Renormalization in Field Theories

UPPSALA UNIVERSITY PROJECT 1 HP Renormalization in Field Theories Author: Alexander Söderberg Supervisors: Lisa Freyhult March 3, 15 Abstract: Several different approaches to renormalization are studied. The Callan-Symanzik equation is derived and we study its β-functions. An effective potential for the Coleman-Weinberg model is studied to find that the β- function is positive and that spontaneous symmetry breaking will occur if we expand around the classical field. Lastly we renormalize a non-abelian gauge theory to find that the β-function in QCD is negative. MASTER PROGRAM IN PHYSICS DEPARTMENT OF PHYSICS AND ASTRONOMY DIVISION OF THEORETICAL PHYSICS

ACKNOWLEDGEMENT I would like to give many thanks to Lisa Freyhult for taking her time to be my supervisor and for the many interesting and rewarding discussions. Moreover, would I like to thank the makers of the website http://fooplot.com/ for its user friendly interface. For creating motivating art to relax my mind on would I like to thank Shigesato Itoi. I

CONTENTS 1. Introduction 1. Perturbative Renormalization 3. Wilsonian Approach 6 4. Callan-Symanzik Equation 1 5. The Coleman-Weinberg Potential 17 6. The β-function in Non-Abelian Gauge Theories 7. Conclusion 3 A. Calculations 35 II

1. INTRODUCTION This project is made to gain more knowledge about quantum field theory. A good way to get a deeper understanding about quantum field theory is to study different approaches to renormalization, since it plays an important role in quantum field theory. Without renormalization, many scattering processes in quantum field theories will diverge when we use perturbation theory, i.e. when we study one order of a series expansion in the coupling constant at a time. Since quantum field theory is a widely used theory, e.g. in particle physics, is it very important to understand how we can avoid these divergent scattering processes. This leads to renormalization being a very important subject to study. The whole process when we convert a divergent process to a convergent process is called renormalization. Some Feynman diagrams may still diverge after a renormalization, but the scattering amplitude for the whole process will not. When we renormalize, we rewrite our theory, i.e. the Lagrangian or the action of the system, in terms of new quantities. We can use different approaches when we renormalize a theory, but the final result should always be the same for physical quantities. Renormalization is not only used in quantum field theory though. We may renormalize any theory where we need to introduce some kind of length or energy scale. Such a theory may appear in statistical physics or in a theory of self similarity in geometrical physics, i.e. a theory of fractals. Since we may be able to renormalize theories that introduce some sort of scale, renormalization is very important to study in e.g. any perturbative quantum field theory. We often use a mathematical tool called regularization when we perform a renormalization. Regularization helps us "control" the infinities which may appear in e.g. a Feynman diagram. There exists different kinds of regularizations and some prove to be more convenient than others for a theory. In section -5 we will study different approaches to renormalization. We will perform a different kind of approach to renormalization in each of these sections. In section -4 we renormalize φ 4 -theory and in section 5 we renormalize the Coleman-Weinberg model 1. φ 4 -theory and the Coleman-Weinberg model are both abelian gauge theories. In section 6 we will renormalize a non-abelian gauge theory, and study the result. We will use the books [1],[],[5] and [7] as well as the three articles [3],[4] and [6] as guidelines throughout this whole project. Since the article [3] is written by the original authors of the Wilsonian approach to renormalization, will we follow this paper in section 3. The article [4] is made by the original authors of the Coleman-Weinberg potential, so this paper will be used as a guideline in section 5. 1 Scalar quantum electrodynamics QED where the scalar mass, m, is considered small. 1

. PERTURBATIVE RENORMALIZATION In this section we will perform perturbative renormalization. That is, we will look at order ν of the couplings λ j and make the amplitude at this order finite using dimensional regularization. Dimensional regularization is when we look at d dimensions and then let d go to four after we have calculated the integrals which appear when we calculate the scattering amplitude. The approach of perturbative renormalization using dimensional regularization for a theory is briefly explained below. For a theory with n fields, φ 1,...,φ n, we rescale the fields in such a way that the divergent residue Z j in the two point Green s functions for the field φ j is removed. In [1] we see that this is achieved by rescaling the fields as φ j Z j φ j,no sum over j..1 We insert the rescaled fields.1 into our Lagrangian, and then define couplings between our old and some new parameters in order to split the Lagrangian into two parts. One part, L will look like our old Lagrangian, but contain the new parameters instead of the old parameters. The other part, the counter-term Lagrangian, L C T, will contain couplings between the new parameters and the old parameters. The counter-term Lagrangian will contain the diverging residues. Changing the Lagrangian in this way will yield new Feynman rules. Most of them, however, will look like the old Feynman rules, but will contain the new parameters instead of the old parameters. We will then be able to find renormalization conditions by defining e.g. coupling constants. Finally we will be able to calculate the scattering amplitude which before the rescaling.1 diverged using perturbation theory, and see that it now converges. Moreover will we find the couplings between the new and old parameters in this way. We will study the simplest case, namely φ 4 -theory. However, the approach of perturbative renormalization and dimensional regularization is generally the same for any theory. So we want to rescale the two point Green s function in φ 4 -theory as Ω T { φx 1 φx } i Z Ω p +...f i ni te... i + m p + m +...f i ni te..., at p m.. Here Ω is an arbitrary energy state, T is the time-ordering operator, φ is a massive field with mass m and Z is a diverging residue. As we can see, the Green s functions. have a pole at p m. In order to rescale the Green s function as. we will rescale our field, φ, in the following way φ Z 1/ φ.3 In this project we will use the Minkowski space-time metric on the form g µν 1,1,1,1, and units such that c 1.

L 1 µ φ m φ λ 4! φ4 1 Z µ φ m Z φ λ 4! Z φ 4 1 µ φ m φ λ 4! φ4 1 }{{} δ Z µ φ 1 δ mφ 1 4! δ λφ 4..4 L +L C T }{{} L L C T Here we have defined the coupling constants, the δ s, which couples the new parameters m and λ to the old parameters m and λ as δ Z Z 1 δ m Z m m δ λ Z λ λ..5 In [1] we see that the new Lagrangian.4 yields the following Feynman rules Figure.1: Feynman rules in φ 4 -theory. Note 1. Two of the Feynman rules in figure.1 is similar to the initial Feynman rules. The only difference is that they contain the new parameters m and λ instead of the initial parameters m and λ. We will use the same renormalization conditions as [1], i.e. Figure.: Renormalization conditions in φ 4 -theory. 3

Here s, t and u are the Mandelstam variables defined as s : p 1 + p t : p 1 p 3..6 u : p 1 p 4 The first renormalization condition in figure. states that the amplitude for the one particle irreducible 1PI diagram is zero if the incoming field has zero spatial momentum 3, i.e. p m. The second of the conditions in figure. define the coupling constant λ. In order to find the δ s will we use the new Feynman rules figure.1 and the renormalization conditions figure.. Let us start by finding δ λ. This is done by using the second renormalization condition in figure., i.e. by studying the following amplitude Figure.3: Scattering amplitude for four φ fields. The one-loop diagram with ingoing momenta p is given by iv p iλ lim ɛ S d 4 k i i π 4 k + m iɛ k + p + m iɛ..7 In this case, the symmetry factor, S, equals two. The amplitude, G, for φφ φφ scattering is given by ig iλ + i V s +V t +V u iδ λ + O λ 3..8 From the second renormalization condition in figure. we get iλ iλ + i V 4m +V +V iδ λ + O λ 3 δ λ V 4m + V + O λ 3..9 3 An 1PI diagram is every Feynman diagram where you cannot cut a line to get two Feynman diagrams. 4

