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Renormalization of Factorially Divergent Series: Quantum Field Theory in Zero Dimensions

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Renormalization of Factorially Divergent Series: Quantum Field Theory in Zero Dimensions Master's Thesis Renormalization of Factorially Divergent Series: Quantum Field Theory in Zero Dimensions Supervisor: Bren Schaap Prof. dr. Ronald Kleiss Dedicated to my personal group Preliminary remarks This research is based almost entirely on information from the book Quantum Field Theory: A Diagrammatic Approach [1] by Ronald Kleiss, and is a continuation of the bachelor's theses [2, 3, 4] of Dirk van Buul, Ilija Milutin and Mila Keijer. The subject of this research was revitalized by Michael Borinsky through his work [5, 6, 7], and he was the one who pointed Ronald Kleiss in the right direction. I would therefore like to thank all of them for introducing me to this fascinating subject. Contents 1 Introduction to zero-dimensional QFT 6 1.1 Random numbers . .7 1.1.1 Green's functions . .7 1.1.2 The Schwinger-Dyson equations . .8 1.1.3 Connected Green's functions . .9 1.2 Feynman diagrams . 10 1.2.1 Diagrammatic equations . 12 2 Renormalization 14 2.1 Renormalizing '3 theory . 15 2.2 Factorially divergent series . 18 2.2.1 Retrieving αf and cf ....................... 20 2.3 The factorially divergent nature of Green's functions . 21 2.4 Asymptotic behaviour of renormalized '3 theory . 22 2.4.1 The improvement factor . 22 2.4.2 Asymptotic behaviour of the tadpole . 24 2.5 Freedom of choice . 26 2.6 Physical interpretation . 29 3 Real fields 30 3.1 '4 theory . 31 3.2 'Q theory for odd Q ........................... 33 3.3 'Q theory for even Q ........................... 35 3.4 Renormalizing '123 theory . 37 4 QED 39 4.1 Bald QED . 40 4.2 Counterterm QED . 43 4.2.1 Computing the counterterm . 43 4.2.2 Everything is in the counterterm . 44 4.2.3 The improvement factor of Counterterm QED . 45 4.3 Quenched QED . 47 4.3.1 The improvement factor of Quenched QED . 47 4.4 Furry QED . 49 4.4.1 The improvement factor of Furry QED . 49 4 5 Combination theories 51 3=4 5.1 'z theory . 52 5.1.1 Unphysical z and the saddlepoint approximation . 53 5.1.2 Physical z ............................. 54 5.1.3 The improvement factor . 55 5.1.4 Determining z ........................... 57 3=6 5.2 'z theory . 58 5.3 Pure '3=4 theory . 61 6 QCD 64 6.1 Asymptotics of H0;0 ............................ 65 6.1.1 Unphysical z > 0 ......................... 66 6.1.2 Physical z = −1.......................... 68 6.2 Renormalizing xQCD . 69 6.2.1 The improvement factor for xQCD . 70 7 Higgs 71 7.1 Asymptotics of H0;0 ............................ 72 7.2 Renormalizing Higgs theory . 73 Conclusion 75 Appendices 76 A Factorially divergent pullbacks . 77 B Computing the counterterm in CQED (alternative method) . 79 C Saddlepoint approximation in QCD . 81 Bibliography 83 5 Chapter 1 Introduction to zero-dimensional QFT \Look yonder," said my Guide, \[...] I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions." | Edwin A. Abbot, Flatland It might come as a surprise to us four-dimensional creatures that zero dimensions can be so interesting. Zero-dimensional spacetimes, also known as \points", contain neither space nor time, and yet hundreds of pages have been written about them. This thesis is an addition to the subcollection of writings about quantum field theory (QFT) in zero dimensions. The sole purpose of this first chapter is to lay the theoretical foundations that are necessary for this research. For those familiar with QFT in zero dimensions, it is merely a demonstration of notation. For others who are familiar with QFT in four dimensions, it is a good reminder of the fundamental basics of QFT. For everyone else, it is a minimal introduction to QFT in zero dimensions. In the first section we introduce the probabilistic interpretation of quantum field theory; this includes random numbers, probability density functions, and expecta- tion values. In the second section we introduce the famous Feynman diagrams, along with their associated Feynman rules. 