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Running of the Coupling in the Φ -Theory and in the Standard Model

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Running of the Coupling in the Φ -Theory and in the Standard Model Running of the coupling in the f4-theory and in the Standard Model Olli Koskivaara Research training thesis Supervisor: Kimmo Kainulainen October 9, 2014 Abstract In this research training thesis the running of the coupling constant as a phenomenon is studied within two different frameworks. First the equa- tion governing the running of the f4-theory coupling at one-loop order is derived. This is done in detail by first calculating the one-loop corrections to the theory and then renormalizing it. The renormalization procedure is treated more generally by e.g. deriving the Callan–Symanzik equation. Using the properties of the renormalization group and the one-loop cor- rections calculated earlier the running of the coupling is obtained. The coupling is found to increase with increasing energy scale. A short dis- cussion on beta-functions is also given. The second example of the running of the coupling is more of a phe- nomenological one. The three gauge couplings of the Standard Model and their scale dependence is studied. A plot of the inverses of the cou- plings as a function of energy is produced. It is noted that the couplings get near each other at an energy scale of approximately 1016 GeV but yet do not meet each other. A search for a theory predicting the unification of the gauge couplings is mentioned. Tiivistelmä Tässä erikoistyössä kytkentävakion jouksemista tutkitaan ilmiönä kahden eri esimerkin kautta. Ensin määritetään f4-teorian kytkinvakion juokse- misen 1-silmukkatasolla määräävä yhtälö. Tämä tehdään yksityiskohtai- sesti laskemalla ensin teorian ensimäisen kertaluvun korjaukset ja sitten renormalisoimalla se. Renormalisaatioprosessi käsitellään hiukan yleisem- mällä tasolla esimerkiksi johtamalla Callanin–Symanzikin yhtälö. Kyt- kentävakion juokseminen saadaan käyttämällä renormalisaatioryhmän o- minaisuuksia sekä aiemmin laskettuja silmukkakorjauksia. Kytkentäva- kion havaitaan voimistuvan energiaskaalan kasvaessa. Lopuksi tehdään lyhyt katsaus beetafunktioiden ominaisuuksiin. Toinen esimerkki on luonteeltaan fenomenologisempi. Standardimallin kolmea mittakenttäkytkentää ja niiden riippuvuutta energiaskaalasta tar- kastellaan tuottamalla kuva käänteisistä kytkentävakioista energian funk- tioina. Kytkentävakioiden havaitaan lähestyvän toisiaan energiaskaalan 1016 GeV lähettyvillä kohtaamatta kuitenkaan toisiaan. Lopuksi maini- taan kiinnostus mittakenttäkytkentöjen yhdistymisen ennustavaa teoriaa kohtaan. i Contents 1 Introduction 1 2 Running of the coupling in the f4-theory 2 2.1 Preliminaries . 2 2.2 One-loop corrections . 4 2.2.1 Propagator . 4 2.2.2 2 ! 2 scattering . 8 2.3 Renormalization . 10 2.4 The Callan–Symanzik equation and the renormalization group 12 2.5 Running of the coupling . 15 3 Beta-functions 18 4 Running couplings in the Standard Model 19 5 Conclusions 22 Appendix A Feynman rules 24 Appendix B Mathematical details 24 B.1 Beta function identity . 24 B.2 Feynman parametrization . 25 Appendix C MATLAB code 26 ii 1 Introduction This work concerns the phenomenon known as the running of the cou- pling constant. This running is a feature of many interacting theories and plays a great role in modern physics. To put it briefly, the phenomenon is about the energy scale dependence of the coupling constant describing the strength of the interaction of a theory. We start by introducing the f4-theory and some basic tools used in quan- tum field theory. We then proceed to calculate in detail the one-loop corrections to the theory, main goal being the equation governing the running of the coupling of the f4-theory at this level. In order to get to this result, we need to visit the realm of renormalization. Renormaliza- tion is a powerful tool used to extract meaningful and finite results from seemingly infinite outcomes. It deals with such fundamental questions as what is really meant by the very notions of electric charge, mass, etc. It also plays a major role in the theory of running couplings. Once we have arrived at our final result concerning the f4-theory, we say a few words about the role of beta-functions1, which we will meet during the f4-calculations, in physics. It turns out that once a beta-function of a given theory is known, the running of the coupling is completely dictated by it. After becoming familiar with the mathematics related to the running of the coupling with f4-theory, we turn to some phenomenology. The en- ergy scale dependence of the Standard Model gauge couplings is studied by producing a plot of their inverses as a function of energy. We finish our discussion by pondering on the possible implications of this plot. 1Not to be confused with the beta function familiar from mathematics, which we will actually also encounter in this work. 