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University of Amsterdam

Institute for Theoretical Physics Amsterdam Faculty of Science

Bachelor Thesis

Renormalization, Regularization and the

Renormalization Group in Quantum Field Theory

by

Tim Blankenstein 11033207

Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between 03-04-2018 and 06-07-2018

Supervisor:

Prof. Eric Laenen

Second examiner: Dr. Juan Rojo July 12, 2018

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Abstract

Renormalization is discussed in the context of quantum field theory and in particular quantum electrodynamics. It is shown how techniques to subtract divergences encountered in loop-order Feynman diagrams are needed and how renormalization performs this subtraction. In this discussion, a variety of regularization and integral evaluation methods are reviewed and special attention is paid to the exposition of divergences using dimensional regularization. Furthermore, it is motivated, after renormalization, when considering loop corrections in the perturbation expansion at different energy scales, there is a necessity for running of the coupling constant to maintain accuracy in predictions. The beta function is introduced from the renormalization group equation and its use for this purpose, as well as the limits of the running of the coupling constant, is discussed.

1

Populaire samenvatting

In de jaren ’20 stond de wereld van de natuurkunde goed op zijn kop: de kwantummechanica, de theorie van de deeltjes, toentertijd een behoorlijke nieuwe theorie, werd volop bestudeerd en Einstein had inmiddels in de jaren ’10 zijn algemene relativiteitstheorie, de theorie van zwaartekracht, gepubliceerd. Des te populairder de kwantummechanica werd, des te meer begon het idee te dagen: een samenvoeging van deze theorie¨en, samen met de 19e eeuwse theorie van elektromagnetisme (elektriciteit en magnetisme), zou natuurkundigen misschien wel in staat stellen over alles in de fysische wereld correcte voorspellingen te kunnen doen, van het grootste tot het allerkleinste. Makkelijk bleek deze samenvoeging niet te realiseren, want tot op de dag van vandaag is er nog geen theorie gevonden die deze drie werelden consistent verenigt, maar in de jaren ’20 deden natuurkundigen een stap in de goede richt-ing: ze begonnen met de ontwikkeling van de zogenaamde kwantumelektrodynamica. Deze theorie verenigt de kwantummechanica en elektromagnetisme met niet Einsteins algemene relativiteitstheorie, maar zijn speciale relativiteitstheorie, een deeltheorie van de algemene relativiteitstheorie dat zich alleen bezighoudt met bewegingen zonder zwaartekracht. Samen met het gebruik van theorie¨en om velden (van deeltjes, ladingen, etc.) te beschrijven, kwam men uiteindelijk uit op een van de meeste precieze theorie¨en in de geschiedenis van de natu-urkunde. Maar gaandeweg bleek er een groot obstakel te zijn. Een obstakel dat binnen andere theorie¨en ook aanwezig was.

De kwantumelektrodynamica is een theorie binnen een verzameling van theorie¨en, genaamd de kwantumveldentheorie¨en, die allemaal de speciale relativiteitstheorie, de kwantummechan-ica en de theorie van velden combineren. Allemaal hebben ze eenzelfde structuur en allemaal

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hebben ze het obstakel van kwantumelektrodynamica gemeen. Dit obstakel toont zich als volgt: in de kwantumveldentheorie worden berekeningen niet op ´e´en manier gedaan waarna dat het definitieve antwoord is, maar worden berekeningen per ’orde’ gedaan. Elke extra orde is een extra berekening om een extra correctie op het antwoord te krijgen, zodat deze nog preciezer wordt. Deze correcties kan je doen door in diagrammen van lijnen die de interactie van deeltjes voorstellen, een extra lijn te tekenen, zodat de lijnen in het diagram een lus (’loop’) vormen, zoals aangegeven in figuren 1 en 2.