When we calculate V p we make use of the following d d l 1 π d l + D n 1n i Γn d/ 4π d/ D n+d/ [n ] Γn +d/ Γ d/ i 1 4π d/ D +d/ i d 4 4π lim logd γ + log4π..1 d 4 d 4 [ɛ d 4] i 4π lim logd γ + log4π ɛ ɛ Introducing a Feynman parameter, x, performing a dimensional regularization and using.1 yields 4 V p λ 1 4π d x lim ɛ ɛ logdx, p γ + log4π..11 Here γ is the Euler-Mascheroni constant γ.577, and Dx, p is given by Dx, p x 1 x p + m..1 If we insert.11 into.9 we get δ λ λ 1 6 4π d x lim ɛ ɛ log Dx,4m Dx, 3γ + 3log4π + O λ 3 λ 1 6 4π d x lim ɛ ɛ log{ x1 x4m + m m 4} 3γ + 3log4π + O λ 3..13 λ 1 6 4π d x lim ɛ ɛ log{ 1 + 4x 4x m 6} 3γ + 3log4π + O λ 3 Inserting δ λ into.8 yields ig iλ + iλ 1 6 4π d x lim logdx, sdx, tdx,u 3γ + 3log4π ɛ ɛ 6 ɛ + log Dx,4m Dx, + 3γ 3log4π + O λ 3..14 iλ + iλ 4π 1 d x log Which is finite for massive fields. Dx,4m Dx, Dx, sdx, tdx, u + O λ 3 Now we want to find to find δ Z and δ m. To do this we will use the first renormalization condition in figure.. We have 4 Integration is in appendix A.1. Figure.4: One particle irreducible 1PI diagram for φ 4 -theory. 5

Which gives us d i Mp 4 k i iλlim ɛ π 4 k + m i e + i p δ Z + δ m + O λ 3 M m d dp Mp δ Z + O λ 3 δ Z O λ 3 p m..15 d M m 4 k i λ lim ɛ π 4 k + m i e + δ m + O λ 3 d 4 k i δ m λ lim ɛ π 4 k + m i e + O λ3 Mp O λ 3 So we need to go to higher orders of λ in order to find the M and δ Z. However, since it is done in a similar way we will not do it here. In appendix A. we see that we can write δ m as δ m λm 4π lim ɛ ɛ log m γ + log4π + O λ 3..16 If we insert the result of the δ couplings.13,.15 and.16 into the Lagrangian.4 we find the renormalized Lagrangian L 1 µ φ m φ λ 4! φ4 + 1 λm 4π lim ɛ ɛ log m γ + log4π φ 1 λ 1 6 4! 4π d x lim ɛ ɛ log Dx,4m Dx, 3γ + 3log4π φ 4 + O λ 3 1 µ φ 1 λ 4π lim ɛ ɛ log m m..17 γ + log4π φ 1 + λ 1 6 4π d x lim ɛ ɛ log{ 1 + 4x 4x m 6} λ 3γ + 3log4π 4! φ4 + + O λ 3 As we may see, the kinetic term is unchanged under renormalization at least to this order, and is therefore a fixed point to the renormalization group flow which we discuss in section 3 and 4. A fixed point is a point where one or several terms of the initial Lagrangian is the same before as well as after renormalization. If we consider O λ 3, i.e. λ << 1, then Z 1 + δ Z 1 making φ the same before as well as after the renormalization see.3. This leads to the kinetic term being left unchanged under renormalization. As with physical quantities, are fixed points the same for every approach to renormalization. 3. WILSONIAN APPROACH In this section we introduce a momentum cut-off, i.e. a maximum momentum, for which the fields equals zero if they have higher momentum than this cut-off. This will cancel out the infinities in diverging diagrams. The physical interpretation of this regularization method is that we are either looking at large, but not infinite momenta, or that we are looking in a 6

neighbourhood of momenta. The momentum cut-off is our regularization since it cancels the infinities in the integrals for the diverging scattering amplitudes. We will then be able to find an effective Lagrangian by integrating out fields with momentum close to the momentum cut-off. A group of transformations which together make up the renormalization group will then be performed in order to make this effective Lagrangian to be on similar form as the original Lagrangian. The transformations in the renormalization group will renormalize the theory. We can then study parameters, such as mass and coupling constants, and see whether they grow or decay after the transformations of the renormalization group. These studies are using perturbation theory, and hence are they valid only if the coupling constant is small. Since there may exist several such fixed points the renormalization group will describe a flow of Lagrangians. This flow of Lagrangians often called the renormalization group flow will be more apparent in the next section. We will consider φ 4 -theory. If we would consider a theory with several different fields, e.g. quantum electrodynamics QED, we would have to let all our fields be zero if they have momentum higher than the momentum cut-off. So we introduce a momentum cut-off, Λ, as follows Figure 3.1: Momentum cuf-off. We will define our fields, φ, in such a way that they equal zero if p Λ, and ˆφ we define as { φ, if p [bλ,λ. ˆφ, else. 3.1 The Lagrangian in φ 4 -theory is given by.4. Inserting φ + ˆφ into.4 yields L 1 µ φ + ˆφ m φ + ˆφ λ φ + ˆφ 4 4! 1 µ φ + µ ˆφ m φ + ˆφ λ φ 4 + 4φ 3 ˆφ + 6φ ˆφ + 4φ ˆφ 3 + ˆφ 4 4! 1 µ φ m φ λ 4! φ4 1 µ ˆφ m }{{} ˆφ λ 4φ 3 ˆφ + 6φ ˆφ + 4φ ˆφ 3 + ˆφ 4. 3. } 4! {{} L L ˆφ L +L ˆφ 7

