys. B 33, 436. Taylor. J . C. (1976). “Gauge Theories of Weak Interactions.” Cambridge Univ. Press, London and New York. ’t Hooft. G. (1971a). Renormalization of Massless Yang-Mills fields. Nuclear Phys. B33, 173. ‘t Hooft. G. (1971b). Renormalizable Legrangians for massive Yang-Mills. Nuclear Phys. B35, 167. ‘t Hooft. G . (1973). Dimensional regularization and the renormalization group. Nuclear Phys. E 61. 455. ’t Hooft, G.. and Veltman, M. (1972a). Regularization and renormalization of gauge fields. Nucleor Phjv. B 44. 188. ‘t Hooft, G . , and Veltman, M. (1972b). Combinatorics ofgauge fields. Nuclear Phys. B50,318. Todorov, I. T. ( I 971 ). *‘ Analytical Properties of Feynman Diagrams in Quantum Field Theory.’’ Pergamon. Oxford. Tomonaga, S. (1946). On a relativistically invariant formulation of the quantum theory of wave fields. Pro.qr. Theorel. Phvs. 2, 27. Tomonaga, S. ( 1972). *‘Development of Quantum Electrodynamics.” Nobel Lect. Phys. 1965, p. 126. Elsevier. Amsterdam. Toussaint. D. (1978). Renormalization effects from superheavy Higgs particles. P h j s Rev. D 18. 1626. Utiyama, R . (1956). Invariant theoretical interpretation of interaction. Ph.vs. Rer. 101, 1597. Vainshtein, A. 1.. and Khriplovich. 1. B. (1974). Renormalizable models of the electromagnetic and weak interactions. Sorier. Ph.vs. Uspekhi 17. No. 2. 263. Velo. G.. and Wightman, A. S., eds. (1976). ”Renormalization Theory, Proceedings of the NATO Advanced Study Institute held at the International School of Mathematical Physics at the Ettore Majorana.” Reidel. Dordrecht. Netherlands. Waller, I . (1930). Bemerkungen uber die Rolle der Eigenenergie des Elektrons in der Quantentheorie der Stranhlung. Z . Phjs. 62,673. Weinberg, S . (1960). High-energy behavior in quantum field theory. Phys. Rer. 118, 838. Weinberg, S. (1964a). Feynman rules for any spin. Phys. R u . B 1336. 1318. Weinberg. S. (1964b). Feynman rules for any spin. 11. Massless particles. Phj~s.Rer. B 1346, 882. Weinberg. S. (1967). A model of leptons. Phj3.s. Rer. Lerr. 19. 1264. Weinberg. S. (1969). Feynman rules for any spin. 111. Phys. R ~ Y .181, 1893. Weinberg. S. ( 1970). Physical processes in a convergent theory ofthe weak and electromagnetic interactions. Phy.s. Rer. Lori. 27. 1688. Weinberg. S. (1973). New approach to renormalization group. Phys. Rev. D 8. 3497. Weinberg. S. (1974). Recent progress in gauge theories of the weak, electromagnetic and strong interactions. Rer. Modern. Phj’s. 46,255. Weinberg, S. (1979). Ultraviolet divergences in quantum theories of gravitation. I n “General Relativity An Einstein Centenary Survey” (S. W. Hawking and W. Israel, eds.). Cambridge Univ. Press. London and New York. Weinberg, S. (1980). Conceptual foundations of the unified theory of weak and electromagnetic interactions. Nobel Lectures in Physics 1979. Ri,r.. Modern Phys. 52, No. 3. 515. “
References Weisskopf, V. (1934a). Uber die Selbstenergie des Elektrons. Z. Phys. 89.27. Weisskopf, V. (1934b). Berichtigung zu der Arbeit: uber die Selbstenergie des Elektrons. Z. Phys. 90, 8 17. Weisskopf, V. S. (1936). uber die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons. K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 15, No. 6,3. Weisskopf, V. S. (1939). On the self-energy and the electromagnetic field of the electron. Phvs. Rev. 56, 12. Weisskopf, V., and Wigner. E. (1930a). Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. Phys. 63, 54. Weisskopf, V., and Wigner. E. (1930b). uber die naturliche Linienbreite in der Strahlung des harmonischen Oszillators. Z. Phys. 65, 18. Wentzel, G. (1973). Quantum theory of fields (until 1947). In “The Physicist’s Conception of Nature” ( J . Mehra. ed.). p. 380. Reidel, Dordrecht, Netherlands. Wess, J., and Zumino, B. (1977). Superspace formulation of supergravity. Phys. Lett. E 66. 361. Wick, G. C. (1950).The evaluation of the collision matrix. Phys. Reu. 80,268. Wilson, K. (1969). Non-Lagrangian models of current algebra. Phys. Rer. 179, 1499. Yang, C. N., and Mills, R. (1954). Conservation of isotropic spin and isotropic gauge invariance. Phys. Rev. 96, I9 I. Zav’yalov, 0. I., and Anikin, A. S. (1976). Counterterms in the formalism of normal product. Theorer. and Math. Phys. 26, 105. Zimmermann, W. (1968). The power counting theorem for Minkowski metric. Comm. Math. Phys. 11, 1. Zimmermann, W. (1969). Convergence of Bogoliubov’s method of renormalization in momentum space. Comm. Math. Phys. 15,208. Zimmermann, W. (1973a). Composite operators in the perturbation theory of renormalizable interactions. Ann. Physics 77, 536. Zimmermann, W. (1973b). Normal products and the short distance expansion in the perturbation theory of renormalizable interactions. Ann. Physics77,570. Zumino, B. (1960). Gauge properties of propagators in quantum electrodynamics. J . Math. Phys. 1, I . Zumino, B. (1975). Supersymmetry. In “Gauge Theories and Modern Field Theory, Proceedings of a Conference Held at Northeastern University, Boston” (R. Arnowitt and P. Nath, eds.), MIT Press, Cambridge, Massachusetts.
LIST OF SYMBOLS
202
List of Symbols
INDEX
A
E
Amputated subdiagram, 79
Elementary sets, 12 Equivalence, 187 External line of, 80 External line to, 80 External momenta, 82 External variables, 82 External vertex, 80 Extra1 vertex, 79
B Bore1 set, 16 Box, 6
C
Canonical decomposition, 83, 89, 91, 92 Canonical variables, 83, 84 Cartesian product, 1 I Class B, , 29, 30 Completion of measure, 16 Convergence of subtractions, 107, 128 Counterterms. I85
F Family of boxes, 6 Feynman integrals, 30, 42, 64 Fubini's theorem, 20, 23 Fubini-Tonelli's theorem, 25
G D Graphs, and subgraphs, 77 kcouphng theorems, 139, 178, 180 Dimensional analysis, I39 Dimension numbers, 45 Direct sum, 26 Disconnected subdiagram, 78 D sets, 102
H Heine-Bore1 theorem, 7, 9 High-energy behavior, 151 203
Index
R
1
Internal line, 80 Internal momenta, 82 Internal variables, 82 Internal vertex, 80
Renormalized Feynman amplitude, 103, 107 Renormalized Feynman integrand. 102
S K k-cell, 8
L Lagrange interpolating formula, 64 Lagrangians, 183, 186 Lebesgue dominated convergence theorem, 20 Lebesgue monotone convergence theorem, 20 Lines, 77 Logarithmic asymptotic coefficients, 30 Low-energy behavior, 138, 171
Schwartz space, 63 Set of maximizing subspaces, 45 Subdiagram, 78 Subgraph, 17 Subtraction scheme, 102
T Taylor operations, 95,97, 102, 103, 108 Tempered distributions, 64n
U Unifying theorem of renormalization, 13 1 Unrenormalized Feynman integrand, 82
M Measure space, 16 Monotone class, 10
V Vertices, 77, 79, 80 0
W Orthogonal complement, 26 Wick ordering, 183 Wick polynomial, 185
P Power asymptotic coefficients, 30 Power counting criterion, 44 Projection on , . . along . . . , 26 Proper subdiagram, 80
Z Zero-mass behavior, 164 Zero-mass limit, 169