Toen de natuurkundigen echter eenmaal hun berekening deden met die correcties, kwam elk antwoord op oneindig uit. Het leek alsof de kwantumveldentheorie¨en, simpel gezegd, ’kapot’ waren en er ergens in het ontwikkelingsproces iets verkeerd was gegaan. Gelukkig kon men echter een verzameling consistente technieken verzinnen om deze oneindigheden te doen verdwijnen uit de theorie¨en. Deze verzameling technieken wordt renormalizatie genoemd. Het is echter moeilijk met oneindigheden te werken, omdat de antwoorden al gauw nergens op slaan (wat is ∞ − ∞?). Om de oneindigheden daarom handelbaar te maken, worden ze bijvoorbeeld voorgesteld als 1/a, waarbij a uiteindelijk naar 0 toe gaat. Er staat dan oneindig, maar er is makkelijker mee om te gaan (1/a − 1/a = 0). Een manier om oneindigheden han-delbaarder te maken heet regularizatie. Zodra de oneindigheden hanhan-delbaarder geworden zijn door deze technieken, worden de oneindigheden door renormalizatie weggehaald. Naast dat berekeningen weer mogelijk zijn, bleek echter ook dat door de methode van renormalizatie toe te passen, we iets betekenisvols kunnen zeggen over op welke energieniveaus van gebeurtenis-sen waarover de kwantumveldentheorie¨en voorspellingen doet, deze voorspellingen nog correct zijn. Een voorbeeld van een gebeurtenis met een laag energieniveau, dus waar relatief weinig energie bij vrijkomt of gebruikt wordt, is het botsen van elektronen. Een met een relatief hoog energieniveau is het ontploffen van een ster. Door renormalizatie slim te gebruiken kan er iets gezegd worden op welke energieniveaus sommige kwantumveldentheori¨en geen correcte voorspellingen meer kunnen doen. Rond de energieniveaus dat de theorie¨en niet meer werken missen we blijkbaar iets in de theorie, dus dit geeft natuurkundigen een goed idee waar te zoeken voor de nieuw soort natuurkunde nodig om weer correcte resultaten te krijgen.

Figure 1: Voorbeeld van kwantumvelden-theorie diagram zonder lus correctie

Figure 2: Voorbeeld van kwantumvelden-theorie diagram met lus correctie

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Contents

1 Populaire samenvatting 2

2 Introduction 5

3 QFT renormalization 7

4 Dimensional regularization 15

4.1 Application to a typical integral . . . 15 4.2 Feynman parametrization and methods of integral evaluation . . . 25

5 The renormalization group 30

6 Summary 37

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2

Introduction

Quantum field theory (often shortened to QFT) has, since its inception in the mid 1920s, quickly become one of the staples of physics. Especially quantum electrodynamics (often shortened to QED), having the honor of being called names like the ”jewel of physics” [3], is held in high regard and it seems rightfully so: a measurement of the electron anomalous magnetic moment ae, that is, the difference between the magnetic moment as predicted by

the 1928 Dirac equation and as predicted by quantum electrodynamics, shows this theory has unbelievable accuracy, see table 1. With its agreement with measurements to up to more than 10 significant figures, this result is one of the most accurate verified predictions in the history of physics. But this precision did not come without a struggle: as quantum field theory developed throughout the 1930s, many calculations of higher order contributions to perturbation series were found to be divergent. Some clarity was brought to the issue by the introduction of Feynman diagrams in 1948 [5]. It was found that loops in Feynman diagrams brought forward the divergences, with calculations from Feynman diagrams containing closed loops diverging [7]. The calculation of the electron anomalous magnetic moment, now one of the most accurate verified predictions in the history of physics, faced this very same problem. Obviously, a solution needed to be found. A collection of techniques to subtract these divergences was invented and this theory of removing divergences would come to be known as renormalization. Some scientists were far from advocates of the use of these techniques to remove divergences. For example, Paul Dirac has been quoted as saying as late as 1975 [6]:

”[Renormalization] is just not sensible mathematics. Sensible mathematics involves dis-regarding a quantity when it is small —not neglecting it just because it is infinitely great and you do not want it!”

This thesis aims to show that renormalization, however talked down to by the words of Dirac, is actually far from unscientific and that when understood well, can be deemed as not only logical, but necessary.

ae= g−22 Value

Measured 0.00115965218073(28) Predicted 0.001159652181643(764)

Table 1: Comparison of the measured and predicted value of the anomalous magnetic moment ae,

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Before subtracting the divergences, a method to expose and manage these infinities is needed first. A technique used for this purpose is called regularization and it is linked in-timately with renormalization. In the first chapter, I will discuss both renormalization and regularization, after which in the second chapter a particular, frequently used method of reg-ularization will be discussed. In the last chapter, the consequences of renormalization for coupling constants at different energy scales will be examined and with that, the limits of theories like quantum electrodynamics and quantum chromodynamics as well.