Terms linear in ˆφ vanish due to the equations of motion, see []. Our generating functional, Z, is given by { } { [Dφ Z ]Λ exp d 4 xl DφD ˆφexp d 4 x L +L ˆφ }. 3.3 [ ] Dφ Λ means that we are integrating over momenta, p Λ. We can integrate out the ˆφ part 5 in this path integral to obtain an effective Lagrangian, L eff { } [Dφ Z ]bλ exp d 4 xl eff 3.4 L eff A µ 1 µφ + B 1 φ + 1 1 + Z µ φ 1 + m + m φ + A µ 3 µ φ 3 + B3 φ 3 + + A 4 µ φ 4 1 + 4! λ + λφ4 + O µ φ 5,φ 5. 3.5 We will now perform a set of transformations which together makes up the renormalization group in order to make this effective Lagrangian look like the original Lagrangian L. Let us first rescale our momenta and position as { x } bx p b 1 p [Dφp Z ] { } 3.6 Λ exp d d x b d L eff b µ φ,φ L eff b d L eff b d A µ 1 b µ φ + B 1φ + 1 1 + Z b µ φ 1 + m + m φ +. 3.7 + A µ 3 4 3 b3 µ φ + B3 φ 3 + A 4 b 4 µ φ 1 + λ + λφ4 + O µ 4! φ5,φ 5 Note. Since the momentum, p, in quantum mechanics may be Fourier transformed as p i, the derivative is also being rescaled in the transformations 3.6 as b 1. We rescale the fields so the kinetic term of L eff is similar to the kinetic term of L φ 1 + Z b d φ φ 1 + Z 1/ b d / φ 3.8 L eff 1 + Z 1/ b d/ A µ 1 µ φ + b 1 B 1 φ + 1 µ φ + + 1 m + m 1 + Z 1 b φ + + 1 + Z 3/ b A d/ µ 3 µ φ 3 + b 3 B 3 φ 3 +. 3.9 + 1 + Z b d A 4 µ φ 4 1 + 4! λ + λ1 + Z b d 4 φ 4 + + O µ φ 5,φ 5 5 In [3] we gain the knowledge that this integration is done by performing the Gaussian integrals. 8

We notice that a general term, C N,M, in L may be written as eff C N,M C 1 + Z M/ b N+Md / d N φ M. 3.1 We rescale an arbitrary term in the following way C N,M C 1 + Z M/ b N+Md / d m m + m 1 + Z 1 b λ λ + λ1 + Z b d 4 A j 1 + Z j / b d+j d/ A j, j 1,3,4,5,... B j 1 + Z j / b d+j d/ j B j, j 1,3,5,6,... 3.11 L eff A µ 1 µ φ + B 1 φ + 1 µ φ 1 + m φ + A µ 3 µ φ 3 + B 3 φ 3 + A 4 µ φ 4 + + 1 4! λ φ 4 + O µ φ 5,φ 5. 3.1 We call the transformations 3.6,3.8 and 3.11 the renormalization group in φ 4 -theory. Since we cannot find an inversion to the integration over ˆφ, the renormalization group is a semigroup mathematically speaking. These transformations leaves the kinetic term of L unchanged, i.e. µ φ x b d b 1 µ b d/ φx µ φx. 3.13 Note 3. There is an factor b d from the measure of the four-vector, see 3.6. The kinetic term µ φ / is therefore a fixed point of the renormalization group. As we saw in the end of section, the kinetic term is unchanged in perturbative renormalization too. The effective Lagrangian equals this fixed point when we look at the free field effective Lagrangian, i.e. when the potential is zero, and all perturbations vanish, i.e. m λ A j B j j. In a vicinity of this point, the perturbations will still be zero, and hence we will have the transformations m m b λ λb d 4. 3.14 A j B j j Since b < 1 we see that m will grow, i.e. m > m, through the transformations 3.11. If we consider 4 dimensions we cannot tell whether λ will grow or decay through the transformations 3.11. If we want to know whether λ grow or decay do we need to go to higher order of λ. However, in [] we see that λ will decay, i.e. λ < λ, if we consider higher orders of λ. As mentioned in the introduction, the choice of approach to renormalization does not affect the physical quantities and hence we know from 3.14 that the mass will grow and that the coupling constant will decay when we renormalize φ 4 -theory at least to this order in perturbation theory. 9

4. CALLAN-SYMANZIK EQUATION Through this approach to renormalization, we derive a partial differential equation for the Green s functions to satisfy. This partial differential equation is called the Callan-Symanzik equation. The Callan-Symanzik equation contains the β-function, which by definition describe the running of the coupling constants, i.e. whether the coupling constant will grow or decay through the renormalization. Moreover, the Callan-Symanzik equation will tell us something about the physical interpretation of the Green s functions. The transformations in the renormalization group, together with the renormalization conditions we obtain for massless fields will be our starting point when we are about to find the Callan-Symanzik equation. We will apply the Callan-Symanzik equation to φ 4 -theory, solve it and study the result. For reasons soon to be discussed, let us start by studying the renormalization conditions which will arise for massless scalars, φ. To set these conditions do we need to study the results that we got in section. For massless fields, the δ m coupling see.16 will vanish as well as the mass term for the fields φ. Since we only studied the couplings up to order λ in section, and since the δ Z coupling see.15 is zero at order λ cannot we say anything about its behaviour when we let the mass go to zero. So the trouble is the δ λ coupling. If we study.13, we see that δ λ will diverge if we let m go to zero. This can be avoided if we introduce an arbitrary momentum scale, M, called the renormalization scale or the mass scale in our renormalization conditions as follows Figure 4.1: Renormalization conditions for φ 4 -theory containing a renormalization scale. Note 4. These renormalization conditions work as a momentum cut-off, similar to the one in the previous section see figure 3.1. Which means that the momentum cannot be larger than the renormalization scale, i.e. p M. In figure 4.1, s, t and u are the Mandelstam variables defined at.6. In a similar way as we did when we performed perturbative renormalization see.9 we find { δλ 3V M + O λ 3 δ Z O λ 3. 4.1 1

Here V p is given by 6 λ V p 4π δ λ 3λ 4π 1 1 ig iλ + iλ 4π iλ + iλ 4π d x lim ɛ ɛ log[ x 1 x p ] γ + log4π d x lim ɛ ɛ log[ x 1 x M ] γ + log4π + O λ 3 1 x 3 1 x 3 M 6. 4. d x log x 3 1 x 3 + O λ 3 stu 1 d x log M 6 + O λ 3 stu Note 5. The minus sign in the logarithm will cancel when we plug in the definition of the Mandelstam variables, see.6. In 4., G is the amplitude for φφ φφ scattering compare with.14. We see that G is finite, meaning that the renormalization conditions at figure 4.1 is valid. These conditions are valid for massive fields too, and hence we will use similar conditions, i.e. conditions which introduce a renormalization scale, M, when we derive the Callan-Symanzik equation. So let us now consider n massive fields, φ j, j {1,...,n}, and renormalization conditions which introduce a renormalization scale, M. In a similar way as in section, we rescale the fields as.1 so the divergent residue Z j is removed from the Green s functions. The Green s function for α φ 1 -fields,β φ -fields,...,ν φ n -fields after we have rescaled each field can then be written as G α,...,ν Z α/ 1...Z ν/ n Ω T { φ 1 x 1...φ 1 x α...φ n y 1...φ n y ν } Ω 4.3 In [] we see that if we assume that G α,...,ν depends on M and the coupling constants, λ j, between the fields, and then perform a small shift in M, each λ and Z will transform as M M + δm λ j λ j + δλ j Z k Z k + δz k φ k 1 + δη k φk,no sum over k. G α,...,ν 1 + δη 1 α... 1 + δηn ν G α,...,ν 1 + α j δη j G α,...,ν + O δη Where α j is the number of φ j in G α,...,ν. The variation of G α,...,ν may be written as 4.4 α j δη j G α,...,ν δg α,...,ν Gα,...,ν Gα,...,ν δm + δλ j M λ j M M + M δm δλ M j α j λ j δm δη j G α,...,ν 4.5 6 See.11 and.1, and set m. 11

