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The effective potential for the Coleman-Weinberg model Bates, Ross Taylor 1982

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EFFECTIVE POTENTIAL FOR THE COLEMAN-WEINBERG MODEL by ROSS TAYLOR BATES B. S c , The U n i v e r s i t y Of Western Ontario, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department Of Physics We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Ross Taylor Bates, 1982 In p resenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: October 18, 1982 Abstract Gauge t h e o r i e s which have a phase t r a n s i t i o n could be u s e f u l i n the study of quark confinement. One of the simplest t h e o r i e s c o n t a i n i n g a phase t r a n s i t i o n i s the Coleman-Weinberg model of massless s c a l a r electrodynamics. The c a l c u l a t i o n of the renormalized e f f e c t i v e p o t e n t i a l f o r the Coleman-Weinberg model i s reviewed i n d e t a i l using the path i n t e g r a l formalism. The e f f e c t i v e p o t e n t i a l i s evaluated at the one-loop l e v e l to show that the model e x h i b i t s dynamical symmetry breaking at zero temperature. The divergent parts are shown to be renormalizable to two-loop order. The temperature dependence of the e f f e c t i v e p o t e n t i a l i s then c a l c u l a t e d to one-loop i n order to demonstrate that the symmetry of the model i s restored at high temperature, i n d i c a t i n g a phase t r a n s i t i o n . F i n a l l y , for models which e x h i b i t t h i s type of behaviour, a p p l i c a t i o n s to SU(n) t h e o r i e s of quarks are discussed. i i i Table of Contents Abs t r a c t i i L i s t of Figures i v Acknowledgement v I . INTRODUCTION 1 I I . FORMALISM 5 2.1 The Path I n t e g r a l 5 2.2 D e f i n i t i o n s 6 2.3 Obtaining The E f f e c t i v e P o t e n t i a l 8 2.4 Dimensional R e g u l a r i z a t i o n 10 I I I . EFFECTIVE POTENTIAL FOR THE COLEMAN-WEINBERG MODEL .12 3.1 P r e l i m i n a r y C a l c u l a t i o n s 12 3.2 One-Loop C a l c u l a t i o n Of V(C) 19 3.3 One-Loop Renormalization Of V(C) 23 3.4 C a l c u l a t i o n Of V(C) In Landau Gauge To Two Loop ...27 3.5 Scalar-Photon "Figure E i g h t " C o n t r i b u t i o n To V(C) .30 3.6 S c a l a r - S c a l a r "Figure E i g h t " C o n t r i b u t i o n To V(C) .32 3.7 Scalar "Hamburger" C o n t r i b u t i o n To V(C) 34 3.8 Photon "Hamburger" C o n t r i b u t i o n To V(C) 36 3.9 "Cracked Egg" C o n t r i b u t i o n To V(C) ...38 3.10 Two-Loop Renormalization Of V(C) 41 3.11 Dis c u s s i o n Of Gauge Invariance 45 IV. FINITE TEMPERATURE EFFECTIVE POTENTIAL 47 4.1 Changes From The Zero Temperature Case 47 4.2 F i n i t e Temperature C a l c u l a t i o n Of V(C) 49 4.3 High Temperature Expansion 51 4.4 Symmetry R e s t o r a t i o n 52 V. SUMMARY AND DISCUSSION 54 5.1 Summary Of The C a l c u l a t i o n 54 5.2 A p p l i c a t i o n s To SU(n) Gauge Theories 55 BIBLIOGRAPHY 58 APPENDIX A - INVERSE AND DETERMINANT OF THE MATRIX M 59 APPENDIX B - SOME FORMULAE REQUIRED IN THE TEXT . ... 64 APPENDIX C - EVALUATION OF I(A,B,C) 66 APPENDIX D - EVALUATION OF J(A,B,C) 72 APPENDIX E - EVALUATION OF K(A,B,C) 81 i v L i s t of Figures 1. The Scalar-Photon "Figure E i g h t " Diagram 30 2. The S c a l a r - S c a l a r "Figure E i g h t " Diagram 32 3. The S c a l a r "Hamburger" Diagram 34 4. The Photon "Hamburger" Diagram 36 5. The "Cracked Egg" Diagram 38 V Acknowledgement I would l i k e to thank my research s u p e r v i s o r , Dr. Nathan Weiss, f o r h i s help and guidance throughout the course of t h i s work. I would a l s o l i k e t o thank the N a t u r a l Sciences and Engineering Research C o u n c i l of Canada f o r i t s support i n the form of f i n a n c i a l 1 I . INTRODUCTION . Recently i n p a r t i c l e physics there has been much i n t e r e s t i n SU(n) gauge t h e o r i e s which d e s c r i b e quarks and t h e i r i n t e r a c t i o n s . I n d i v i d u a l quarks have not been observed and hence i t i s b e l i e v e d that they e x i s t only i n bound s t a t e s . However, at s u f f i c i e n t l y high temperatures i t i s thought that these bound s t a t e s could p o s s i b l y undergo a phase t r a n s i t i o n i n t o an unconfined phase of fre e quarks. One can de s c r i b e the confined nature of the quarks at zero temperature i n terms of a symmetry i n the underlying gauge theory. This symmetry w i l l not be present i n the high temperature unconfined phase. (When the s t a t e of a system does not possess the same symmetry as the underlying theory d e s c r i b i n g i t , the symmetry i s s a i d to have been spontaneously broken. A d e t a i l e d review of symmetry breaking {2} can be found i n Abers and Lee.) Thus i f we wish to inclu d e both phases i n our theory, we must con s t r u c t a model c o n t a i n i n g an i n t a c t symmetry at zero temperature which can be broken at high temperature. With the eventual c o n s t r u c t i o n of such a model as the mo t i v a t i o n , we need to develop some c a l c u l a t i o n a l techniques. In p a r t i c u l a r we w i l l look at the e f f e c t i v e p o t e n t i a l method. The e f f e c t i v e p o t e n t i a l d e s c r i b e s the ground s t a t e of the system, and from i t many p h y s i c a l r e s u l t s can be d e r i v e d . The e f f e c t i v e p o t e n t i a l method i s a s e m i - c l a s s i c a l approximation i n that i f one, c a l c u l a t e s i t p e r t u r b a t i v e l y , the lowest order term i s the c l a s s i c a l s o l u t i o n with the higher order terms being the quantum c o r r e c t i o n s . We i l l u s t r a t e the techniques involved 2 through an example by doing a d e t a i l e d c a l c u l a t i o n of the e f f e c t i v e p o t e n t i a l f o r the Coleman-Weinberg model {6}. This c a l c u l a t i o n represents a review of work done i n the l i t e r a t u r e by s e v e r a l authors {6,9,13,14} on vari o u s aspects of the model, and c o n s t i t u t e s the main part of t h i s t h e s i s . The Coleman-Weinberg model describes a photon f i e l d which i s minimally coupled to a charged, massless, s e l f - i n t e r a c t i n g s c a l a r f i e l d . I t was chosen because i t possesses the two phase property we d e s i r e , and yet i s simple enough to be used f o r an example. At zero temperature the symmetry of the model i s broken dynamically by electromagnetic r a d i a t i v e c o r r e c t i o n s to the lowest order ( c l a s s i c a l ) approximation. Note however that the techniques developed w i l l be eq u a l l y a p p l i c a b l e to models where the symmetry i s broken e x p l i c i t l y i n the lowest order. The temperature dependence of the model i s such that the broken symmetry i s res t o r e d i n the high temperature l i m i t . Although there are some d i f f e r e n c e s , t h i s i s the same behaviour we need for the theory d e s c r i b i n g quarks, and the techniques developed for the Coleman-Weinberg model should be a p p l i c a b l e f o r both. An o u t l i n e of the t h e s i s i s presented below. Chapter Two summarizes some of the formalism which w i l l be needed, and l i s t s the main r e s u l t s . For the reader u n f a m i l i a r with the subject matter, references are su p p l i e d to the appropriate area i n the l i t e r a t u r e where d e t a i l s may be found. 