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3

QFT renormalization

While some predictions of quantum field theory have an unprecedented level of accuracy, the theory inherently has the problem that closed loops in Feynman diagrams will give rise to momentum integrals that are not finite [7]. These divergences are not cancelled by contribu-tions of higher-order diagrams, so in the increasingly higher order perturbation expansions, these divergences will remain. For an example, consider the electron self-energy in quantum electrodynamics, which describes the contribution of the energy of the particle considered due to interactions between the particle and its self-generated photons. The one-loop order expression for this function is

Σ(p) = −ie 2 (2π)4 Z ∞ −∞ d4q γµ −i/p + m − i/q q2((p + q)2+ m2)γν  ηµν− (1 − λ−2)q µqν q2  , (1)

with the corresponding Feynman diagram shown in figure 3. In this expression we can recog-nize two integrals in both of which the largest power of q in the denominator is 4 and where one has a −i/q term in the numerator. Ignoring terms with lower powers, we can see that, since we are integrating in four dimensions, we will therefore end up with a logarithmic and a linear term respectively that will both diverge (see also ’Power counting’ for a more in-depth discussion on how to spot a divergent integral). Obviously, a way to handle these infinities will need to be devised. This is were renormalization comes in. To make a first step towards the solution of our problem, let us take a look at the source of our functions: the Lagrangian of quantum electrodynamics,

L = −1

4(∂µAν− ∂νAµ)

2− ¯ψ( /∂ + m)ψ − ieA

µψγ¯ µψ −12(λ∂µAµ)2, (2)

where −e denotes the electron charge, λ an arbitrary gauge-fixing parameter and m denotes the mass of the fermion field. Important for the understanding of our method of renormal-ization is the realrenormal-ization that the parameters in the Lagrangian do not have an immediate connection with the physical quantities they denote. For example, the mass parameter m in the Lagrangian, denoting the mass of the fermion field, is only the mass of the fermion field at the tree-level. The mass of the fermion field is defined as the position of the pole in the full

p p + q p

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fermion propagator, which includes all the higher-order corrections, not the mass parameter in the Lagrangian, which does not include these corrections. The same goes for the electron charge parameter −e. The link between the actual physical parameters is still there, but it is hindered by the infinities that pop up. For a physical theory, we of course demand that the physical parameters and coupling constants are finite. Since functions coming from the Lagrangian diverge, we are forced to view the original parameters in the Lagrangian as being infinite, to avoid interpreting the physical ones as being infinite.

An infinite Lagrangian seems unphysical as well however. We therefore interpret the parameters and fields in this Lagrangian as the result of one particular parametrization, which only gives physical, finite results in the tree-level approximation. This interpretation also shows us that the fact that the original parameters and fields in the loop-order are infinite is not so unusual, because we see this parametrization was not suited for those calculations to begin with. Now, since in the loop-order these parameters do not correspond to the physical parameters anyway (as was explained for the mass parameter), and since fields are not

Power counting

Often, it is not immediately obvious whether an integral is convergent or not. There exists a nice method however, called ’power counting’, to easily spot an integral that is divergent. First of all, it should be noted that an integralR∞

−∞dx x

−(1+)will not diverge

for any  > 0. If  is equal to or smaller than zero, the integral diverges, since this yields a logarithmic term for  = 0 and a linear or even faster diverging term if  < 0. We can see from this that in n dimensions this condition holds as well. In that case R∞

−∞dnx x

−(n+) will not diverge for any  > 0, since R dnx ∝ R∞

0 d|~x||~x|(n−1). Realize

as well that, as x goes to infinity, the terms with smaller powers become negligible in comparison to the higher power terms. As x becomes bigger and bigger, the x with the largest power will dominate and determine whether the integral diverges or not. If the function is a fraction, the largest power of x has to be determined separately for the numerator and denominator and added up to get the largest power of x in the integrand (since R dx (a + bx)/c = R dx a/c + R dx bx/c, terms in the numerator independent of the variable of integration should be watched out for as well). With the condition of divergence in n dimensions in the back of our minds, this all put together means that determining divergence is a matter of seeing if for the term with the largest power p in an n dimensional integral the condition p + n < 0 holds. If this is not the case, the integral will diverge. If it is the case, the integral will not diverge. This method provides a quick way of determining whether an integral is divergent or not, something often useful to know in QFT and especially in light of renormalization.