Now let us define the so called β- and γ-functions as β j λ : M δλ j δm γ j λ : M δη j δm. 4.6 Here we see that the coupling λ and the residue η has a dependence on the renormalization scale, M, and therefore we will have a flow of Lagrangians. Inserting this β j and γ j in 4.5 yields M M + β j + α j γ j G α,...,ν 4.7 λ j This partial differential equation is known as the Callan-Symanzik equation. Let us use the Callan-Symanzik equation in order to find expressions for the β- and γ-functions. Starting by finding an expression for the γ-function. According to [] we may write the two-point Green s function for the field φ j as iω γ pg p iω γp p 1 +V p + i iχγ pδ Z χ γ p j + O λ 3 j G p 1 p 1 +V p δ Z j + O λ 3 j. 4.8 Here p is the ingoing momentum, 1 is from the diagram without a loop, V p is the sum of all one-loop diagrams and δ Z j is the vertex counterterm for field φ j one incoming and one outgoing. The terms ω γ p and χ γ p will vary for different theories. In φ 4 -theory ω γ p 1 and χ γ p p. The dependence of M is solely in δ Z j. Solving the Callan Symanzik equation 4.7 for γ yields γ j G M M + β j G + O λ 3 j λ j 1 p γ j + O λ j 1 1 p M δ Z M j + O λ 3 j + β j O λ j + O λ 3 j. 4.9 γ j M δ Z j M + O λ j Let us now find the β-function in a similar way. According to [] may we write the n-point Green s function for α j φ j -fields as n iω β G α,...,ν p i iω β λ j +V p i iχ β ɛδ λj + k1 χ β G α,...,ν p + i k i p k p k n W l p λ j α l δ Zl + O λ 3 j l1 λ j +V p + ɛδ λj + i l W l p λ j α l δ Zl + O λ 3 j. 4.1 1

Here p k are the momenta of the external legs, λ j is the tree-level diagram, V p is the sum of all 1PI-diagrams, δ λj is the vertex counterterm and Σ l W l p + λ j α l δ Zl are external leg corrections. The terms ω β, χ β and ɛ will vary for different theories. In φ 4 -theory ω β 1, χ β 1 and ɛ 1. The dependence of M exist only in the vertex counterterms. Solving the Callan-Symanzik equation for β yields β j k β j i p k G α,...,ν M λ j M + α l γ l 1 + O λj [ 4.9 i M k p k G α,...,ν M M δ Z j + α l M + O λ j β j M λ j α l δ Zl ɛδ λj λ j α l M β j M M ɛδ λj λ j α l δ Zl + O λ 3 j + ] λ j + O λ j + O λ 3 j + O λ j δ Z l λj α l δ Z l ɛδ λj + O λ j So to lowest order of λ j, we find the γ- and β-functions through β j λ M λj α k M δ Z k ɛδ λj γ j λ M. 4.11 δ Z j M Let us apply the Callan-Symanzik equation to φ 4 -theory, and find β as well as γ, so we can study the solution of the Callan-Symanzik equation. In φ 4 -theory the Callan-Symanzik equation will look like M M + βλ λ + nγλ G n. 4.1 By using 4.11 and the renormalization conditions in figure 4.1, we find β and γ from 4.1 and the results we got in section, see.13 and.15. We find δ λ 3λ 4π δ Z O λ 3 1 β 4.11 M δ λ M + O λ3 3λ 4π 3λ 4π γ 4.11 O λ 3 1 d x lim ɛ ɛ log[ x 1 x M + m ] γ + log4π + O λ 3 1 + O λ 3. 4.13 x1 xm d xm x1 xm + m [ 1 m ] d x 1 + m /x1 xm + O λ3 M << 1 3λ 4π + O. 4.14 λ3 13

Note 6. Since M is arbitrary did we chose M in such a way so we could perform the calculations. If we assume that λ λm and solve the expression of β for λ we get M dλ d M 3λ 4π dλ λ 3 d M 4π M 1 λ 3 logm + ξ 4π 4π λm 3 logm + ξ [ ξ ξ ] 4π 3 ξ logm. 4.15 Here ξ is a constant of integration. Since the coupling constant cannot be negative must ξ logm >, i.e. logm < ξ. From 4.15 we the see that λ will grow when logm ξ, i.e. λ will grow as M grow, and decrease when M is considered small. Note 7. If the β-function for a theory is negative, the coupling constant, λ, will be given by λm β Aλ, A > 1 A logm ξ, ξ > logm. 4.16 As we can see, λ will decrease when M is considered large, and grow as logm ξ, i.e. λ grows as M decreases. 4.14 gives us the Callan-Symanzik equation for φ 4 -theory M M + 3λ 4π G n O λ 3 4.17 λ Let us solve this partial differential equation and study the result. If we let all incoming and outgoing fields have momenta p, then we can write the Green s function as i n G n p, M ig n p p /M i n+1 p G n n p /M. 4.18 Here G is the scattering amplitude for the process. We can exchange M / M to p / p in 4.17 in the following way M Gn M i n+1 p n M p d M 3 d x G n x i n 1 d xp /M M p d x G n x 4.19 xp /M p Gn p i n+1 p n G n p n+1 p /M + i n+1 p d p n M d x G n x xp /M n i n+1 p G n n p /M i n 1 d M p d x G n x xp /M 4.19 n + M G n M 4. 14

Inserting this result into 4.17 yields p p M Gn M n + p G n. 4.1 p n + 3λ 4π p p 3λ 4π λ λ + n G n O λ 3 G n. 4. In [] we see that we can compare this equation to the partial differential equation describing the density, D, of bacteria in a pipe t + vx x ρx Dt, x. 4.3 Here t is the time, x is the position, vx is the flow velocity and ρx is the rate of growth. This comparison can be made if we perform the following replacements logp/m t λ x βλ vx n ρx G n p,λ Dt, x Proof to why we can replace logp/m with t is below p [ p log p Gn M t p Me t. 4.4 ] Me t 1 Me t Gn Me t t Met t Gn t Gn. 4.5 The analogy 4.3 and 4.4 tell us that the β-function describes the flow of Lagrangians which depends on the coupling constant, n for any order of λ, then γ n describes the rate of growth of Lagrangians and that the Green s functions, G n, describes the density of Lagrangians as a function of coupling constants. In [] we see that 4.3 has the following solution { t } Dt, x ˆD xt, xexp d t d ρ xt, x, xt, x v x, x, x x. 4.6 d t Here ˆD is the initial density of bacteria, meaning that once we know the initial density, we can find the density of bacteria at any given time and distance. Applying this solution to 4.17 gives us { G n p,λ Ĝ n λp,λexp n p p p M } d logp /M Ĝ n λp,λexp { n logp/m } M n Ĝ n λp,λ p 4.7 15