3 The path i n t e g r a l formulation of the theory and the e f f e c t i v e p o t e n t i a l are introduced, and a general o u t l i n e of the procedure used to c a l c u l a t e the e f f e c t i v e p o t e n t i a l i s g i v e n . A l s o included i s a b r i e f d i s c u s s i o n of the dimensional r e g u l a r i z a t i o n techniques used i n e v a l u a t i n g Feynman i n t e g r a l s . Chapter Three contains a d e t a i l e d c a l c u l a t i o n of the e f f e c t i v e p o t e n t i a l f o r the Coleman-Weinberg model at zero temperature. I t i s evaluated and renormalized at the one-loop l e v e l i n a general Lorentz gauge. The equivalence of the conventional and minimal s u b t r a c t i o n r e n o r m a l i z a t i o n schemes i s demonstrated. The broken symmetry of the theory i s a l s o shown. Then the renormalized two-loop e f f e c t i v e p o t e n t i a l i s c a l c u l a t e d i n Landau gauge to show the e x p l i c i t c a n c e l l a t i o n of the divergent p a r t s . Having i l l u s t r a t e d the method of o b t a i n i n g them, the remaining f i n i t e terms are l e f t i n i n t e g r a l form and w i l l not be evaluated here. F i n a l l y the gauge dependence of the e f f e c t i v e p o t e n t i a l i s d i s c u s s e d , as w e l l as i t s e f f e c t on the p h y s i c a l r e s u l t s of the model. Chapter Four then s t u d i e s the temperature dependence of the Coleman-Weinberg model at the one-loop l e v e l . The changes i n procedure from the zero temperature case are mentioned, and the temperature dependence of the e f f e c t i v e p o t e n t i a l c a l c u l a t e d . The r e s u l t i s i n i n t e g r a l form and must be evaluated by approximation techniques. A high temperature expansion i s performed which shows that the symmetry of the model i s r e s t o r e d at high temperature, i n d i c a t i n g a phase t r a n s i t i o n . Chapter F i v e summarizes the techniques developed i n the 4 preceding chapters. I t then gives a d i s c u s s i o n of t h e i r a p p l i c a t i o n to other gauge t h e o r i e s where new research i s p o s s i b l e , which was the o r i g i n a l m o t i v a t i o n behind reviewing the Coleman-Weinberg c a l c u l a t i o n . 5 I I . FORMALISM This chapter w i l l b r i e f l y review some of the basic d e f i n i t i o n s and procedures needed as background to the developments i n subsequent chapters. Only the main r e s u l t s w i l l be g i v e n , w i t h the reader r e f e r r e d to the l i t e r a t u r e f o r more d e t a i l e d d i s c u s s i o n . I t w i l l be assumed that the reader i s at l e a s t p a r t i a l l y f a m i l i a r with p e r t u r b a t i o n theory. Green's f u n c t i o n s , Feynman diagrams and f u n c t i o n a l methods. These t o p i c s may a l l be found d e s c r i b e d i n v a r i o u s references {4,5,12,16}. 2.1 The Path I n t e g r a l A ba s i c t o o l we w i l l need i s the fundamental path i n t e g r a l over f u n c t i o n a l space. Consider a f i e l d theory with degrees of freedom Q, (x) .Q^tx),.... which i s described by an a c t i o n S(Q), where the a c t i o n i s qu a d r a t i c i n the f i e l d s Q ^ x ) . For t h i s theory consider the generating f u n c t i o n a l Z(J) i n the prescence of a source J ( x ) . The path i n t e g r a l expression f o r Z(J) i s ( where the symbol "o3Q" denotes the measure on the f u n c t i o n a l space. To e x p l i c i t l y show the quadratic form of S(Q) one can w r i t e t h i s as (2.1. 2) 1(^ =JDQ exp t ^ ( ^ ( ^ Q P M v^w) 6 One obtains Z(J) by performing the f u n c t i o n a l i n t e g r a l over a l l f i e l d c o n f i g u r a t i o n s Q. By analogy with simple gaussian i n t e g r a t i o n , the path i n t e g r a l can be evaluated to give and the determinant i s to be taken i n the f u n c t i o n a l sense. The i d e n t i t y w i l l be u s e f u l i n c a l c u l a t i o n s . The above e v a l u a t i o n of the fundamental path i n t e g r a l forms the basi s f o r c a l c u l a t i n g the generating f u n c t i o n a l f o r t h e o r i e s with more general a c t i o n s . A f u l l d e s c r i p t i o n of f u n c t i o n a l methods and the path i n t e g r a l formalism can be found i n many f i e l d theory textbooks { 1 2 , 1 6 } . 2 . 2 D e f i n i t i o n s Consider the simple quantum f i e l d theory of i n t e r a c t i n g s c a l a r f i e l d s Q^(x) described by an a c t i o n S(Q). For n o t a t i o n a l s i m p l i c i t y some i n d i c e s and v a r i a b l e s w i l l be suppressed, except where needed f o r i l l u s t r a t i o n . Thus f o r example J»Q w i l l be used to abbreviate J d ' x J , * ( x ) Q A ( x ) . S o l u t i o n s to the theory may (2.1.3) 1(J) where (2.1.4) (2.1.5) Ad\ 7 be obtained from a knowledge of the Green's f u n c t i o n s (2.2.1) • = < o | T ( Q Q L - - Q n ) h > which are the vacuum expectation values of the time ordered product of the f i e l d s {12}. Using the path i n t e g r a l r e p r e s e n t a t i o n , the ground s t a t e vacuum amplitude f o r the theory i n the prescence of a source J ( x ) i s given by^ (2.2.2) = JSDQ expjf (SfaWT.QJj The f u n c t i o n a l Z(J) may be used to generate the Green's f u n c t i o n s of the theory. The complicated bookkeeping involved i n c a l c u l a t i n g these Green's functions can be t r a n s l a t e d i n t o the simpler g r a p h i c a l form of Feynman diagrams {12}. The Green's f u n c t i o n s can be found by summing the graphs according to r u l e s obtained from the f u l l c a l c u l a t i o n with Z(J) i n (2.2.2). The complexity of the diagrams, as measured by the number of loops, r e f l e c t s the order i n p e r t u r b a t i o n theory to which one i s c a l c u l a t i n g . Those graphs i n which a l l p a r t s are j o i n e d are r e f e r r e d to as connected Feynman diagrams, and they may be summed to give the connected Green's f u n c t i o n s . Instead of using Z(J) from (2.2.2), i t i s b e t t e r to work with the connected generating f u n c t i o n a l (2.2.3) 8 because only connected Green's f u n c t i o n s c o n t r i b u t e to the S-matrix {1,12}. The p e r t u r b a t i v e expansion of W(J) i n powers of ri corresponds to a loopwise expansion of connected Feynman diagrams. Hence i t generates only connected Green's f u n c t i o n s . Next the e f f e c t i v e a c t i o n i s defined by the Legendre transform (2.2.4) W ( T ) - T . Q where (2.2.5) -The e f f e c t i v e a c t i o n generates connected Green's fu n c t i o n s which are one p a r t i c l e i r r e d u c i b l e i n Q. This means that t h e i r Feynman graphs cannot be made disconnected by c u t t i n g a s i n g l e s c a l a r propagator. F i n a l l y the e f f e c t i v e p o t e n t i a l i s defined at constant f i e l d C by (2.2.6) \J(C) = - P ( C ) (5<Px) -I 2.3 Obtaining The E f f e c t i v e P o t e n t i a l The procedure used by Jackiw {13} to c a l c u l a t e the e f f e c t i v e p o t e n t i a l V from the a c t i o n S(Q) w i l l be followed. One s h i f t s the f i e l d Q(x) by an x-independent constant f i e l d C to o b t a i n S(Q+C). Then the terms which are l i n e a r i n Q are 9 omitted. Using the d e f i n i t i o n s (2.2.2) and (2.2.3) y i e l d s the connected generating f u n c t i o n a l i n the form of (2.3.1) Note that constant, quadratic and higher order r e f e r only to the dependence on the f i e l d Q. I t has been shown by Jackiw {13} that the performance of the Legendre transform (2.2.4) on (2.3.1) gives an e f f e c t i v e p o t e n t i a l i n the form of (- V ~ (cons k n-0 ~ C i U €X p ^ u o c M i c ) (2.3.2) ^ 5)Q €xp ^ ( ^ a c \ r a j . c + r V i ^ e r or<Lr ) p r where only connected one p a r t i c l e i r r e d u c i b l e graphs are to be included i n the l a s t term. Since only gaussian type path i n t e g r a l s can be done, the l a s t term i n (2.3.2) must be evaluated by expanding part of the exponential i n a power s e r i e s . The d e t a i l s of the procedure w i l l become c l e a r e r when i t i s a p p l i e d to a s p e c i f i c example {13,14} i n the next chapter. 