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physical, we might as well re-express these in terms of new parameters and fields, of which we demand they are finite. The choice for the new parameters and fields is not unique, it only has to be self-consistent and has to remove the divergences, but the new parameters can be chosen such that they correspond directly to observable quantities. For example, we can express the amplitudes arising from the QED Lagrangian in terms of the physical mass instead of the original Lagrangian mass parameter. Next, we perform calculations and check if the results are finite. If this is the case, we have successfully renormalized the theory through this particular parametrization.

Before we can carry out the renormalization procedure, we have to deal with the infinities that arise immediately upon the actual calculations first. Only then can we explicitly remove the divergences for good. A procedure to momentarily keep the involved expressions finite is called regularization. After this procedure we can always take a certain limit again to obtain the original, infinite expression. Of course, by then, we would have removed those divergent terms through some chosen renormalization procedure.

There are many ways to regularize the loop diagrams, and each regularization method expresses infinities in their own, characteristic way. The most straight-forward method is cutoff regularization, wherein we simply limit the interval over which we evaluate the integral to some finite value Λ2 [8]. We would thus perform the substitution

Z ∞ 0 → Z Λ2 0 ,

after renormalization retaking the limit Λ2 → ∞. The downside of this method is however that it does not respect the symmetries (spin- and Lorentz symmetry, to name a few) of QFT unless we continuously throughout the calculation make sure it does. This keeping track of symmetries becomes especially unworkable in higher-loop order, certainly with more than one symmetry in the system. This method is therefore mostly used as an extra check for dimensional regularization, on which we will go into more detail later on.

A different, less intuitive, but more sophisticated method of regularization is Pauli-Villars regularization. In this procedure, the idea is to introduce some term in the propagators appearing in the functions, such that they vanish fast enough at larger momenta [7]. We would for example perform the substitution,

1 q2+ m2 → 1 q2+ m2 − 1 q2+ Λ2 = q2+ Λ2− q2− m2 (q2+ m2)(q2+ Λ2) = Λ2− m2 (q2+ m2)(q2+ Λ2), (3)

where Λ2 is unrelated to cutoff regularization. Notice we recover the original expression of terms involving the propagator, complete with divergence, once we take the limit Λ → ∞. After calculations, the Λ term will show up as Λ2, Λ2ln Λ2 or ln Λ2 terms, which we can

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remove using renormalization. This method of regularization, like cutoff regularization, has the downside of not respecting symmetries (gauge invariance, for example) inherent in QFT. Another regularization procedure is lattice cutoff regularization, in which space-time is replaced by a discrete lattice of points. Usually a cubic lattice with a lattice spacing of a is taken for this. Because of the discrete spacing, the momentum of the particles will be restricted to be less than π/a, preventing them from reaching the momentum approaching infinity that cause the divergences in the loop corrections of the theory. When taking the limit a → 0, we see that the maximum momentum will return to infinity. This method is not useful in calculations involving scattering.

Finally, a method of regularization most often used is dimensional regularization. For now, we will not go too in-depth into this method, since the entire next chapter will be devoted to this regularization procedure. It will be mentioned however that it is based on the observation that certain functions (in particular, the so-called vertex function) are finite in less than four dimensions, in which we normally evaluate the functions, as QFT incorporates special relativity. We thus formally continue the Feynman integrals to an arbitrary number of dimensions, that is, n dimensions to n = 4 + , where  some small, complex number (in the next chapter it will be shown why this number is complex). The divergences will then show up as 1/ poles (1/n in general for nth-loop order diagrams). The nice thing is that since we are only changing the dimensions the integrals are evaluated in and nothing of the integrand, we do not have to worry much about symmetry and physical principle violations, since very little symmetries in QFT depend on the number of space-time dimensions. The price we pay, however, is intuition: considering physical phenomena in a non-integer number of space-time dimensions is highly abstract.