1 2 Running of the coupling in the f4-theory 2.1 Preliminaries Let us start by defining the theory we are dealing with. The f4-theory consists of a massive real scalar field f(x) with a quartic self-interaction. The Lagrangian of the theory is thus that of the Klein–Gordon theory supplemented with the interaction term, 1 2 1 l L = ¶ f − m2f2 − f4. (1) 2 m 2 4! This toy model is often used as an example in quantum field theoretic calculations since it is quite simple and exhibits many features present in more complicated theories. The Feynman rules for the theory are given in Appendix A. Before rushing into calculations some important concepts should be in- troduced. The information about scattering events is given by correlation functions also known as Green’s functions and n-point functions. The n- particle propagator is a time-ordered vacuum expectation value of the fields f(xi), (n) G ≡ hW jT ff(x1)f(x2) ··· f(xn)gj Wi . (2) The simplest nontrivial connected Feynman diagrams, from which all other diagrams can be constructed, are the so called one-particle irreducible (1PI) diagrams. These diagrams are determined by two requirements. They are amputated, which means that all external lines have been re- moved. Additionally, they are not allowed to have any cutlines, i.e., they cannot be split into two separate diagrams by cutting one of the internal lines. Since only the 1PI diagrams are significant for our calculations, it is use- ful to define for such diagrams the proper vertex function G(n), which is obtained from the n-particle momentum space correlation function by re- moving all the propagators on the external legs and omitting the delta function indicating the conservation of momentum. The relation be- tween the correlation function G(n) and the proper vertex function G(n) 2 can hence be written as " n # n ! (n) (2) D (D) (n) G (p1,..., pn) = ∏ G (pi) (2π) d ∑ pi G (p1,..., pn). (3) i=1 i=1 The fact that 1PI diagrams can be regarded as the basic building blocks of Feynman diagrams simplifies calculations tremendously. For instance in the f4-theory the exact propagator can be written diagrammatically as a geometric series of the form = + 1PI + 1PI 1PI + ··· , where 1PI ≡ + (4) + + ··· = G(2) consists of all 1PI diagrams. The exact four-point can be represented as a similar series where each term is constructed from the collection of 1PI diagrams G(4) = 1PI 0 1 (5) ≡ + @ + perm.A + ··· , where perm. includes the two other topologically nonequivalent one-loop diagrams which are obtained by permuting the incoming and outgoing momenta. 3 2.2 One-loop corrections In this section we will calculate the one-loop corrections to the propagator and to the four-point function. These results will be needed later when we start to renormalize our theory. 2.2.1 Propagator The one-loop correction to the propagator is given by the first 1PI diagram in equation (4), this is the so-called tadpole diagram. Using the Feynman rules listed in Appendix A we have2 il Z d4 p i = − , 2 (2π)4 p2 − m2 + i# 1 where the factor 2 in front of the integral is a symmetry factor originating from all possible contractions when forming the tadpole diagram. For example by introducing a cutoff one readily sees that the integral above is quadratically divergent. This is why we shall use dimensional regularization, i.e., we calculate the integral in a general dimension D. Our integral becomes il Z dD p i − m4−D , 2 (2π)D p2 − m2 + i# where m is an arbitrary parameter with dimensions of mass introduced to keep the dimensions of the integral unchanged. This step is quite crucial because this scale, entering our theory via the parameter m, will play a key role in the behaviour of the coupling constant. In order to calculate the regularized integral we shall use a trick known as the Wick rotation. We start by integration over p0. Due to the Feynman prescription i# the integrand has poles above and below the real axis, as shown in Figure 1. The Wick rotation corresponds to a change of variables D D 2 2 p0M −! ip0E ; d pM −! id pE, pM −! −pE, (6) where in the subscripts M stands for Minkowskian and E for Euclidean. 2The factors 2π appearing in integrals like this are mere conventions essentially aris- ing from Fourier transformations between momentum and position space integrals [1]. 4 Figure 1: The Wick rotation accounts for changing the integration along the real axis to integration along the imaginary axis. The figure shows the whole integration contour together with the two poles above and below the real axis. We will further need an important result from complex analysis called the Cauchy’s integral theorem, which essentially states the following [2]: If f (z) is an analytic function, and f 0(z) is continuous at each point within and on a closed contour C, then I f (z) dz = 0. C Since the contour in Figure 1 encloses no poles, the theorem above tells us that our integral over the contour vanishes.
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