10 2.4 Dimensional R e q u l a r i z a t i o n When e v a l u a t i n g Z(J) and the Green's f u n c t i o n s i n p e r t u r b a t i o n theory, one f i n d s divergent i n t e g r a l s of the form (2.4.1) where t y p i c a l l y F(k) behaves f o r lar g e k l i k e k*1 or k~^ , i n which case the i n t e g r a l has a quadratic or l o g a r i t h m i c divergence. To solve the problem of e v a l u a t i n g loop i n t e g r a l s such as (2.4.1) which are u l t r a v i o l e t d i v e r g e n t , one uses the techniques of dimensional r e g u l a r i z a t i o n . A d e t a i l e d d i s c u s s i o n of these methods i s provided i n a paper by 't Hooft and Veltman { 1 9 } . To solve divergent i n t e g r a l s l i k e I above, one considers (2.4.2) which i s evaluated, i n an a r b i t r a r y n-dimensional space. In order f o r the dimension of l ' to be the same as I , an a r b i t r a r y s c a l e f a c t o r /JL should a l s o be included i n (2.4.2). However, p h y s i c a l r e s u l t s w i l l be independent of t h i s f a c t o r and the convenient choice >4_=1 i s made. The i n t e g r a l lf w i l l be f i n i t e i n some domain, u s u a l l y f o r n<4. I t can be evaluated i n t h i s domain and the r e s u l t a n a l y t i c a l l y continued to include the region of i n t e r e s t (n=4). F i n a l l y the Lim n -^4~ i s taken, y i e l d i n g an answer which w i l l c o n t a in poles i n (n-4). These must be removed by r e n o r m a l i z a t i o n and/or c a n c e l l a t i o n with other terms. 11 In p r a c t i c e one dispenses w i t h going through the procedure in d e t a i l each time, and uses g e n e r a l i z e d formulae i f p o s s i b l e . The formulae f o r some di m e n s i o n a l l y r e g u l a r i z e d i n t e g r a l s which w i l l be needed are given i n Appendix B. The i n t e g r a l s are over n-dimensions with the Lim n—*4~ understood. 12 I I I . EFFECTIVE POTENTIAL FOR THE COLEMAN-WEINBERG MODEL In t h i s chapter the e f f e c t i v e p o t e n t i a l V((j)) i s c a l c u l a t e d to 0(ti 2) f o r the Coleman-Weinberg model {6} discussed i n Chapter One. In order to show the gauge dependence, a general Lorentz type gauge term i s included i n the Lagrangian d e n s i t y . As discussed i n Chapter Two, the expansion of the e f f e c t i v e p o t e n t i a l i n powers of h i s r e l a t e d to a loop-wise expansion of the a s s o c i a t e d Feynman diagrams. The one-loop c a l c u l a t i o n i s s u f f i c i e n t to i l l u s t r a t e that the e f f e c t i v e p o t e n t i a l i s gauge dependent. The two-loop c a l c u l a t i o n i s then s i m p l i f i e d by working with the p a r t i c u l a r choice of «*=0 known as Landau gauge. 3.1 P r e l i m i n a r y C a l c u l a t i o n s I t i s convenient to describe the complex s c a l a r f i e l d i n the model i n terms of two r e a l f i e l d s (^(x) and ( j ^ x ) . I n i t i a l l y the Minkowski metric g M v ( g O Q =1 ,9LJ ) w i l l be used. The Coleman-Weinberg Lagrangian d e n s i t y i s then given by the f o l l o w i n g expression {6}. (3.1.1) where -(2«0 ' Q^A/J*" i s a general Lorentz type gauge term, added in order to trace the gauge dependence, i n which the parameter °^will f i x the gauge choice. 13 Expanding (3.1.1) gives (3 • i • A ) -4 w ^ A H C U A A ) . where i s the antisymmetric 2x2 matrix w i t h HT a»1. The a c t i o n i s then given by S=^d"x . (3.1.3) 5C43<U= j&\k*A* 4e>/U/U<M>« F o l l o w i n g the procedure {13} discussed i n Chapter Two, we s h i f t the s c a l a r f i e l d C^(x) by an x-independent constant f i e l d Cq_ ( i e . (JX(X)—Kjii x^ + c<x)» a n d then omit the l i n e a r terms from the r e s u l t i n g expression. Note the a b b r e v i a t i o n C 2=C QC C L. (3.1.4) - eeifVVtk^. +^*fl*4<k•-fe^ULc1 - ^ c 4 14 I n t e g r a t i n g (3.1.4) by p a r t s and dropping the surface terms gives r~ e~QXf\A >^ -fc^fU^fck *H^ ft><kA (3.1.5) -r«v4k 404^ -^HM C o l l e c t i n g terms together and d e f i n i n g Q y i e l d s (3.1.6) • ± * n ^ A - ^&4<HJ At t h i s p o i n t we introduce a more compact n o t a t i o n f o r those terms i n (3.1.6) which are quadratic i n the f i e l d s . 15 Consider the row vector QT(X) (3.1.7) Q"V)= P.W R,« At6c) ftjft and the x-re p r e s e n t a t i o n of a 6x6 matrix M given by (3.1.8) \^\ -With t h i s n o t a t i o n (3.1.6) can be w r i t t e n i n the form (3.1.9) S u b s t i t u t i n g (3.1.9) i n t o (2.2.2), one obtains (3.1.10) 16 where (3.1.11) and we have defined the symbol <B> where (3.1.12) As discussed i n Chapter Two, there i s a correspondence between a loop-wise expansion of Feynman diagrams and an expansion of Z i n powers of t i . In order to make t h i s correspondence c o r r e c t l y {13}, we must r e s c a l e the f i e l d s i n Z with and A-^h^A. Then ( 3 . 1 . 1 1 ) and ( 3 . 1 . 1 2 ) may be w r i t t e n as f o l l o w s . (3.1.13) and (3.1.14) 17 Now we expand the exponential of (3.1.13) i n a power s e r i e s . Since <odd number of fields>»0, terms i n the expansion w i t h h a l f - i n t e g e r powers of "h w i l l not c o n t r i b u t e to Z^. o.t.15) *?cA&AiOfi» -^c^c^W^M^ll Performing the m u l t i p l i c a t i o n w i t h i n the double i n t e g r a l gives (3.1.16) ^ e E^CA < ( y ^ & W & M <to%W<t>f W> where the l i n e a r property <A+B> = <A> + <B> has been used. 18 We can now s u b s t i t u t e (3.1.10) i n t o (2.2.4) to obt a i n the e f f e c t i v e a c t i o n . R e c a l l that only terms corresponding t o connected diagrams are to be included from Z^. o...7> r(c)= ra +ir; + *x + o(t3') <3.1.18) P 0 = - ^ C ^ ^ ) (3.1.20) The e f f e c t i v e p o t e n t i a l may be obtained from the e f f e c t i v e a c t i o n using (2.2.6), with a s i m i l a r correspondence f o r (3.1.2D v ( 0 = v 0 -+ W , f ^ V 2 +Of-k 3) (3.1.22) V0 = ^ r C ^ (3.1.23) 19 3.2 One-Loop C a l c u l a t i o n Of V(C) In the l a s t s e c t i o n the e f f e c t i v e p o t e n t i a l was found to be given by the f o l l o w i n g expression. (3.2.1) where constant r e f e r s to terms independent of C. The f u n c t i o n a l i n t e g r a l i n (3.2.1) can be expressed i n terms of a f u n c t i o n a l determinant using (2.1.3). (3.2.2) V(c> h.c"+*^f[-i^(ortn| + M ) +*X +o(tf) The f u n c t i o n a l determinant can be expressed i n terms of an ordinary determinant {13} using (2.1.5). ( 3 . 2 . 3 ) =fti\)\QfJU(ie\n) where M i s expressed i n the momentum re p r e s e n t a t i o n (3.2.4) The determinant of M i s evaluated i n Appendix A and i s given by (A.37) with r,s defined by (A.35),(A.36). 