Once we have chosen our method of regularization, we can apply a renormalization pro-cedure to remove the divergences. To show how this works in practice let us consider the one-loop order self-energy function (equation (1)) that we have dimensionally regularized. It becomes, Σ(p) = mA(p2) + iB(p2)/p , where A(p2) = −e 2µ 8π2(3 + λ −2)h1  + 1 2γE− 1 2ln 4π i + Af(p2) and B(p2) = −e 2µ 8π2λ −2h1  + 1 2γE− 1 2ln 4π i + Bf(p2) ,

where Af and Bf are terms in A(p2) and B(p2) respectively that are free of ultraviolet divergences, γE the Euler-Mascheroni constant (0.577...) and µ an arbitrary reference mass

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however, will become useful later on when we discuss the renormalization group). The term [1/ +12γE −12ln 4π] comes from −12(4π)−/2Γ(−12) = 1  + 1 2γE− 1 2ln 4π + O() ,

an expression of which we will see the origin when we evaluate a typical QFT integral in the next chapter. To begin our procedure of renormalization for this one-loop order function, we can introduce counterterms in the Lagrangian to render it finite. There will have to be two in total, each for one 1/ term. To cancel, we can introduce

∆LA= − e2µ 8π2(3 + λ −2) ¯ψψh1  + 1 2γE− 1 2ln 4π i , (4)

which will give rise to the contribution to A(p2) of

∆A(p2) = e 2µ 8π2 m(3 + λ −2)h1  + 1 2γE− 1 2ln 4π i

and the counterterm

∆LB= − e2µ 8π2λ −2ψ /¯∂ψh1  + 1 2γE− 1 2ln 4π i , (5)

which will bring about the contribution to B(p2),

∆B(p2) = e 2µ 8π2λ −2h1  + 1 2γE− 1 2ln 4π i ,

where we can see that the extra contributions to A(p2) and B(p2) cancel the divergent terms. They also cancel the γE and ln4π term, something which is called modified minimal

subtrac-tion, commonly written as MS. It is also possible to solely subtract the divergent 1/ term, called minimal subtraction, but more often modified minimal subtraction is more convenient. This shows there is some ambiguity in what finite part to subtract through the counterterms. This ambiguity will not get in the way of our ability to make predictions however, as will be explained at the end of this chapter.

With the one-loop order divergences removed, we have verified that for this function at this loop-order, QED is renormalizable. If we would want to prove that QED in its entirety is renormalizable, we would have to prove that we can remove the divergences for all orders of diagrams, not only for this function, but every possible function coming from the QED Lagrangian. The fact that all counterterms are of the same structure, however, is actually a characteristic property of a renormalizable theory and thus already a sign that QED is renormalizable.1 This is a property of a renormalizable theory since it will allow us to absorb the infinities into the parameters and fields of the original Lagrangian, only adding a few

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extra terms of a set form per higher order to counter the additional divergences that arise at that particular order. In particular, in dimensional regularization, the poles will take the form of a power series in the coupling constants and in 1/ when we consider higher-order terms, poles ∼ N X m=1 e2  m , (6)

where m the loop order and N the total number of loops in the diagram considered. Higher-order poles encountered when considering additional loops in this case arise from integrating over logarithmic functions encountered in the lower orders as well as functions in lower-order combining in such a way they generate additional loops. These two factors in general give poles with multiplicity m and lower for m loops present. We will discuss these higher-order poles more in-depth in the next chapter. Note that with how renormalization works, the counterterms for every order lower than the order considered, will be present when considering diagrams with additional loops. It is therefore important to not immediately get rid of linear  terms, since these may cancel 1/ poles later on.

A theory that is partially divergent and partially convergent at loop-order will not be consistent. Therefore, we will discuss, for consistency’s sake, the renormalization for QED in its entirety from now on, that is, for the vertex function, vacuum polarization and self-energy at one-loop order. To remove the divergence coming from the vacuum polarization diagram, we will need to insert the counterterm

∆Lvac= − e2µ 24π2(∂µAν− ∂νAµ) 2h1  + 1 2γE− 1 2ln 4π i (7) into the Lagrangian. For the vertex function this will be the counterterm

∆Lver = − ie3µ 8π2 λ −2ψγ¯ µψA µ h1  + 1 2γE− 1 2ln 4π i . (8)

To keep the focus of this chapter on renormalization only, I will refrain from going into details of the vacuum polarization and the vertex function and will just incorporate these counterterms in our process of renormalization. This only means we will have two additional counterterms to work into the parameters and fields.

