REGULARIZATIONS AND DIVERGENT DIAGRAM IN GAUGE THEORES Thesis Submitted To The University of Calicut in partial fulfilment of the requirements for the Degree of DOCTOR OF PHILOSOPHY IN PHYSlCS BY P. C. RAJE BHAGEERATHI DEPARTMENT OF PHYSICS UNIVERSITY OF CALICUT KERALA - 673 635 August - 1999 DECLARATION I hereby declare that this thesis entitled "Regularizations and Divergent diagrams in Gauge Theories" is a bonafide record of research work done by me and that no part of this thesis has been presented before for the award of any dcgrce or diploma. University of Calicut, 1% August 1999 .Kaje Bhageerathi I Department of Physics Grams: UNICAL n (0490) 400273,401 144 University of Calicut Fax- (0490) 400 269 Calicut University P. 0,Kerala, India. 673 635 Email: kn @ unical.ac.in Dr. Kuruvilla Eapen Reader CERTIFICATE Certified that this is the bonafidc work can-icd out by Ms.P.C.Rajc Bhageerathi under my supervision and guidance during 1994-1999 for the award of the degree of Doctor of Philosophy in Physics, of University of Calicut, Kerala and that no part of this work has been presented for any other examination or degree of this or any other University. / Dr.Kuruvilla Eapen Abstract The quantum theory of fields was invented many years ago to give expression to the relativistic quantum mechanics of the radiation field. Within one or two decades, the theory was extended to include the nuclear as well as the weak interactions. In the past, there were several occasions when the quantum theory of fields appeared to be inconsistent and were at the verge of abandonment. But with the advent of the renoimalization theory, this problem was more or less solved. Instead of completely eliminating the inconsistencies, one was able to reformulate the theory in such a way, that the embarrassing infinities did not appear explicitly. Several regularization techniques were devised earlier to eliminate the infinities appearing in the field theories, the most important ones being the Pauli-Villars and the dimensional regularizations. Recently, a new regularization technique was developed by Evens etal, which they have named nonlocal regularization. Chaptcr 1 describes the methods of dimensional and nonlocal regularizations as applied to QED. In chapter 2, the nonlocal regularization method is extended to evaluate the vertex part of QED. Chapter 3 gives the Ward identity satisfied by nonlocal QED. In chapter 4, the Yang-Mills theory is studied using the nonlocal regularization method. This meihod of regularization being new offers immense scope for development. Acknowledgements It is a great pleasure to express my profound gratitude to Dr.Kuruvilla Eapen,Department of Physics,University of Calicut for his encouragement and gu~dance throughout the course of this work .I am very thankful to him for the great patience with which he explained things to me and for providing all freedom one requires for research. I am very much grateful to Pr0f.K Neelakandan,Head of the Department of Physics, for providing me all the facilities and aIso to the personal interest he took forward in helping me in certain matters. I can never forget the timely advice rendered to me by Prof.R.P.IVoodasd, Department of Physics,University of Florida ,USA. I am thankful for the valuable suggestion put forward by him. My friends here have provided me with good company. Thanks a lot to every one of them, though it is not enough just to say passing thanks like this. It will be ungrateful on my part if I missed to mention Sashi- who got Ine all the publications as and when I required, without which this work would not have been what it is I express my deep sense of gratitude to my parents, sister and brother for all the moral support they provided me and also to one whose very thought brushed aside all the gloominess and elevated my moods. Finally I thank the Council of Scientific and Industrial Research. for providing financial assistance in the form of fellowship during the period in which this work was accomplished. Raje Bhageerathi Contents Page Chapter 1 Introduction 1 .l Primitive Divergences in Gauge Theories 1.2 The Technique of Dimensional Regularization 1.3 Ward Identity 1.4 Limitations of the technique of Dimensional ~e~ularization Nonlocal Regularization Nonlocal Quantum Electrodynamics The Electron Self - energy and Vacuum Polarization in Nonlocal Regularization Chapter 2 QED Vertex Part in Nonlocal Regularization 2.1 Introduction 2.2 Feynman Rules for Nonlocal QED 2.3 The QED Vertex Part in Nonlocal Regularization 2.4 Furry's Theorem 2.5 Discussion Chapter 3 Ward ldentity For Nonlocal QED 3.1 Introduction 3.2 The Ward Identity to order e2 3.3 Fourth - order electron self - energy 3.4 Fourth - order electron self - energy in nonlocal QED 3.5 Ward Identity to order e4 in nonlocal