20 S u b s t i t u t i n g these r e s u l t s i n t o (3.2.2), the e f f e c t i v e p o t e n t i a l expression becomes (3.2.5) X ( 1 ^ A S C - ) ) + (coy,5+.) U \ + 0 ( V ) The i n t e g r a l i n (3.2.5) i s more conveniently performed i n Euclidean space. This i s p o s s i b l e since the f i n a l r e s u l t f o r the e f f e c t i v e p o t e n t i a l i s unaffected by a switch from Minkowski to Euclidean space. The change i s a f f e c t e d by now using the Euclidean metric ( g A v = ), and making the s u b s t i t u t i o n kg-^iko i n the zero component of the momentum v e c t o r s . The equivalent Euclidean space expression for (3.2.5) i s then (3.2.6) where a minus sign from w i t h i n each In term has been f a c t o r e d out and included with the terms independent of C. The i n t e g r a l s i n (3.2.6) w i l l be solved using dimensional r e g u l a r i z a t i o n techniques {19}. Consider the n-dimensional i n t e g r a l ( 3'2 - 7 ) r = ; 0 ^ U " + f l ) 21 Taking the p a r t i a l d e r i v a t i v e of (3.2.7) w i t h respect to A gives (3.2.8) i i _ ( sCA L IA ~ ) (Lrrf (AV / O E v a l u a t i n g t h i s with (B.1) from Appendix B gives (3.2.9) i i - T P - V P A % I n t e g r a t i n g the r e s u l t y i e l d s - I where the i n t e g r a t i o n constant r e f e r s to terms independent of A, S u b s t i t u t i n g n=4-2£ (3.2.1D J _ - ^ n + (<*n*f) Expanding (3.2.11) i n a Taylor s e r i e s i n £ and using (B.5) to expand P(c_~1) r e s u l t s i n (3.2.12) 1 = £ +lV(2Ht-JU(A/4n-) +0(0 +(con5+.) 22 Applying (3.2.12) to (3.2.6) (3.2.13) + (va)+i - X n ( A c y ^ ) ) + V wH-ih + [ V - M A s c V w ^ + +# v*. + O f t * ) Since the e f f e c t i v e p o t e n t i a l may only be defined up to an a r b i t r a r y constant, the terms independent of C may be dropped. Noting that r 2 + s 2 = 4-48<*e2A~', we combine terms i n (3.2.13) to obtain 23 3.3 One-Loop Renormalization Of V(C) We w i l l take the l i m i t as fc^O4" i n (3.2.14), and hence the £ 1 term diver g e s . At t h i s order i t i s s u f f i c i e n t to renormalize the mass and the c o u p l i n g constant A, as t h i s w i l l remove the Gh) divergent piece from (3.2.14). The usual mass and c o u p l i n g constant r e n o r m a l i z a t i o n c o n d i t i o n s {6} f o r the theory are given by (3.3.1) (3.3.2) Ay C-O c-a 1 A* I t i s not p o s s i b l e i n t h i s case to define the renormalized c o u p l i n g constant A R with (3.3.2), due to the l o g a r i t h m i c s i n g u l a r t y i n V(C) at C=0. Instead one must choose some a r b i t r a r y point C=CQ away from zero at which t o d e f i n e AR. (3.3.3) k S u b s t i t u t i n g (3.2.14) i n t o (3.3.1) and (3.3.3) gives (3.3.4) + t-i11^ * iXA) + T M£) + $ = o C-o 24 and (3.3.5) + ^ [ l l e 4 i n € ^ ^ ^ ) + ^ ^ 4 ^ ^ ( 4 1 ) Since both the bare and renormalized masses are zero, there i s no need to use a mass counterterm. However the co u p l i n g constant A d i f f e r s from Expressing A i n terms of >Aft from (3.3.5) (3.3.6) - i a TLe^e 1 + (4*) + t i f k j L ( ^ ) + £ | B L 4n ^ 5 j j where r R and s^ are merely the expressions r and s using Aft instead of A. S u b s t i t u t i n g (3.3.6) i n t o (3.2.14) (3.3.7) 25 Since A R was defined at an a r b i t r a r y p o i n t , one may red e f i n e i t as f o l l o w s (3.3.8) L _ S u b s t i t u t i n g (3.3.8) i n t o (3.3.7) (3.3.9. W>= S [ A R + ^ (|Ai+1e»--«^A.c^ ) +0(4?) This i s renormalized e f f e c t i v e p o t e n t i a l t o Qh) . While the above r e n o r m a l i z a t i o n procedure was simple enough at t h i s l e v e l , d i f f i c u l t i e s a r i s e at higher orders. Expressing \ i n terms of AR from (3.3.5) to the next order would be an onerus task. In a d d i t i o n , the charge and the f i e l d s may need to be renormalized. Things are g r e a t l y s i m p l i f i e d by using an a l t e r n a t e r e n o r m a l i z a t i o n procedure c a l l e d minimal s u b t r a c t i o n . With t h i s method the c o u p l i n g constant i n (3.2.14) i s d e f i n e d as a power s e r i e s i n "h. The divergent pieces i n V(C) can then be ca n c e l l e d by an appropriate choice of the c o e f f i c i e n t s i n the power s e r i e s . S u b s t i t u t i n g A= A 0 +f»A| +'h 2Ai + 0("h3) i n t o (3.2.14) + + f £ -^<?AS + 0 W ) 26 where r 0 and s Q are the expressions f o r r and s using A0 i n s t e a d of A. Making the f o l l o w i n g choice f o r A( ( 3 . 3 . 1 1 ) and s u b s t i t u t i n g (3.3.11) i n t o (3.3.10) gives (3.3.12) v(c> ^ [ A 3 + i ^ ( ! ^ + V - ^ ^ c 5 ) + CW) The i n t e r p r e t a t i o n of }\Q i n terms of the usual renormalized coupling constant may be made by comparison with (3.3.3). Since (3.3.12) i s the same as (3.3.9), the two r e n o r m a l i z a t i o n procedures are e q u i v a l e n t . Henceforth only the simpler minimal s u b t r a c t i o n method w i l l be used. When i t becomes necessary, the charge and the f i e l d s w i l l a l s o be renormalized i n t h i s way. Through simple algebra one may obtain the s t a t i o n a r y p o i n t s of V(C) from (3.3.12). where C=0 i s a maximum and the other two p o i n t s are minima. Since the minimum i s no longer at C=0, we see that the OCh) c o r r e c t i o n to V(C) has indeed broken the symmetry dynamically. ( 3 . 3 . 1 3 ) c = o ) c - ±exp-27 3.4 C a l c u l a t i o n Of V(C) In Landau Gauge To Two Loop The expression f o r the CKti2) c o r r e c t i o n to the e f f e c t i v e a c t i o n was found i n Se c t i o n 3.1 and i s obta i n a b l e from (3.1.20) and (3.1.16). C t < ft*Wj* (fcWO^x) A v(y) A v ( y ) < K t y > (3.4.D + i Ae £ Q d C t < / U ( * M l ^ ^ We define the f o l l o w i n g n o t a t i o n (3.4.2) Gqt(x,y) = <*q(x)(pb(Y)> (3.4.3) G ^ ^ v ) = - X / L t t f l v t o ) (3.4.5) Gqv(x,y) - ~A<4^X)A VW)> 28 Next we expand (3.4.1) using Wick's theorem {12}. As discussed i n Chapter Two, only the terms corresponding to connected diagrams which are a l s o one p a r t i c l e i r r e d u c i b l e i n Cj) are t o be kept. A l s o note that £qcj <(Pq(Pj>=0 since 6"a<j i s antisymmetric while <0q0<i> i s symmetric under a**d. Thus we have ( 3 . 4 . 7 ) - i e x e qj G*v(p*4) i L p v G^ CWGdb<?) - e 4 C , C k Gqfc(p+A) G A v ( f t -1 c « c b G Q G # fl) 6 W ^ A ) - C q C b G Q f G ^ ( 0 G j b f e 7 | - Iff + ^ A a ^ ^ H C a C W G ^ > ) ^ ^ Fou r i e r transforming to 'momentum space afl 29 R e c a l l from (2.1.2) and (2.1.3.) the f o l l o w i n g expressions. 0 . 4 . 8 ) 7 ( T ^ = ^£>q e^p JL ^\({qrnq + T T Q ) (3 .4.9) 1 (J) = ?(o) exp - i - J ^ I T ' T d ^ From these one obtains r e s p e c t i v e l y (3.4.10) (3.4.11) •5^QQ<6c)Q»6y)gxft^QTnQ - i Equating (3.4.10) and (3.4.11) y i e l d s <Q Q i(x)Q^(x)>=i^ <' (x-y)M^ . In keeping with the previous n o t a t i o n t h i s becomes Go<p (x»y) = V (x—y)Mj^J . F o u r i e r transform to momentum space to obtain G^(k)=Mj"p . In t h i s case the matrix M i s the one given by (3.1.8). The momentum re p r e s e n t a t i o n of M~' i s c a l c u l a t e d i n Appendix A. In oi = 0 Landau gauge the r e s u l t i s given by o.4.,2) G,t(i> [S > b(F-^)+|c^(i a- ^ ( J F - i c ^ 30 The 0($i2) c o n t r i b u t i o n to the e f f e c t i v e p o t e n t i a l V 2 may now be determined. The terms i n (3.4.15) correspond to the f i v e two-loop Feynman diagrams which f o l l o w . The i n t e g r a l s can be evaluated with dimensional r e g u l a r i z a t i o n techniques. 3.5 Scalar-Photon "Figure E i g h t " C o n t r i b u t i o n To V(C) \ \ P f 1 \ i . / / Figure 1 - The Scalar-Photon "Figure E i g h t " Diagram 31 Consider the term i n (3.4.15) given by which corresponds to the Feynman diagram i n Figu r e One. S u b s t i t u t i n g (3.4.12) and (3.4.13) i n t o (3.5.1) (3 5 2) T -(3.5.2) ^ ^ V - j - ^ As discussed p r e v i o u s l y , we may evaluate the i n t e g r a l s using the equivalent Euclidean space expression ( 3 . 5 . 3 ) ~ r ~ ~ ) ( ^ v r CM* flMc?) (k\\C) E v a l u a t i n g the i n t e g r a l s with (B.1), and s e t t i n g n=4-2£. gives ( 3 . . . . . i , = t o - u v ^ c c ^ r ( * < ) Expanding t h i s in a Taylor s e r i e s i n £ and using ( B . 5 ) to expand T(£-1) r e s u l t s i n I i = ^ t * £ f t ? + 1 ( j * m v f t : i _ V U h € * 4 / > - ^ ( V t l - 3 J ^ ( U ) - w J ^ C ^ + F ; ( A A c ) + O f e ) (3.5.5) 32 where (3.5.6) 3.6 S c a l a r - S c a l a r "Figure E i g h t " C o n t r i b u t i o n To V(C) Figure 2 - The S c a l a r - S c a l a r "Figure E i g h t " Diagram 33 Consider the term i n (3.4.15) given by ( 3 . 