Now that we have calculated, for one-loop order, the terms needed in the Lagrangian to remove the divergences, we are going to express the old parameters and fields in terms of new

non-renormalizable theory is not capable of making predictions. A non-renormalizable theory is for example encountered in general relativity: if we perform a perturbation of the metric tensor gµν around the Minkowski

metric, we obtain a series that is divergent at every term. We can then still make predictions, but we would have to manually insert new counterterms in the Lagrangian at every higher order to cancel the divergences [9].

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parameters and fields containing these calculated counterterms to incorporate them into the Lagrangian (as mentioned before, we can do this since the old parameters did not correspond to physical quantities anyway). We will denote the original, infinite terms as A0µ, ψ0, e0, m0

and λ0, after which we express them in terms of new, finite parameters,

A0µ=pZAAµ, ψ0=pZψψ , e0= Zee , m0 = Zmm , λ0 = Zλλ , (9)

where Aµ and ψ are called the ’renormalized’ fields and e0, m0 and λ the ’renormalized’

parameters. These new, ’renormalized’ parameters will not directly correspond to the physical quantities, but then again, neither did the old parameters. If, however, the renormalization is successful, all physically meaningful results will be able to be expressed in terms of finite functions of these new parameters. In higher-than-lowest-order diagrams, the physical mass mP and charge of the electron eP will then be a function of e and m, that is,

mP = m f (e2, m/µ) , eP = e g(e2, m/µ) ,

where f and g are functions that can be determined order by order in e2. Of course, the parametrization can be done in such a way that f and g both equal unity and m and e actually do equal the physical mass and charge of the electron. However, the new parameters will be always be finite, no matter the parametrization. This is because the redefinitions of equation (9) are done in such a way that the divergences are absorbed by the extra Z factors for each of their respective field or parameter (ZA for Aµ, Ze for e, etc.). These Z factors

will be infinite, as they will serve to cancel the infinities encountered in the the loop-order diagrams. In lowest order, the Z factors will therefore generally be equal to unity, since tree-level diagrams contain no divergences. For our dimensionally regularized functions in QED, the Z factors will contain a power series in 1/ and e (shown in equation 6), together with coefficients that will be determined from the Lagrangian with its counterterms. With the relabeling, the original Lagrangian was

L = −1 4(∂µA 0 ν− ∂νA0µ)2− ¯ψ0( /∂ + m0)ψ0− ie0A0µψ¯0γµψ0−12(λ0∂ µA0 µ)2,

which, with the redefinitions of the parameters and fields of equation (9) and using Zi =

1 + (Zi− 1), becomes the ’new’ Lagrangian,

L = −1 4(∂µAν− ∂νAµ) 2− ¯ψ( /∂ + m)ψ − ieA µψγ¯ µψ −12(λ∂µAµ)2 −1 4(ZA− 1)(∂µAν − ∂νAµ) 2 1 2λ 2(Z2 λZA− 1)(∂µAµ)2 − (Zψ− 1) ¯ψ /∂ψ − m(ZmZψ− 1) ¯ψψ − ie(ZeZψZA1/2− 1)Aµψγ¯ µψ . (10)

If we compare this Lagrangian with the original one of equation (2) with added counterterms of equations (4), (5), (7) and (8), we can determine the Z factors for the renormalization of

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QED at first order as ZA= 1 + e2µ 6π2 h1  + 1 2γE− 1 2ln 4π i + O(e4) , Zψ = 1 + e2µ 8π2 λ −2h1  + 1 2γE − 1 2ln 4π i + O(e4) , Ze = 1 − e2µ 12π2 h1  + 1 2γE− 1 2ln 4π i + O(e4) , Zm = 1 + 3e2µ 8π2 h1  + 1 2γE − 1 2ln 4π i + O(e4) , Zλ = 1 − e2µ 12π2 h1  + 1 2γE − 1 2ln 4π i + O(e4) ,

where, just as with the counterterms, we can choose to either perform minimal subtraction, letting the Z factors only contain the divergent 1/ term, or modified minimal subtraction, MS, including the logarithmic and γE term as well, as was done here. Aside from this it is

important to notice that the parameters with an obvious physical interpretation like e and m do not depend on the arbitrary λ parameter, removing ambiguity from these parameters that have a clear link with physical, observable quantities. Note that the ’new’ Lagrangian of equation (10) that includes the Z factors holds for renormalization of any order diagram. The only thing that would change would be what is inserted into the Z factors, or, in terms of dimensional regularization, at what order we would cut off the power series in 1/ and e.