d λp,λ β λ, λm,λ λ. 4.8 d logp/m The partial differential equation 4.8 describes the flow of a modified coupling constant λ, often called the running coupling constant, as a function of momentum. The running coupling constant tell us about the behaviour of the coupling constant as a function of the renormalization scale. We can find Ĝ n by comparing 4.7 with 4.18 M n Ĝ n λ i n+1 p p G n n λ Ĝ n λ i n+1 M G n n λ M n G n i n+1 p M G n n λ i n+1. 4.9 p G n n λ In the case when n 4 we have G 4 i 5 p 8 λ + O λ i λ p 8 + O λ. 4.3 Let us study the solutions to the differential equation 4.8, when we assume that the β- function to the lowest order is given by β ±Aλ α, A >,α > 1 d λ d log α ±A λ p/m λ λ λ τ p d λ λ α ±A p M d log p/m λ α+1 λ α+1 ±A log p/m α + 1. 4.31 1 λ A α 1log p/m + 1 α 1 λ α 1 τ α 1, τ > λ τ 1 Aτλ τ log p/m As we may notice is λ of similar form as 4.15. We see that if β A λ τ λ τ λ. 4.3 λ In this trivial case, the coupling constant is finite before renormalization as well. If β is positive, λ will decay for small momenta and grow for large momenta, and vice versa if β is negative. Since the renormalized coupling decreases as momentum grows for a negative β- function, is a theory with a negative β-function called asymptotically free. The behaviour of λ follows from m p log < lim log. 4.33 M p M M 16

Here p m is the minimum value of the momenta rest frame, and p M is the maximum value of the momenta. Important to remember, for a positive β-function is λ always smaller than λ, and vice versa for a negative β-function, since p p M < 1 log <. 4.34 M This means that the renormalized coupling will be smaller than the bare coupling in φ 4 - theory, which coincides with the result in the previous section. We see from 4.31 that we can write the bare coupling constant, i.e. the un-renormalized coupling constant, λ, as As we can see, has λ τ a pole at λ λ τ τ p 1 ± Aτ λ τ log p/m. 4.35 p M exp 1 A λ. 4.36 τ Since p cannot be greater than M see note 4, this pole will disappear for a negative β- function, i.e. when we have a plus sign in 4.36. However, it will not disappear for a positive β-function. This pole is where the bare coupling constant diverge, making it possible for us to calculate the renormalization scale for the system when the particles is at rest, i.e. p m. 1 M m exp A λ. 4.37 τ Note 8. In QED, the pole 4.36 is called the Landau pole in [1] we see that the β-function is positive in QED. If we were to calculate M in QED through 4.37, we would find that it is somewhere around 1 GeV. which is much higher than the experimental energy scale. This makes it impossible to calculate all observables in QED if we introduce a momentum cut-off like we did in figure 4.1. However, this pole might be a result of perturbation theory, which could mean that for high enough energies we cannot use perturbation theory. See [5] for more information. 5. THE COLEMAN-WEINBERG POTENTIAL Until now we have studied different approaches to renormalization of φ 4 -theory. Let us now consider another kind of approach to renormalization, and renormalize a different theory, namely the Coleman-Weinberg model. The Coleman-Weinberg potential is an effective potential, which takes into account first order loop corrections, i.e. first order quantum corrections. We find the effective potential by expanding the scalar field, φ, around the classical field, φ cl, up to first order quantum corrections. If we would consider quantum corrections of a higher order than the first, would 17

neither of the terms of order two or higher greatly contribute to the first order terms of the effective potential. This can be seen by using units with instead of units such that 1. The expansion around the classical field is then given by φ φ cl + n n φ n. 5.1 Here φ n is the complex valued, n th order quantum correction. We note that the classical limit, i.e., of 5.1 is the classical field. The effective potential is found by taking the classical potential, V cl, and adding quantum corrections from the kinetic and mass terms from the different fields. These quantum corrections will only depend on φ cl V eff φ cl V cl φ cl + n n V n φ cl. 5. Here V n is the n th order quantum corrections from the kinetic and mass terms for the different fields. The order of leads to higher order terms being much smaller than lower order terms, and thus does higher order terms only make small contributes to the first order terms in the effective potential. The effective potential will depend on the gauge we chose to work in, but the physical quantities which are derived from the effective potential will be independent of this gauge. We will work in the Landau gauge and study the effective potential in a simple case, namely the Coleman-Weinberg model. A renormalization scale, M, will be introduced, in order to renormalize the effective potential which we will find. The Lagrangian for the Coleman-Weinberg model is given by L 1 Fµν Dµ φ D µ φ V φ 4 V φ m φ φ + λ φ φ, 5.3 6 D µ φ µ + i e A µ φ where A µ is an abelian vector field. The Landau gauge is given by µ A µ, 5.4 It is easy to check that the Lagrangian 5.3 has a symmetry under an U1-transformation φx e iα φx. 5.5 In order to find the effective potential do we need to expand around the classical field, φ cl. We will expand up to first order quantum corrections, i.e. φx φ cl + 1 σx + iπx. 5.6 18

Since the fields A,φ cl,σ and π commute with each other will we get the Lagrangian 7 L 1 1 [ µ Fµν σ + µ π ] eφ cl A µ µ π e A µ σ µ π + π µ σ 4 e φ cl Aµ e e A µ φ cl Aµ σ + σ + π m φ cl + φ cl σ + 1 σ + π. 5.7 λ φ 4 cl 6 + 1 σ + π + φ 3 cl 4 σ + φ cl 3σ + π + φ cl σ σ + π Note 9. As we can see in the Lagrangian 5.7, the vector field has acquired a mass given by eφcl. Later in, we see that this is just a temporary mass for the photon, i.e. the vector field. We will define the classical potential, V cl, as all terms in the Lagrangian 5.7 which only contain φ cl, i.e. V cl φ cl m φ cl + λ 6 φ4 cl. 5.8 From this potential we may define the mass, m, and the coupling constant, λ, in a similar way as in [4] m : d V cl dφ cl 4λ : d 4 V cl dφ 4 cl φcl φcl M. 5.9 Here M is the renormalization scale. We see that M has the same dimension as φ cl 8. We will define the effective potential, V eff φ cl, as the potential 5.8 where quantum corrections from the kinetic and mass term for the fields A,π and σ have been added. In appendix A.4 and A.5 we see that we can write the effective potential as i { d 4 xv eff i d 4 x m φ cl + λφ4 cl 6 + 3 e φ cl [ log e 4π φ ] cl ξ + + 1 m + λφ cl [ log m 4π + λφ ] cl ξ + + 1 m + λφ cl /3 [ log m 4π + λφ cl /3 ξ ] } 5.1 7 Derivation is in appendix A.3. 8 I.e. [M] [φ cl ] mass. One can find [φ cl ] by using the fact that the action, S, is dimensionless, see [] for more information. 19