6 1.1 Random numbers Zero-dimensional quantum field theory concerns itself with everything there is to know about all possible zero-dimensional spacetimes that are filled with random (complex) numbers. For any given spacetime, these numbers are generated by a set of stochastic variables f'ig, called “fields”, which we can group together into one vector ~'. These fields obey a (spacetime-specific1) probability density which we will write as 1 1 P(~') = exp − S(~') (1.1) N ~ where the action S(~') is a function of the fields and N is the normalization constant. Every theory is fully specified by its corresponding action. When we observe the values of the fields in one zero-dimensional spacetime, the resulting numbers look random and nothing can be said about the underlying prob- ability. This is in contrast to a spacetime where these numbers change over time and where we can probe the underlying probability density by repeated measure- ments.2 To emulate this behaviour in zero dimensions and get a good definition of probability, we can observe a collection of independent and identically distributed spacetimes. In other words: we cannot say anything about a single value of a field but we can say things about expectation values. 1.1.1 Green's functions Q ni We will look at the expectation values G~n = h i 'i i which are called the Green's functions. Trivially, G~0 = h1i = 1. To make life a little easier we define Z 1 Y ni ~ H~n ≡ ( 'i ) exp − S(~') d' (1.2) i ~ with the integral implicitly running from −∞ to +1 so that H~n G~n = (1.3) H~0 and H~0 is an explicit way of writing N. In general, the integrals H~n can not be solved analytically. In this research we solve these integrals perturbatively to get a series expansion in ~. To be able to say things about the collection of Green's functions we define the path integral 1For comparison: to our best understanding the four-dimensional spacetime we live in is gen- erated by the action of the Standard Model. 2This is completely analogous to how one would test if a die is fair or not. 7 1 Z 1 1 Z(J~) ≡ exp − S(~') + J~ · ~' d'~ N ~ ~ Z ki 1 Y X 1 Ji'i 1 = ( ) exp − S(~') d'~ N ki! ~ ~ i ki≥0 ki X Y 1 Ji = G~k (1.4) ki! ~ ~k i ~ where bookkeeping device Ji is called a source and the sum runs over all vectors k with integer entries ki ≥ 0. This is our first of many encounters where we use a P 1 k series expansion (in this case exp(x) = k≥0 k! x ) to simplify an integral. Every Green's function G~n can be retrieved from the path integral by applying the correct operator: " ni # Y @ ~ G~n = ~ Z(J) : @Ji i J~=~0 1.1.2 The Schwinger-Dyson equations∗ We will now deduce properties of the collection of Green's functions from the path integral. Let us define @ as the vector with ith component @ and consider the @J~ @Ji equality Z @S @ ~ 1 @S @ 1 1 ~ ~ ~ − Ji Z(J) = ~ − Ji exp − S(~') + J · ~' d' @'i @J~ N @'i @J~ ~ ~ Z 1 @S 1 1 ~ ~ = (~') − Ji exp − S(~') + J · ~' d' N @'i ~ ~ 1 Z @ 1 1 = −~ exp − S(~') + J~ · ~' d'~ N @'i ~ ~ = 0 where in the last step the integral over 'i vanishes because the probability density goes to zero at its end points. From this we find a Schwinger-Dyson equation (SDe) @S @ ~ ~ ~ Z(J) = JiZ(J) (1.5) @'i @J~ for every field in the theory. From the SDe's we can obtain a relationship between the individual Green's functions. For this we need to plug in the path integral as expansion of Green's functions and group together the terms with the same powers of Ji . This method is very powerful in cases where computing H~n is computationally heavy: instead of having to compute an integral for every Green's function you consider, it suffices to calculate a few integrals and use the (in general relatively simple) SDe's to compute the rest of the Green's functions. ∗This subsection is not essential to understanding the main line of reasoning in this thesis. 8 1.1.3 Connected Green's functions While the Green's functions fully describe P(~'), the average quantum field theorist will use connected Green's functions since these are directly related to quantities we can measure, such as cross sections and decay times. The connected Green's functions are defined by2 ki ~ X Y 1 Ji ln(Z(J)) = C~k (1.6) ki! ~ ~k i where this time normalization requires C~0 = 0. Every Green's function can be written in terms of connected Green's functions, and vice versa. For instance, for a theory with only one field, C1 = G1 = h'i corresponds to the mean, C2 = 2 2 2 G2 − G1= h' i − h'i corresponds to the variance, etc. The connected Green's functions are the main subject of study in this research. Just as the Schwinger-Dyson equations yield a set of equations for the Green's functions, it is also possible to obtain a set of equations for the connected Green's functions. We shall do this in section 1.2.1, but first we must climb onto the shoulders of Richard P. Feynman. 2Using the series expansion of ln(1 + x). 9 1.2 Feynman diagrams If we consider only real-valued fields3 we can write the most general expression of the action as X Y 1 ki S(~') = λ~k 'i ki! ~k i and distinguish the terms we encounter:4 • The constant term λ~0 does not contribute to the theory because it gets ab- sorbed by the normalization constant.
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