6 . . ) 1 ^ ^ . k , ( 0 G k l ( « nG^jk) which corresponds to the Feynman diagram i n Figure Two. S u b s t i t u t i n g (3.4.12) i n t o (3.6.1) (3.6.2) Changing to the equivalent Euclidean space expression for and c o l l e c t i n g terms y i e l d s (3.6.3) 1A 4( 4( •1 E v a l u a t i n g the i n t e g r a l s w i t h (B.1), and s e t t i n g n=4-2£ gives 1 34 Expanding i n a Taylor s e r i e s i n £ and using (B.5) to expand F(£-1) r e s u l t s i n where (3.6.5) ^ 4 H 4 J U ( M -114/n(C^ 4 F 2 ( A , e \ c ) 4 0(e) 4 J W ( A TcVh)+(AcVi) 4lje^(Ac%) 4i4>£v?(M (3.6.6) - L+oXn ( M (A cM) - <3 ^  CM (A cV*) « W^ftq JG* (M/r) -ia£~ (ACV2) - 4i^A CV6)J 3.7 Scalar "Hamburger" C o n t r i b u t i o n To V(C) ?7>-i ^ ™ J \ Figure 3 - The Scalar "Hamburger" Diagram 35 Consider the term i n (3.4.15) given by ^ " " ^ ) ( I n f C« Gq(/p) Gjf (A) Gfd( (3.7.1) 4 - 2 G q f ( ? a ) G c d ( ? ) G j t a ) J which corresponds to the Feynman diagram i n Figure Three S u b s t i t u t i n g (3.4.12) i n t o (3.7.1) V - J f e fir [ ( f - 4 JC1- 4 -Changing to the equivalent Euclidean space expression (3.7.3) 4 - 3 ( H t + M U l * M ' ( f + 4 c * ) " ] ] Thus (3.7.4) where I(A,B,C) i s evaluated i n Appendix C and i s given by (3.7.5) B,C) - C p. & ( | f t f f 4 W ( P*+ 0 " 36 S u b s t i t u t i n g (C.24) i n t o (3.7.4) g i v e s (3.7.6) -\5j^(V«/r)-l6A,(c i)ft+ F,(A,«\c) +0(0 where and F (A,B,C) i s given by (C.25). 3.8 Photon "Hamburger" C o n t r i b u t i o n To V(C) Figure 4 - The Photon "Hamburger" Diagram 37 Consider the term i n (3.4.15) given by (3.8.,) I 4 = _ e 4 $ 0 „ ^f[c^Ga^k)G^)G^U)j which corresponds to the Feynman diagram i n Figure Four. S u b s t i t u t i n g (3.4.12) and (3.4.13) i n t o (3.8.1) gives (3.8.2) Now we change to the equivalent Euclidean space expression for 1^ given by (3.8.3) and s u b s t i t u t e n=4-2£to obtain (3.8.4) where I -- - « 4 c l [ ( > * 0 K ^ « v , ee)+J(4cx, ^ ^ ) ) (3.8.5) W . C ^ ) ^ (Iff)" ( ^ ) V ^ ) ( F ^ X ^ ) and J(A,B,C) i s evaluated i n Appendix D. 38 S u b s t i t u t i n g (C.24) and (D.31) i n t o (3.8.4) g i v e s (3.8.6) where (3.8.7) (A>e\C> -*Vfz £ (4^eV; <?tf) + F x( k2> e*c* eV)J and F r(A,B,C),F T(A,B,C) are given by (C.25),(D.32) r e s p e c t i v e l y . 3.9 "Cracked Egg" C o n t r i b u t i o n To V(C) Figure 5 - The "Cracked Egg" Diagram 39 Consider the term i n (3.4.15) given by J-5 (3.9.1) which corresponds to the Feynman diagram i n Figure F i v e . S u b s t i t u t i n g (3.4.12) and (3.4.13) i n t o (3.9.1) gives (3.9.2) * 5 Now we change to the equivalent Euclidean space expression (3.9.3) X ( ( r a f + ec1)" ( A 1 + i ( p * +i c)" to o b t a i n (3.9.4) I 5 = ' t . « L K ( « L C L ) ^ C l ) 4 c l ) 40 where (3.9.5) and K(A,B,C) i s evaluated i n Appendix E. S u b s t i t u t i n g (E.24) i n t o (3.9.4) gives (3.9.6) where and F K(A,B,C) i s given by (E.23). 41 3.10 Two-Loop Renormalization Of V(C) The e f f e c t i v e p o t e n t i a l i n <x=0 Landau gauge may be obtained from (3.2.14). (3.10.1) ^ ( b H ^ ) " [ ? l JJU ? *(,£JU ( 4) + \FU ( i ) S u b s t i t u t i n g the r e s u l t s of Sections 3.5 through 3.9 i n t o (3.4.15) gives (3.10.2) JL-\ The f i n i t e p a r t s F^(A,e2,C) of V 2 have already been st a t e d i n Sections 3.5 through 3.9. The term f (A,e 2 )C* (1 Srr1 f 1 i s an ab b r e v i a t i o n for the numerous divergent pieces of V2 which have only a simple C 4 dependence on C. They are not st a t e d e x p l i c i t l y here because t h e i r exact form w i l l not be needed i n the r e n o r m a l i z a t i o n procedure. 42 Proceeding with the minimal s u b t r a c t i o n method, we make the f o l l o w i n g power s e r i e s expansions i n "h. (3.10.3) A ~ Ao+'HA l + \ ' ' (3.10.4) (3.10.5) C ~ C{\ Vtf^Jf- ) S u b s t i t u t i n g these expressions i n t o (3.10.1) leads a f t e r some simple algebra to (3.10.6) V(0 = % A 0 \*i\0i, -(%^[c\m+i +> W e„l e,1 JU ei + U A0A, JU (%) + f U M¥l * ¥ A» A 43 Make the f o l l o w i n g choices f o r A, and /\^ . (3.10.7) and (3.10.8) S u b s t i t u t i n g (3.10.7) and (3.10.8) i n t o (3.10.6) g i v e s (3.10.9) +^F t(A.>%\C) + 0(tf) 44 where (3.10.10) S u b s t i t u t i n g (3.10.2) i n t o (3.10.9) (3.10.11) x-I Make the f o l l o w i n g c h o i c e s f o r \x>z\ » ef ( 3 . 1 0 . 1 2 ) A 1 = A , " ( l ^ T 1 f ( A o ^ o " ) ( 3 . 1 0 . U ) ^ 3e 0 x (it/7*y* r 1 ( 3 . 1 0 . 1 4 ) e ^ - e 0 4 ( i t S / r ^ V 1 45 P u t t i n g these choices i n t o (3.10.11) y i e l d s V C O = % ( A . + * (i*f ( I A? +.1 tf) A * c W A . ) (3.10.15) + F x ( A . , « . \ c ) 4-OW) The divergent 0(ti 2) pieces have thus c a n c e l l e d . The choice of \ x i s l e f t open so that i t may be used to cancel those terms i n F/(Ao' eo'C) which have a simple C* dependence on C. 3.11 D i s c u s s i o n Of Gauge Invariance The one-loop renormalized e f f e c t i v e p o t e n t i a l was found i n Section 3.3 for the Coleman-Weinberg model i n a general Lorentz gauge. (3.11..) v<cV§S[A + i ( f A V k ^ e \ \ ) ^ From the presence of the gauge parameter ot i n (3.11.1), i t can be seen that the e f f e c t i v e p o t e n t i a l i s gauge dependent. The question has been r a i s e d {7,13,14} as to how t h i s w i l l a f f e c t the p h y s i c a l consequences of the model. Any p h y s i c a l q u a n t i t i e s obtained should a p r i o r i be independent of gauge ch o i c e . There can be no d i r e c t p h y s i c a l i n t e r p r e t a t i o n of a gauge dependent e f f e c t i v e p o t e n t i a l , and thus the v a l i d i t y of any approximation to the complete V(C) must be looked a t . The le a d i n g term of the s c a l a r - v e c t o r mass r a t i o f o r the Coleman-Weinberg model has been c a l c u l a t e d by Kang {14} at the 46 two-loop l e v e l i n a general Lorentz gauge. To the extent that the approximations made have the same behaviour f o r a l l gauge choi c e s , the r e s u l t i s gauge independent as expected. One assumes that i t i s p o s s i b l e to work with the e f f e c t i v e p o t e n t i a l i n a convenient gauge to o b t a i n v a l i d p h y s i c a l r e s u l t s . Although the intermediate steps may be gauge dependent, the a p r i o r i assumption of gauge in v a r i a n c e f o r p h y s i c a l q u a n t i t i e s w i l l ensure that the f i n a l r e s u l t i s c o r r e c t . 47 IV. FINITE. TEMPERATURE EFFECTIVE POTENTIAL This chapter w i l l study the temperature dependence at the one-loop l e v e l of the e f f e c t i v e p o t e n t i a l f o r the Coleman-Weinberg model j u s t looked a t . For s i m p l i c i t y the o(=0 Landau gauge i s worked i n . The Boltzmann constant k s w i l l be set equal to one. The Euclidean metric (g,nv = £><*v) * s used and the temperature parameter w i l l be the inverse temperature =^T~'. The main r e s u l t i s to show that the broken symmetry of the theory i s restored at high temperature. 4.1 Changes From The Zero Temperature Case The method of c a l c u l a t i n g the e f f e c t i v e p o t e n t i a l V(^~') c l o s e l y p a r a l l e l s that done i n Chapter Three f o r V($~'=0). One begins with the p a r t i t i o n f u n c t i o n where the tra c e i s taken over a l l p o s s i b l e s t a t e s of the system described by the Hamiltonian H. With t h i s exception, the d e f i n i t i o n s of Section 2.2 are unalt e r e d . The c a l c u l a t i o n then proceeds as described i n Section 2.3 with only a few changes. A d e t a i l e d d e s c r i p t i o n of the f i n i t e temperature path i n t e g r a l formalism may be found i n the l i t e r a t u r e {3,8,11}. (4.1.1) ~ L ( T V 48 For c a l c u l a t i o n a l purposes, one need only note the f o l l o w i n g few simple r e s u l t s . In the f i n i t e temperature case one i s r e s t r i c t e d to the s e c t i o n of Euclidean space bounded by 0<t<£. The volume element thus becomes (4.1.2) The boson f i e l d s d e f i n e d on t h i s space must s a t i s f y the p e r i o d i c boundary c o n d i t i o n s ( 4 - K 3 ) A A ( ^ x ) = AM(oj) <*•'•«> $,(?,!