V eff m φ cl + λφ4 3e4φ4 cl cl [ + log e 6 4π φ ] cl ξ + m + λφ cl [ + log m 44π + λφ ] cl ξ +. 5.11 m + λφ cl + /3 [ log m 44π + λφ cl /3 ξ ] Here ξ is given by ξ lim ɛ ɛ γ + log4π + 3. 5.1 We see that ξ is dimensionless. Applying the definition 5.9 of λ to V eff, and only considering first loop order, i.e. λ e 4 << 1, we find the following renormalization scale 9 M π e { 1 lim exp ɛ ɛ + γ + 1 } /. 5.13 Insert M in the effective potential 5.11 to yield the following result 1 V eff m φ cl + λφ4 cl 6 + 1 4π m + λφ cl + Here M is given by 4 log [ φ 3e 4 φ 4 cl log cl M + m + λφ cl /3 m + λφ cl e M + 4 log m + λφ cl /3 ]. 5.14 e M M M exp5/. 5.15 Note 1. If we study the transition from 5.11 to 5.14, we see that the dimensionless ξ is "replaced" by the renormalization scale, M, which has the same dimension as mass. Important to note is that the dimensions are always correct. By studying 5.11 we see that expξ has the dimension of mass, and since M expξ the dimensions are correct. When we perform such a transition, i.e. when we "replace" a dimensionless quantity with a quantity which has a dimension of some kind, we perform a dimensional transmutation. For more information about dimensional transmutations, see [6]. Since the renormalization scale is arbitrary, we may define a new coupling, λ, to a new renor- 9 Derivation is in appendix A.6. 1 Derivation is in appendix A.7.

malization scale, M, in a similar way as 5.9, i.e. λ M : 1 d 4 V eff 4 dφ 4 cl φcl M [ λ + 3e4 M { 44π 4!log M + d 3 dφ 3 cl [ log [ M M exp5/ ] λ + 9e4 8π [ λ + 9e4 M 8π log M 3 ] + O λ φ 4 φ cl cl φ cl M M } φcl M 5 + 1 4! ] + O λ ] + O λ 5.16 If we fix the bare coupling λ, i.e. the un-renormalized coupling λ, we may find the beta function, β λ, for the coupling λ through the definition 4.6. Doing this yields β λ β λ 4.6 : M λ M 5.16 9e4 8π M M M + O λ 9e4 4π + O λ. 5.17 As in φ 4 -theory see 4.14, β λ is positive, which leads to the running of the coupling to be of similar form as in φ 4 -theory. So the renormalized coupling constant in the Coleman- Weinberg model will decay for small momenta and grow for large momenta. Since the Coleman- Weinberg model, just as scalar QED, can be seen as an approximation to QED where the spin of the fermions are neglected see [5], the result 5.17 strengthen the fact that the β-function is positive for QED as mentioned in note 8 in the previous section. Note 11. Since the β-function is positive in QED, the electric charge, i.e. the coupling constant, will decay at low momentum and grow at high momentum. This can be explained by screening, i.e. that the electron is surrounded by photons and electrons/positrons, which are continuously created and annihilated in vacuum. Since the positrons tends to be closer to the electron, they "screen" its charge and thus, at greater distances from the electron its charge seems to be smaller and vice versa at small distances. Smaller distances corresponds to higher energies, and greater distances corresponds to lower energies. Figure 5.1: Creation and annihilation of photons and electron/positron pairs. Let us now plot the effective potential 5.14 to see whether spontaneous symmetry breaking occur or not when we perform the expansion 5.6 around the classical field. In order to do 1

this do we have to make some assumptions. First off, let us consider only first order loop corrections, i.e. λ e 4 << 1. This yields that the effective potential 5.14 may be written as V eff m φ cl + λφ4 cl 6 3e4φ4 cl φ + 4π log cl M + O λ m φ cl + λφ4 cl 6 3e4φ4 cl φ + 4π log cl M. 5.18 If we use units such that λ/6 1 and 3e 4 /4π 1, and make the substitution x φ/m we can plot a rescale of the effective potential 5.18, namely V eff M 4 m M x + x 4 [ 1 + log x ]. 5.19 With help of http://fooplot.com/ can we plot 5.19 for different values on m / M. The resulting graphs is below. Figure 5.: Plots of the effective potential 5.19. Starting from the left, is m / M.15,.1,.5. As shown in these graphs, new ground states will appear when we expand around the classical field. This means that spontaneous symmetry breaking has occurred. 6. THE β-function IN NON-ABELIAN GAUGE THEORIES In this section we will find the β-function for a non-abelian gauge theory. We will then apply our result of the β-function to quantum chromodynamics QCD. A gauge theory is nonabelian if the generator matrices, T α, does not commute, i.e. let us consider that the Lagrangian is invariant under the following SUN Special Unitary transformation φ j x U j k xφ k x. 6.1 Here U is a N N matrix. We can write U in terms of its generator matrices U j k x δ j k i g θ α x T α j k + O θ, [ T α,t β] i f αβγ T γ. 6. If the anti-symmetric structure constants f αβγ, is non-zero, it is a non-abelian gauge theory.

In [] we see that the quantized Lagrangian in a non-abelian gauge theory is given by L L YM +L GF +L GH +L Ψ. 6.3 Here L YM is the Yang-Mills Lagrangian, L GF is the gauge-fixing to L YM Lagrangian, L GH is the ghost Lagrangian or the Faddeev-Popov Lagrangian and L Ψ is the fermion/quark Lagrangian. L YM contains the free gluon term and the interaction terms between several gluons, L GF introduces our gauge condition ξ which is dependent on the gauge we chose to work in, L GH introduces the interactions between gluons and ghost fields and L Ψ contains the free fermion term and the interaction terms between fermions and gluons. We will denote gluons as A µ A α µ T α, ghost fields as c α and the j th fermion as ψ j. In appendix A.8 we see that we can write the Lagrangian as L 1 Aγ µ g µν µ ν A γ ν g f αβγ A αµ A βν µ A γ ν g f αβγ A α µ ν 4 Aβ + + 1 ξ Aα µ µ ν A α ν + cβ c β + g f αβγ c β µ A α µ cγ + i ψ j ψ j + g ψ j A α T α j k ψ k. 6.4 In order to find the β-function we will perform perturbative renormalization. Similar as in section, we rescale the Green s functions for two A α µ s and two ψ j s as Ω T { } A α µ x 1A α µ x Ω i Z A p Ω T { ψ j x 1 ψ j x } Ω i Z ψ j p +...f i ni te... i +...f i ni te... p, at p M. 6.5 i +...f i ni te... +...f i ni te... p Here M is a renormalization scale. Unlike section, Z A and Z ψj they still diverge. are not residues. However, Note 1. Since the ghost fields, c α, only appear as internal fields in Feynman diagrams, we do not need to rescale those. This can be seen if we look at the Lagrangian before the gauge fixing, i.e. L L YM +L Ψ 1 Aγ µ g µν µ ν A γ ν g f αβγ A αµ A βν µ A γ ν g f αβγ A α µ ν 4 Aβ + i ψj ψ j +, 6.6 + g ψ j A α T α j k ψ k which does not contain ghost fields. The rescaling 6.5 can be achieved if we rescale the fields as.1, i.e. { A α µ Z 1/ A Aα µ ψ j Z ψj ψ j,no sum over j. 6.7 3