regularization 3.6 Discussion Chapter 4 Nonlocal Yang - Mills 4.1 Introduction 4.2 Nonlocalization of QCD 4.3 Feynman Rules for nonlocal QCD 4.4 Quark Self - energy in nonlocal regularization 4.5 QCD vertex part in nonlocal regularization 4.6 Discussion References Introduction P. C. Raje Bhageerathi “Regularizations and divergent diagram in gauge theories” Thesis. Department of Physics, University of Calicut, 1999 Chapter - 1 Introduction 1.l Primitive divergences in Gauge Theories ) It has been observed for a long time that there arise certain divergences while tackling many problems in field theories. Many methods were devised to isolate the divergences appearing in the field theories. This was necessary for the theories to be considered as physically acceptable ones. The divergences that are chiefly encountered while co~isideringany field theory - the Abelian, non-Abelian or 44 theory - are the ultraviolet and the infrared ones. Consider an integra! of the form g [l].There are four powers of q in the numerator and two in the denominator, so the integral diverges quadratically at large q. It is ultraviolet divergent. The problems of infrared divergence arise when q + 0. g is the order of the particular diagram represented by the above integral. A lot of effort has gone into the study of ultraviolet divergences in Abelian and non-Abelian gauge theories during the last four decades. To tackle divergent diagrams in these theories, ~t is necessary to regularize the divergences in suitable ways. Regularization is a method of isolating the divergences in Feynman integrals. It makes the task of renormalization much more explicit. There are several techniques of regularization. The most intuitive one is to introduce a cut-off A in the momentum integrals. In QED, it .% is required to modify the free photon propagator The Pauli-Villars regularization [2] is also similar to the one above. Here, a fictitious field of mass M is introduced. In both cases the limit A + CO (M + CO ) is taken. The renormalized quantities then become independent of A (M). These methods create problems while dealing with non- Abelian gauge theories. The Pauli-Villars regularization does not preserve gauge invariance in non-Abelian gauge theories. So inorder to tackle these problems, the technique of dimensional regularization [3] was developed mainly by 't Hooft and Veltman [4], (see Ashmore [5], C.G.Bollini and J.J. Giambiagi [6]). The idea is to treat the loop integrals which cause the divergences as integrals over d - dimensional momenta, and then take the limit d + 4. It turns out that the singularities of one - loop graphs are simple poles in d - 4. In the next section the method of dimensional regularization is given in the context of QED. l.2 The Technique of Dimensional Regularization In this section, one can see how the method of dimensional regularization is applied to a field theory, by considering QED as an example. QED is the theory of interaction of light and matter. It is also one of the rare parts of physics which is known for sure, a theory that has stood the test of time. It is an Abelian gauge theory which is renormalizable [7-1 l].It comes under the U(l) group - the unitary group of order one [l2, 131. The three primitively divergent diagrams in QED are the electron and photon self energy graphs and the vertex graph [l, 14, 151 shown in fig. 1.l. (a) Electron self-energy diagram (b) Photon self-energy diagram (c) Vertex graph. Fig.1.1. The general formula for the superficial degree of divergence D of a Feynman graph in d - dimensional space - time is [I], where n = number of vertices, pe = number of external photon lines, E, = number of external electron lines, d = dimension of space - time When d = 4 this yields 3Ee D -4--- - Pe 2 showing that D is independent of n. Consider the two self-energy diagrams in fig.1 .l-The electron self-energy diagram has Ee = 2, P, =O, so D = 1. The Feynman rules give Here there are four powers of k in the numerator and three in the denominator. So D = l,as predicted. The photon self - energy, which is also called vacuum polarization denoted as n, is This integral is quadratically divergent, as anticipated. Now for the vertex graph, we have E, =2, P, = 1. Hence D = 0 which indicates a logarithmic divergence. The Feynman rules give Although the electron and photon self energy graphs are superficially linearly and quadratically divergent respectively, they both turn out to be logarithmically divergent only. The two self energy graphs and the vertex graph, all have the property that the removal of their infinities results in a redefinition of various physical quantities, ie; electron mass and wave function normalization, and electric charge. ie; no extra terms in the Lagrangian are required of a type which are not there already.