() = Q>M*) The time i n t e g r a t i o n i s now over a f i n i t e range and hence the F o u r i e r transform to momentum space i s d i s c r e t e rather than continuous. This r e s u l t s i n the f o l l o w i n g changes from the zero temperature formalism. (4.1.5) <4.1.6) i>o > ^ = 7 . / r h f ' With these r e s u l t s one may now c a l c u l a t e the e f f e c t i v e p o t e n t i a l at f i n i t e temperature. Note that f o r zero temperature *°o and the r e s u l t s w i l l reduce to those i n Chapter Three. 49 4.2 F i n i t e Temperature C a l c u l a t i o n Of V(C) R e c a l l from (3.2.6) that the one-loop e f f e c t i v e p o t e n t i a l foro(.=0 Landau gauge i s given by (4.2.1) ' fo r the 9*=0 case. Noting the changes de s c r i b e d i n Section 4.1, the r e s u l t f o r the general case i s then given by ^L(t\^^) *JL(£*^*~z)J + o(**) Consider the i n t e g r a l (4.2.3) I ( ^ ) = f f ' J g i D where (4.2.4) D ~ Z ^ ( • T ^ W ) Taking the p a r t i a l d e r i v a t i v e of D with respect to t2 gives (4.2.5, j f = I ( l l r f ^ ) H 50 The s e r i e s (4.2.5) may be found i n reference t a b l e s {10} and summed to y i e l d I n t e g r a t i n g gives (4.2.7) D - 7 J U ( S i ^ f l t t V / f l P ] ) + D' with a constant of i n t e g r a t i o n D y. W r i t i n g s i n h i n terms of exponentials i n (4.2.7) and s u b s t i t u t i n g i n t o (4.2.3) gives i ( r ) - - ^ g ? k i l 4 ' - 2 i i (4.2.8) 4-D ' 4- 7£n( l - e * p H ( f V The term i n v o l v i n g ln2 may be dropped as i t i s independent of A 2. At zero temperature I ( $ H ) i n (4.2.8) must reduce to i t s equivalent I( 6^=0) c a l c u l a t e d i n Section 3.2. Thus (4.2.8) becomes (4. 2.9) z(r)-~ Kf--o) 0,JL[\-^[-mh^'}] 51 F i n a l l y the i n t e g r a t i o n s over angles may be performed and the v a r i a b l e change x 2»£ 2k 2 made to y i e l d (4.2.10) =I(?T'=o) + ^ ( T * J > X-lv,fl-e»p(-(xH^f]} a S u b s t i t u t i n g (4.2.10) i n t o (4.2.2) t o ob t a i n the f i n i t e temperature e f f e c t i v e p o t e n t i a l gives + X * i \ - e x p [ - ( x * + ^ C * ) V l ] } V O f t 1 ) The term V($~' = 0,C) was found i n Chapter Three and i s given by (3.10.15). The x - i n t e g r a t i o n s i n (4.2.11) cannot be evaluated e x a c t l y and some approximation technique must be used. 4.3 High Temperature Expansion Consider the i n t e g r a l piece of (4.2.10) i n the high temperature l i m i t as $—*0. 0 One may expand the integrand i n a Taylor s e r i e s about p =0, and subsequently perform the i n t e g r a t i o n over x. Dolan and Jackiw have discussed t h i s expansion i n d e t a i l {8}, and t h e i r r e s u l t i s 52 given by (4.3.2) ^ 0 ^ Yin- 64 S u b s t i t u t i n g (4.3.2) i n t o (4.2.11) and r e t a i n i n g only the lea d i n g high temperature terms y i e l d s ( 4 . 3 . 3 ) V(f) = V(f=oHfcr-^p + * ^ ( > + | A > - ^ The term i s independent of C and may be dropped. Thus the f i n i t e temperature e f f e c t i v e p o t e n t i a l at the one-loop l e v e l for high temperature i s ( 4 . 3 . 4 ) \i(r) v ( r = ° ) + H ^ 0 * H \ A ) + • • • 4.4 Symmetry R e s t o r a t i o n R e c a l l the zero temperature e f f e c t i v e p o t e n t i a l from equation (3.3.12) f o r o£=0 Landau gauge. (1.4.,) V ( f ; o ) = g f A + i ( | A + V ) i n C ^ O ^ ) 53 Also r e c a l l i t s s t a t i o n a r y p o i n t s from (3.3.13) (4.4.2) (4.4.3) C - ± exo -P L ^ M f A - ^ e * ) As discussed i n Chapter Three, the prescence of the two s t a t i o n a r y p o i n t s other than C=0 r e f l e c t s the broken symmetry of the model at zero temperature. At high temperature the e f f e c t i v e p o t e n t i a l from (4.3.4) has i t s dominant C-dependent behaviour i n the form of (4.4.4) •V(r')= k r - C \ 3 e V ^ ) / w and C=0 i s the only s t a t i o n a r y p o i n t . Hence the symmetry i s re s t o r e d at high temperature. From (4.3.4) i t can be seen that the f i r s t d e r i v a t i v e of V( ^H,C) i s continuous, so the phase t r a n s i t i o n from the symmetric to the broken case i s not f i r s t order. 54 V. SUMMARY AND DISCUSSION This chapter summarizes the e f f e c t i v e p o t e n t i a l techniques j u s t developed. I t then considers t h e i r a p p l i c a t i o n to studying models other than Coleman-Weinberg, c o n c e n t r a t i n g i n p a r t i c u l a r on SU(n) gauge t h e o r i e s . 5.1 Summary Of The C a l c u l a t i o n The e f f e c t i v e p o t e n t i a l was introduced using the path i n t e g r a l formalism i n Chapter Two. The method of i t s c a l c u l a t i o n was then o u t l i n e d f o r the general case. A knowledge of the e f f e c t i v e p o t e n t i a l a l l o w s one to d e r i v e many p h y s i c a l r e s u l t s . The d e t a i l s i n v olved i n the c a l c u l a t i o n were given by example through the study of the Coleman-Weinberg model i n Chapter Three. E v a l u a t i n g the one-loop renormalized e f f e c t i v e p o t e n t i a l demonstrated the dynamical symmetry breaking of the model, and a l s o i n d i c a t e d that the r e s u l t was gauge dependent. I t was concluded that the a p r i o r i assumption of having gauge i n v a r i a n t p h y s i c a l q u a n t i t i e s should ensure that one can work with the e f f e c t i v e p o t e n t i a l method i n any convenient gauge and s t i l l o b t a i n the c o r r e c t p h y s i c a l r e s u l t s . The two-loop c a l c u l a t i o n i n Landau gauge was then performed which e x p l i c i t l y demonstrated the r e n o r m a l i z i b i l i t y of the e f f e c t i v e p o t e n t i a l to that order. F i n a l l y i n Chapter Four the temperature dependence of the e f f e c t i v e p o t e n t i a l was determined to one loop order. The important r e s u l t was that the symmetry of the model which was 55 dynamically broken at T=0 could be r e s t o r e d at high temperatures. I t i s t h i s two phase nature, where a dynamically broken gauge theory can have i t s symmetry r e s t o r e d , that w i l l be of great use when a p p l y i n g e f f e c t i v e p o t e n t i a l methods to other models. 5.2 A p p l i c a t i o n s To SU.(n) Gauge Theories The techniques developed above may be used to study more complex t h e o r i e s than Coleman-Weinberg. P a r t i c u l a r l y i n t e r e s t i n g are SU(n) gauge t h e o r i e s d e s c r i b i n g quarks which i n t e r a c t through gluon gauge f i e l d s . Both quarks and gluons c a r r y an e x t r a quantum number known as colour i n a d d i t i o n to the usual set of quantum numbers. Since an i n d i v i d u a l quark has never been observed, i t i s b e l i e v e d that only colour n e u t r a l s t a t e s e x i s t . The quarks are bound i n such a way that the r e s u l t i n g colour quantum number of the s t a t e i s zero. At s u f f i c i e n t l y high temperature, i t i s thought that the quarks could undergo a phase t r a n s i t i o n to an unconfined phase of free quarks. That such a t r a n s i t i o n i s p o s s i b l e has been demonstrated i n l a t t i c e c a l c u l a t i o n s of quark confinement at high temperature {15,18}. Thermal f l u c t u a t i o n s cause the gluon f i e l d s to screen the c o n f i n i n g forces between the quarks. I t i s p o s s i b l e to show {15} that the c o n f i n i n g phase and the deco n f i n i n g phase correspond to s t a t e s of the system where a symmetry of the un d e r l y i n g theory i s , r e s p e c t i v e l y , i n t a c t or broken. This symmetry i s the Z(n) symmetry, where Z(n) i s the centre of the SU(n) gauge group of transformations ( i .e. J 7.