Inserting these rescaled fields into the Lagrangian 6.4 yields L 1 Z A A γ µ g µν µ ν A γ ν g Z 3/ A f αβγ A αµ A βν µ A γ ν g 4 Z A f αβγ A α µ ν Aβ + + 1 ξ Z A A α µ µ ν A α ν + cβ c β + g Z 1/ A f αβγ c β µ A α µ cγ + i Z ψ ψ j ψ j + + g Z ψ Z 1/ A ψ j A α T α j k ψ k L +L CT Here L and the counter-term Lagrangian, L CT, are given by. 6.8 L 1 Aγ µ g µν µ ν A γ ν g f αβγ A αµ A βν µ A γ ν g f αβγ A α 1 µ 4 Aβ ν + ξ Aα µ µ ν A α ν + + c β c β + g f αβγ c β µ A α µ cγ + i ψ j ψ j + g ψ j A α T α j k ψ k L CT 1 δ A A γ µ g µν µ ν A γ ν g δ 1 f αβγ A αµ A βν µ A γ ν g 4 δ f αβγ A α µ ν Aβ + + 1 ξ δ A A α µ µ ν A α ν + cβ c β + g δ 3 f αβγ c β µ A α µ cγ + iδ ψ ψ j ψ j + + g δ 4 ψ j A α T α j k ψ k. 6.9 Here the couplings, δ, between the new coupling constant, g, and the original coupling constant, g, are given by δ A Z A 1 δ ψ Z ψ 1 δ 1 Z 1 1 g g Z 3/ A 1 δ Z 1 g g Z A 1 δ 3 Z 3 1 g g Z 1/ A 1 δ 4 Z 4 1 g g Z ψz 1/ A 1. 6.1 In [1] we see that the interaction terms in the Lagrangian 6.8 yields the following Feynman rules 4

Figure 6.1: Feynman rules in a non-abelian gauge theory. We will use the following renormalization conditions Figure 6.: Renormalization conditions for a non-abelian gauge theory. With these renormalization conditions may we find the β-function through the formula 4.11. In [] we see that 4.11 translates to the following to lowest order of g for a non-abelian 5

gauge theory β g M M δa + δ ψ j δ 4 δ ψj γ ψj M M γ A M δ A M. 6.11 Note 13. When we derive the γ-functions see 4.8 for a non-abelian gauge theory, we have ω γ 1 and χ γ p p if it is a two-point Green s function for two fermions, as well as ω γ q g µν q q µ q ν δ αβ and χ γ 1 if it is a two-point Green s function for two photons. When we derive the β-function see 4.1 for a non-abelian gauge theory, we have ω µα β γµ T α, χ β 1, ɛ g and δ g j δ 4. For a three-point Green s function for two fermions and one photon we have α l δ Zl δ A + δ ψj. Let us first find δ ψj. It can be found by looking at the following amplitude Figure 6.3: 1PI diagram for two fermions in a non-abelian gauge theory. The amplitude, Σ, for the process above is given by iσ p i pv p + i pδ ψj + O g 3. 6.1 Here pv p is the amplitude for the one-loop diagram above. Using the first renormalization condition in figure 6. we get When we calculate V p it is convenient to use 1 1 k k + p 1 1 1 i MV M + i Mδ ψj + O λ 3 δ ψj V M + O g 3. 6.13 1 d x xk + xkp + xp + k xk [ l k + xp ] 1 d x xkp + xp + l xkp x p [ Dx, p x1 xp ], 6.14 1 d x l + D 6

and d d l l µ π d l + D. 6.15 In appendix A.9 we see that the one-loop diagram is given by i pv p i p g δ ψj g 8π C r 8π C r 1 1 d x1 xξ 1 x, p d x1 xξ 1 x, M + O g 3 Here C r is the Casimir operator in representation r, and ξ 1 is given by. 6.16 ξ 1 x, p lim ɛ ɛ log{ x1 xp } γ + log4π. 6.17 Let us now find δ A. It can be found by studying the following amplitude Figure 6.4: 1PI diagram for two gluons in a non-abelian gauge theory. The amplitude Π for the whole process is given by i g µν q q µ q ν δ αβ Πq i g µν q q µ q ν δ αβ V 1 q +V q +V 3 q + +V 4 q δ A + O g 3. 6.18 Here V j is the amplitude for the j th one loop diagram starting from upper left in figure 6.4. The second renormalization condition in figure 6. yields V 1 M +V M +V 3 M +V 4 M δ A + O g 3 δ A V 1 M +V M +V 3 M +V 4 M + O g 3. 6.19 7

When we calculate the V j s we will use d d l l π d l + D dd d x1 xq d 4 d d l 1 π d l + D [ D x1 xq ] d d l 1 π d l + D 6. d d l π d 6. d 4 l µ l ν l + D g µν g µν 4 d x1 x q d d l l π d l + D d d l 1 π d l + D x1 x g µν q d d l 1 π d l + D 6.1 f αγδ f βγδ δ αβ C G. 6. In appendix A.1 we see that V 1 g π n f C r and in appendix A.11-A.14 we see that 1 g µν q q µ q ν V q +V 3 q +V 4 q g 4π C G 1 d xx 1 xξ 1 x, p, 6.3 [ d x 9 x1 x + 1 x + 1 1 + x + 6x1 x 31 x + 1 x1 x ] g µν q [ x1 + x 1 x x1 x ] q µ q ν ξ 1 x, p. 6.4 Here C G is the Casimir operator in an adjoint representation G, and n f are the number of species of fermions. Note 14. Whenever we integrate over the Feynman parameter, x, may we exchange x for 1 x. The following is proof to this symmetry 1 d x f x [ y 1 x ] 1 d y f 1 y [ y x ] This means that we can rewrite terms linear in x in the integrals as x x + x x + 1 x 1 d x f 1 x. 6.5 1. 6.6 8

Since we can exchange x to 1 x in the integrals see 6.5, we can study the g µν q -terms and the q µ q ν -terms in 6.4 to see that they differ by a minus sign. For simplicity, let us not write out the integrals over x. Let us start studying the g µν q -terms 9 x1 x + 1 x + 1 1 + x + 6x1 x 31 x + 1 x1 x x1 x + 1 x + 1 1 + x 31 x x x + x + 1 x + 1 + x + 1 x 3 + 6x 3x 1. 6.7 + 7x 4x 6.6 1 + 7 4x 3 4x A similar study on the q µ q ν -terms in 6.4 yields [ x1 + x 1 x x1 x ] x + x + 1 4x + 4x + x x. 6.8 1 4x + 4x 6.6 1 + 4x 3 + 4x Which differ from 6.7 by a minus sign. If we use 6.7 and 6.8 we can write 6.4 as g µν q q µ q ν V q +V 3 q +V 4 q g µν q q µ q ν g 4π C G V q +V 3 q +V 4 q g 1 4π C G We find δ A if we insert 6.3 and 6.9 into 6.19 δ A g 4π 1 d x 3 4x ξ 1 x, p 1 d x 3 4x ξ 1 x, p. 6.9 d x [ 3 4x C G 8x1 xn f C r ] ξ 1 x, M + O g 3. 6.3 Let us now find δ 4. It can be done by studying the following amplitude Figure 6.5: Scattering Amplitude for two fermions and one gluon in a non-abelian gauge theory. 9