->exp(t2iTi/n)i^). At high enough temperature the Z(n) 56 symmetry i s broken and the quarks are fre e i n an unconfined phase. What has j u s t been described i s a gauge theory c h a r a c t e r i z e d by a high temperature phase with a broken symmetry (deconfining) and another low temperature phase where the symmetry i s i n t a c t ( c o n f i n i n g ) . The s i m i l a r i t i e s to the Coleman-Weinberg model suggest that the e f f e c t i v e p o t e n t i a l techniques reviewed above could be used to study the c o n f i n i n g -deconfining phase t r a n s i t i o n i n the continuum f o r SU(n) gauge t h e o r i e s . As a s p e c i f i c case we take the SU(2) gauge theory which i s described by the p a r t i t i o n function-(5.2.1) 2 3 )«%*A 0 ff i ) J A * ( * , * ) ^ o where (5.2.2) and (5.2.3) with a , i , j - 1 , 2 , 3 . Working from (5.2.1) i n a gauge AQ (X)=£ , (J?(x), and using the same techniques shown i n the 57 Coleman-Weinberg example, Weiss {20} c a l c u l a t e d the f i n i t e temperature one-loop e f f e c t i v e p o t e n t i a l o b t a i n i n g (5.2.4) V ( C ) -4 I 45 - i i * ; -V-This p e r t u r b a t i o n c a l c u l a t i o n i s v a l i d at high temperature and shows that the symmetry i s broken. The two-loop c a l c u l a t i o n has not yet been done, and i t i s hoped that doing so might a i d i n the understanding of how the symmetry i s r e s t o r e d at lower temperatures. The f e a s a b i l i t y of such a study i s c u r r e n t l y being i n v e s t i g a t e d . 58 BIBLIOGRAPHY 1. Abbott, L.F. 1981. Report Ref.TH.3113-CERN (unpublished) 2. Abers, E.S. and Lee, B.w. 1973. Physics Reports, 9,1. 3. Bernard, C.W. 1974. Phys. Rev. D, 9 ,3312. 4. Bjorken, J.D. and D r e l l , S.D. 1964. R e l a t i v i s t i c  Quantum Mechanics (McGraw-Hill, New York! 5. Bjorken, J.D. and D r e l l , S.D. 1965. R e l a t i v i s t i c  Quantum F i e l d s (McGraw-Hill, New York) 6. Coleman, S. and Weinberg, E. 1973. Phys. Rev. D, 7 , 1888 . 7. Dolan, L. and Jackiw, R. 1974. Phys. Rev. D, 9 ,2904. 8. Dolan, L. and Jackiw, R. 1974. Phys. Rev. D, 9 ,3320. 9. E l i a s , V. and Mann, R.B. 1982. U n i v e r s i t y of Toronto P r e p r i n t 10. Gradshteyn, I.S. and Ryzhik, I.M. 1980. Table of  I n t e g r a l s , S e r i e s , and Products (Academic, New York) 11. Gross, D.J., P i s a r s k i , R.D., and Y a f f e , L.G. 1981. Rev. Mod. Phys., 53 ,43. 12. Itzykson, C. and Zuber, J . 1980. Quantum F i e l d Theory (McGraw-Hill, New York) 13. Jackiw, R. 1974. Phys. Rev. D, 9 ,1686. 14. Kang, J.S. 1974. Phys. Rev. D, j_0 ,3455. 15. Polyakov, A.M. 1978. Phys. L e t t . B, T2 ,477. 16. Ramond, P. 1 9 8 1 . F i e l d Theory A Modern Primer (Benjamin, Reading) 17. Selby, S.M. 1972. CRC Standard Mathematical Tables (CRC, Cleveland) 18. Susskind, L. 1979. Phys. Rev. D, 20 ,2610. 19. 't Hooft, G. and Veltman, M. 1972. Nucl. Phys. B, 44 ,189. ~~ 20. Weiss, N. 1981. Phys. Rev. D, 24 ,475. 59 APPENDIX A ~ INVERSE AND DETERMINANT OF THE MATRIX M In t h i s appendix the inverse and determinant w i l l be c a l c u l a t e d f o r a matrix M i n the momentum r e p r e s e n t a t i o n . In what f o l l o w s k 2 i s de f i n e d using the Minkowski metric g ^ (g 0 0 5 8 1 >9ij. ) where M,V=0,1,2,3. The d e f i n i t i o n C2=C,2+C2 i s als o used. F i r s t consider the 2x2 matrix F given by <»•» F a b = S a t ( i T - K ) - ^ c , c k where a,b=1,2 and S i s the Kronecker d e l t a f u n c t i o n . The determinant and inverse of F are then given by (A.2) M F = ( * - 4 e ) ( # - k * } (A.3) Consider the 4x4 matrix N given by <A.4> KL = - 9 A V ( ^ " A ) + where/U,V=0,1 ,2,3. The inverse and determinant of N are then given by ( A . s ) d ^ N - -{AX-AT(*Z-BA1-A) 60 Consider the nxn matrix (A.7) 1 A B D f o r a r b i t r a r y , i n v e r t i b l e matrices A,B,E,D. Let us w r i t e the inverse of M as PT' = (A.8) 2 The equation MM"'=I leads to the f o l l o w i n g four c o n d i t i o n s . (A.9) (A.10) (A. 1 1 ) (A.12) W = ( A - R P H E ) 2 = ( D - r T / T B ) X = -A"'B2 —i Take the f o l l o w i n g s p e c i f i c case f o r M <A-'4> B„V = /.e £ q f Cf £ v (A.I6) D-uv = - 9 * v ( ^ ^ l ) + ( | - * ' ) M v where a,b,f ,g=1 ,2 and M,V=0,1,2,3. 61 Now A and D" can be found from (A.3) and (A.6) r e s p e c t i v e l y . The r e s u l t i s ... W ^ - ^ M ^ b (A-,7) A'b ~ (K-keU^e) <A.lt, D - = Combining (A.13) through (A.18) The determinants may be evaluated to give ( A . 2 D M ( A - B D - E ) = [&-kci£-*ecV\\ (A.22) ae+( D - E A " B ) = - o r»(if - e i c ) \ A l - i c irK where (A.23) ft = (£_ ^ C " Xe 1^ 1) + lOCM? Using (A.19) through (A.23) one may obtain the inverses W and Z defined by (A.9) and (A.10). The r e s u l t s are as f o l l o w s 62 (A.24) W q t = (A.25) ^ = (~ 9 > a v A ^ ^ U K - c L ( A x - i C ^ ^ C x ) ^ A v S u b s t i t u t i n g the expressions f o r A,D,B,E,Z,W i n t o (A.11) and (A.12) gives -1 ( A - 2 6 ) X a v = ^ O L £ Q ^ C F A v K (A. 27) YuV, ^  " ; C € ^ ^ £ t ^ 9 K ' The inverse of M i s now given by (A.8) with W,X,Y,Z given by (A.24) through (A.27). Note that for oC=0 the r e s u l t s reduce to the simple form (A.29) ~2AVU--°) = ( - g « v + J U v A " 1 ) ^ ' e ^ ~ ' (A.30) X a v ( ^ = Y*k(*a0) = ° F i n a l l y the determinant of M may be c a l c u l a t e d . R e c a l l the f o l l o w i n g matrix algebra property {17}. (A.3,1 de-V h = M ( J l) = O e t - / f l M ( D - F / r , B ) 63 S u b s t i t u t i n g i n t o (A.31) from (A.2) and (A.22) gives (A.32) ae+fi = -^(^-eK^^-MK^-^C^K^- iC 1 ) (A.33) ue+M ~ - ^ ' ( ^ - ^ ( A ^ ^ C ^ K Now def i n e (A.34) r = \ 4 ( \-^ 4oteW-,y/,-(A.35) 5 = \ - ( l - m ^ A " ) ^ Using the d e f i n i t i o n of K i n (A.23), the determinant of M i s given by (A.36) d e i n = - o C ' ^ e ^ ^ ^ X ^ - A ^ ^ - ^ 5 ^ ) 64 APPENDIX B ~ SOME FORMULAE REQUIRED IN THE TEXT Below are the formulae f o r some dimen s i o n a l l y r e g u l a r i z e d i n t e g r a l s which use the Euclidean metric ( g A t v = S M V ) . They are taken from Ramond {16}. The methods f o r o b t a i n i n g these formulae are explained i n a paper by 't Hooft and Veltman {19}. (B.1) (B.2) P(/0 (nVT (B.3) +1 -fl A l s o taken from Ramond {16} i s the f o r m u l a where the cC:, a r e known as Feynman p a r a m e t e r s . 65 The expansion of the f - f u n c t i o n {16} about i t s poles at non-positive i n t e g e r s i s given by (B.5) r(-.m + £)= fclj U *iv(">+0+1 f^+vyWi).vj;WoJ where, using the Euler constant V, we de f i n e by (B.6) l|Xm+|) = -J- 1 * -I- • • • \ -k ) = 66 APPENDIX C ~ EVALUATION OF I(A,B,C) In t h i s appendix we evaluate the i n t e g r a l ( c . D i v w ) = \ & & ( ( mt+A)-'($>B)>*c) ,-» with p 2 , k 2, and (p+kj 1 defined using the Euclidean metric ( 9 M V = $ M V ) « Introducing Feynman parameters from (B.4) gives 0 6 6 Applying (B . 1 ) to the i n t e g r a t i o n over p (C.3) xf«C4. Applying (B . 1 ) to the i n t e g r a t i o n over k (C.4) I 67 S u b s t i t u t i n g n=4-2e Note that the integrand contains poles i n the of, ,0^,0/3 parameter space. The i n t e g r a l s can be evaluated by f o l l o w i n g the technique of E l i a s and Mann { 9 } . Consider the pole where *,=c(x = 0 and ^3 = 1 and d e f i n e , for small p o s i t i v e £ (c.6) = ) < k \ < k U 3 <bw.+^+4,-i) ,77777777^ Changing v a r i a b l e s «<(= toiv,</-i= C^, 0(^=1-^3 and using the i d e n t i t y S(ax)=a~'£(x) gives (C.7) P ^ U ) ^ c' ^ 0 0 0 where the oi^ have been r e w r i t t e n as <JL±. A f t e r a p p l y i n g binomial expansions to (C.7) and expanding ZL i n a Taylor s e r i e s , one has (c.8) p a y^ki^y*! ^(m^-^ C1'k(*xh^ 0 0 o 68 With the v a r i a b l e change oiv=Ox,<^x=^( 1-x) (C.8) becomes o O 0 An a l t e r n a t e form of the pole w i l l be needed. By making the s u b s t i t u t i o n ^3=1-0(3 (C.8) becomes (C.10) ?V1 ^ ° >^^P4^3 W ^ ^ O ^ ' t W ^ 0 o 0 The leading behaviour of the poles at <\ = «<3=0,<*x= 1 and <i^ = at3 = 0,ol| =1 can be obtained from (C.5) by a s i m i l a r d e r i v a t i o n . More simply, one can r e l a b e l the i n t e g r a t i o n v a r i a b l e s i n (C.6) and obtain the r e s u l t s below by comparison with (C.9) and (C.10). The two a l t e r n a t e forms f o r each pole are given by (cm ^ s ± o _ ^ £ - ' B l - a t (C.12) > ^ , ^ p w 3 ^u+^T1-O Q O (c.,3, ? ^ - ^ ^ r ' A ' ^ (c ,4) P^-**"-* W-uS*, $(«.«vM»-0 A'"" ( i x * - , ) ' 0 0 0 One can r e w r i t e (C.5) by adding the poles i n the form of (C.9),(C.11),(C.13) and s u b t r a c t i n g them i n the form of (C.10),(C.12),(C.14). 69 This r e s u l t s i n o o b (C.15) (/k^+G,) _ J B C 1 ^ + ^ + - ^ " (^ ,r ^ - K ^ " Changing v a r i a b l e s o f ^ ^ x ^ s ^ l - x ) and performing the ^ i n t e g r a t i o n gives T r a i n ) /\\-l£ 5t-:i£ ,t-l£ /' /' O o (C.16) where we define a=x(1-x)-1. Performing a Taylor expansion i n £ i n s i d e the i n t e g r a l r e s u l t s i n the expression o o (C.17) "AQX VB^d^Ua^ A _ _B c 70 where ° o (C. 18) -1JU [ A ? x + ( l - x) + C ( l - ^ - f\ (h^f[jU 0-?x) -2JM] With the a i d of i n t e g r a l t a b l e s {10} consider (C.19) (C.20) 'AxjV^HO^C"! Ax -c (1 + q) ° o ° = A+B - ( A + r t f Jf S u b s t i t u t i n g (C.19),(C.20),(C.21) i n t o (C.17) (C.22) I : 71 Now perform a Taylor expansion i n £ on a l l but r ( 2 t - l ) 1 = 0 M X (C.23) Using (B.5) to expand f ( 2 l - 1 ) f i n a l l y y i e l d s J (C.24) 7e x + E(M,0 4 0^) where (C.25) 4(A+&+c)j^H^ 4i^4/r(A+P+C-2AirxA-l&>n5,-2C^c) 72 (D APPENDIX D - EVALUATION OF J(A,B,C) In t h i s appendix we evaluate the i n t e g r a l where the Euclidean metric (g>»v = £,uv) has been used. Consider the i d e n t i t y (D.2) |- B~t-{fkX^(k\BX^c)-^^)'^(f^c]\ I n s e r t i n g (D.2) i n t o (D.l) one obtains (D.3) - ( ^ c ) f ^ k f i ft)( Jf+BX (?t c)) Vf*f Thus (D .4) T (^B ) C^6"C H [T l (^B ) c )4 -T ( ( f l Jo )ol - T ^ . o ^ - t ^ B ^ where 73 Introducing Feynman parameters w i t h (B.4). gives o o 6 J (/ ""1 V T7; (D.6) Applying (B.2) to the i n t e g r a t i o n over p -3 (D.7) X + A p p l y i n g (B.2) and (B.3) t o the i n t e g r a t i o n over k (D.8) S u b s t i t u t i n g (D.8) i n t o (D.4) -1 r 1 r> 0 0 0 (D.9) 74 P u t t i n g n=4-2£ gives d 0 o ( D . 1 0 ) ^jj^^-j- ^  ^ + T <*,J3 4«^_o<^] £'~^ X As i n Appendix C the i n t e g r a l s can be solved by f o l l o w i n g the technique of E l i a s and Mann {9}. Consider the pole where «=/, » °^=0, o/3= 1 and def i n e for small p o s i t i v e £ ,ex^+o(jo^ -t0(20(3) ( D . 11 ) ft Changing v a r i a b l e s <l=Z°<-\,<*[= ^ = 1 -£=<3 and using the i d e n t i t y S(ax)=a~' S(x) gives O o O (D.12) ' 75 where the o(i have been r e w r i t t e n a s ^ . Applying binomial expansions to (D.12) and expanding 5^ i n a Taylor s e r i e s , one obtains the f o l l o w i n g expression f o r the l e a d i n g term i n £. (D. 13) PQ > \ <U, ^ < k ^ 3 ^ ( a ^ - oij) O O 3-E With the change of v a r i a b l e s ^ f ^ X j O ^ ^ i - x ) equation (D.13) becomes An a l t e r n a t e form of P_ w i l l be needed. S e t t i n g ^dx=2^ dx(1-x) in (D.14), changing o^ '=i-°<3 and changing v a r i a b l e s back to e<t = ^ x ,°<2 1-x) gives (D.15) ^ - ^ ^ ^ [ k ^ ^ , ^ ^ - 0 ^ " 2 £ ) e C " ^ (c/ ( 4«X J One can r e l a b e l the i n t e g r a t i o n v a r i a b l e s i n (D.11) and obtain the l e a d i n g £->0 behaviour f o r the pole at = <=^= o,0^= 1 by comparison with (D.14) and (D.15). The two a l t e r n a t e forms are (D.16) P l s > ^ C S - ^ V C B ^ (D.17) fig ^ > 0 o o 2 - € 76 The l e a d i n g £-?>0 behaviour of the pole at «/t= -<3=0, *,= 1 f o l l o w s from (D.10) by a s i m i l a r d e r i v a t i o n , with the r e s u l t (D. 19) P 2 3 — U ^ ^ ^ - . d l ^ o ^ - l ) d o o One can r e w r i t e (D . 1 0 ) by adding the poles i n the form of (D.14),(D.16),(D.18) and s u b t r a c t i n g them i n the form of (D.1 5 ),(D.17),(D.19). This w i l l r e s u l t i n the f o l l o w i n g expression r qec CM 4"* (D.20) o o o 77 Changing v a r i a b l e s c^^x,^*^(1-x) and doing the i n t e g r a t i o n over dlt gives J - 2 £ — (D.21 ) where again a = x ( l - x ) - 1 . Performing a Taylor expansion i n £ i n s i d e the i n t e g r a l r e s u l t s i n 4 f q - T r ^ 2£ - 2 £ 'x(l-x)fl-f) L ? ( H O O ^ P J ' " L 0 < M > ) 3 J (D.22) 78 where <5 O -2 (D.23) v ( W ^ ^ s c * + 3 8 c \ k c - %Bc zi^(i-pfr*)) With the a i d of i n t e g r a l t a b l e s {10} consider the"" f o l l o w i n g ?c\i*$i P A 3 (D.25) r (D.26) WW ^BxH-Q:i-x))xf/-x)6-pj (D.27) 5^ 1 o o x£l-x)(Bx-fC(i-x))p 7x( |-x) J - i -2 . <k X 1 79 (D.28) -%c\^[fyr (0.29) ^ ( U ^ i-p S u b s t i t u t i n g (D.24) through (D.29) i n t o (D.22) (D.30) (4 -2e ) r (2e-3) 4 ( " h r r * 6E fT*-' Now r(2g.-3)=r(2e-1 )/(2£-3) (2e-2). We can thus perform a Taylor expansion i n C o n a l l but T(2e-1) and expand P(2£.-1) using (B.5) to o b t a i n (D.31) iti + ^ (te)-vqJLLa fa)-sBJL( 80 where r E ^ c V [ H W ^ C ) + I2AJU * / ) t l B i n ^ (D.32) s > 81 APPENDIX E - EVALUATION OF K(A,B,C) In t h i s appendix we evaluate the i n t e g r a l where the Euclidean metric (9m.v = ^,M),) has been used. Equation (E.I) can be w r i t t e n i n the form Consider the numerator of the integrand (E.3) \io = - U - p ) 2 + f ^ - P ^ ( f c p ) -2. One can expand and i n s e r t 0=(k 2-p 2)(C-B)+Bk 2-Bp 2-Ck 2+Cp 2 to obtain I n s e r t i n g 0=2(k 2+p 2)-2(k 2+p 2) and regrouping terms gives 82 One can combine appropriate c a n c e l l i n g f a c t o r s of A,B,C to w r i t e (B.5) i n the form k c = ( M % A V - A - * - < r « $ t ^ (E.7) - ^ ( J ^ B ) - * Y P V ) _ I f we add 0= (k+p)^ - (k+p?" to A i n the numerators, K 0 becomes S u b s t i t u t i n g (E.7) i n t o (E.2) - f l T Y c - s f f l ^ B j C ) - 1(^0,8,c)] (E.8) 83 where Wi t h the v a r i a b l e change It' =p+k ,p' = -k the e x p r e s s i o n s f o r , K 3 and become ( E - , 4 ) K J f t . e ) =\<,(A ,6) (E.16) J W J fay1 \ C , ( A B ^ - B K 1 ( f l 1 6 ) ^ j g ^ ( k V f t ) S u b s t i t u t i n g (B.14),(E.15),(E.16) i n t o (E.8) K(ft,B,c) = VC,(B,c^  -K,(fl,c) -K,(A,8) +(2B+2c-A)T(aB,c) (B . i 7 ) +A"(B-c)[K,(ac)-K,(o)c)-K,(fl/e) + K,(O,B\ 84 Applying (B.1) to (E.9) and then s e t t i n g n=4-2£. gives S u b s t i t u t i n g (E.18) i n t o (E.17) Expanding a l l butp(£_-1) i n a Taylor s e r i e s i n 81 y i e l d s -v Tfc-i) [itTr - r f (36C -Ac - A8 -6 l -C 1-} -v efeA-ic)6irv(a/^r) (E.20) ^ where lL (A&c5= EC ia^gc -Ac_^Ac -Afii^flB +$cJUx - c z A ^ c -BU^£ +gck?B - ( i - c ) ^ A (E.2D -^A(-6>cJUe t cUac + g\UB -BCJkfi) +-6> + P - X C ) X B -t C(C4A-zg) JUc] 85 One can expand w i t h (B.5) to obtain (E 22) ^ C ' j a ^ C s K - f t c ^ B - B ' - c 7 ) ' 4Bf64A-^in(e/^7r) + 0 + A B where (E.23) L and F-j-(A,B,C) i s given by (C.25). F i n a l l y one s u b s t i t u t e s (C.24) i n t o (E.22) with the r e s u l t (E.24) 

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