The amplitude Γ for the whole process is given by iγ µα p, p i g γ µ T α 1 +V 1 p, p +V p, p + δ 4 + O g 3. 6.31 Here V j is the j th one-loop diagram starting form the upper left in figure 6.5. The third renormalization condition in figure 6. yields i g γ µ T α i g γ µ T α 1 +V 1 M, M +V M, M + δ 4 + O g 3 δ 4 V 1 M, M V M, M + O g 3. 6.3 In appendix A.14 we see that V 1 p, p i g C r C G lim d 4 d d k 1 π d k + p k + p, 6.33 and in appendix A.15 we see that V p, p 3i g C G lim d 4 d d k 1 π d k p k p 6.34 6.33 and 6.34 gives us δ 4 i g C r 1 C G + 3 C G i g C r +C G lim d 4 + O g 3 lim d 4 g 4π C r +C Gξ M + O g 3 d d k 1 π d k M k + M + O g 3 d d k 1 π d k M i g C r +C G i 4π ξ +. 6.35 Here ξ is given by ξ M lim ɛ ɛ log M γ + log4π. 6.36 3

We may now use 6.16,6.3 and 6.35 to find the β-function through 6.11 β g M 1 g 1 M 4π d x [ 3 4x C G 8x1 xn f C r ] ξ 1 x, M g 1 8π C r d x1 xξ 1 x, M + g 4π C r +C Gξ + O g 3 [ M M ξ j M M ], j {1,} M g 3 1 4π d x [ 3 4x C G 8x1 xn f C r ]. 6.37 4C r 1 g 3 [ 1 4π 1 8n f d xx1 x + 4 d x1 xξ 1 x, M + C r +C Gξ + O g 3 d x 3 4x + C G 1 ] d x1 x C r + O g 3 The integrals which appears above has the following values 1 d x 3 4x [3x 4x3 3 [ x d xx1 x 1 1 β g 3 [ 5 4π 3 + g 3 4π ] 1 ] 1 3 4 3 5 3 x3 1 3 1 3 1. 6.38 6 ] 1 d x1 x [x x 1 1 1 1 C G 8n f 11 3 C G 4 3 n f C r 6 + 4 1 ] C r + O g 3. 6.39 In [] we see that for a SUN gauge theory with fermions in the fundamental representation this β-function becomes β g 3 11 4π 3 N 4 3 n f 1 g 3 4π In QCD, the gauge group is SU3, giving us the following β-function 11 3 N 3 n f. 6.4 β g 3 34π 33 n f. 6.41 Since the number of different species of fermions, n f, is less than 17, this β-function will be negative. As noted in section 4, a negative β-function imply that the theory is asymptotically free. That is, the coupling constant will grow for small momentum and decay for large 31

momentum, which is the opposite as what we found for the abelian gauge theories which we have studied in the previous sections, see the results in section 3-5. Similar to QED, this result of the β-function can be explained by anti-screening. In QCD, a colour charge, i.e. a quark, is surrounded by gluons and quarks/antiquarks, which are continuously created and annihilated in vacuum similar to figure 5.1. However, since gluons can interact with each other, and since gluons also has a colour charge, the colour charge of a quark seems larger at greater distances, and smaller at shorter distances. Note 15. This result of the β-function is only valid for small values on the coupling constant since we have used perturbation theory throughout the calculations of this β-function. That is, the β-function in QCD is negative when the quarks are close to each other and the coupling between them are weak. We cannot say anything about the β-function from the calculations in this section when the quarks are far away from each other and the coupling between them is strong, i.e. when the g 1. To find the β-function in this case is one part of the unsolved and infamous confinement problem in QCD. One way to solve it would be too renormalize QCD without using perturbation theory, see []. Note 16. It is possible to introduce so called QCD strings or QCD flux tubes as degrees of freedom to illustrate the force between two or more colour charges, i.e. quarks. These flux tubes grows as the distance between the quarks grows, indicating that the force between them gets bigger. As seen in [7], the flux tubes in QCD are analogous to the magnetic flux between one imaginary monopole and antimonopole in a superconductor. Figure 6.6: The colour flux between two quarks, which can be compared to magnetic flux tubes in superconductor. 7. CONCLUSION Since we expect the same result of a physical quantity no matter how we derive it from a theory, the choice of approach to renormalization should not matter. This was shown in examples in section -4, where we have performed three different approaches to renormalization perturbative, the Wilsonian approach and by using the Callan-Symanzik equation of φ 4 -theory. In section and 3 we found that the scalar field φ is not renormalized to the lowest order of the coupling constant, and therefore the kinetic term of the Lagrangian is a fixed point of the renormalization, and in section 3 and 4 we found that the coupling constant will grow through renormalization. The Callan-Symanzik equation, which the Green s functions satisfy, was derived in section 4. By studying this partial differential equation we found that the Green s function describes the density of Lagrangians as a function of the momenta and the coupling constants, which 3

we gain when we renormalize a theory. The starting point for the derivation of the Callan- Symanzik equation was an arbitrary momentum scale, called the renormalization scale, which we first introduced in order to renormalize massless fields. However, it was noted that we could renormalize a massive theory with this renormalization scale too. This renormalization scale can be seen as a momentum cut-off. The Callan-Symanzik equation contains the important β-function, which describes the flow of Lagrangians or coupling constants which we gain when we renormalize a theory. It tells us about the running of the coupling constants. A positive β-function means that the renormalized coupling constant decay for small momenta and grow for large momenta, and a negative β-function means that the theory has asymptotic freedom, i.e. that the renormalized coupling constant grow for small momenta and decay for large momenta. We showed in section 4, that if the β-function is positive, the bare coupling constant will contain a pole, from which we can find a value on the renormalization scale for theory. The Coleman-Weinberg potential was derived in section 5. It is an effective potential which takes into account quantum corrections to the classical field. If we plot the Coleman-Weinberg potential, we find whether spontaneous symmetry breaking will occur or not when we expand around the classical field. This effective potential will depend on the gauge we chose to work in, but the physical quantities, e.g. the β-function, which may be derived from it will not. It is important to keep in mind that in all of the different cases where we have studied the behaviour of the coupling constants have we used perturbation theory, i.e. we have assumed that the coupling constant is small. The coupling constant, however, does not need to be small. If the coupling constants were to be large, we would not be able to use perturbation theory. This leads to some unsolved problems, e.g. the confinement problem in QCD which we discussed at the end of section 6. Some other renormalization problems are the Landau pole in QED discussed at the end of section 4 and in theories of quantum gravity, i.e. in grand unification theories GUT s where we try to construct a theory consisting of both quantum mechanics and gravity. As in many quantum field theories, some scattering processes does diverge in GUT s, and therefore we want to renormalize the theory. Since gravity is perturbatively nonrenormalizable, we cannot use perturbation theory in order to renormalize the theory. 33