Now that we have removed the divergences, one question remains: how to make physical predictions at this point? This seems hard to do, since at loop-order, our expressions will be full of ambiguous terms: m and e are not necessarily physical, unless we choose them to be, the Z-factors that absorb the divergences have an arbitrary finite component and we have introduced an arbitrary mass term µ along the way. To make predictions, however, is actually not that difficult. Now that we have made the theory workable again, we will make physical predictions with our theory and fit those to experimental data once. Particularly, we will choose µ to be near the energy scale at which the event takes place (does it require a lot, or little energy?). There is not much of a reason for specifically choosing µ to take this role, aside from it having the same dimensions as energy since we are working in natural units. Once the calculations have been fitted to the experimental data, the theory will work just as desired, giving finite physical predictions from that point on. The theory was not wrong, but the parametrization of the theory at loop-order was.

Now that we have an idea how to deal with divergences arising from diagrams containing loops, we will take a look at how these divergences exactly arise and how we deal with them using dimensional regularization in the next chapter.

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4

Dimensional regularization

As previously mentioned, to calculate the integrals that arise in quantum field theory it is necessary to use a method of regularization at an intermediate stage of calculation to avoid infinities, since keeping these divergences make the theory unworkable. A widely used method for regularization was dimensional regularization, which alters the number of dimensions in which the integrals are evaluated by a small number . Normally, since QFT incorporates special relativity, the integrals are evaluated in four dimensions. The trick of dimensional regularization is however to, for the time being, evaluate the integrals in n dimensions writing n as n = 4 + , where  a small complex number [7].

4.1 Application to a typical integral

To show how this method works in practice, let us first evaluate a typical integral I(n, α) =

Z

dnq

(q2+ m2− iε)α (11)

and use dimensional regularization to show how the divergences arise. Note that normally in these types of integrals the factor −iε is not present. This factor has been added to make the existence of poles clearer. Namely, when integrating over q0 in qµ= (q0, q1, ..., qn−1), we can see that, when using the (− + + +) Minkowski metric signature, the integrals have poles near q0 = ±pq2+ m2. These poles now do not lie exactly on this position, since they have been

displaced slightly into the upper-left and lower-right quadrant of the complex plane thanks to the i contribution in the denominator of the integrand, see figure 4. The existence of these

Figure 4: Moving of the poles away from the real axis due to the −iε term in the denominator of the integrand, where ω the original (real) position of the pole.

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poles prohibits us from evaluating the integral without running into divergences. To avoid these poles, we perform a so-called Wick rotation: we take a different complex contour and include the original real contour as shown in figure 5. This is allowed because of Cauchy’s theorem in complex analysis: a path in the complex plane can be changed as much as we needed, but the path cannot cross over any poles. Because of the particular placement of ε in the denominator of the integral, no poles are crossed in the Wick rotation, therefore, this procedure is allowed. Now, since this closed contour C in the Wick rotation does not include singularities, Cauchy’s integral theorem states that the integral of the entire contour must be equal to zero, I C dq0 = Z +∞ −∞ dq0+ Z −i∞ +i∞ dq0+ Z C2 dq0+ Z C4 dq0 = 0 . (12)

This equality must remain no matter how far we extend the arcs (contours C2 and C4). Since

the integrands we encounter in QFT grow slower than 1/x (or we will make sure they do), the integrand goes to zero when sent to plus or minus infinity and so will the integral.2 This means that if we extend these arcs to infinity, their contribution will tend to zero and we are

Figure 5: The curve taken by the Wick rotation, where ω the original (real) position of the pole. Notice that the path on the imaginary axis does not contain any poles.

2This ’growing demand’ is a general demand for the integrand of a finite integral over the entire line. See

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left with Z +∞ −∞ dq0+ Z −i∞ +i∞ dq0 = 0 , which in turn means that

Z +∞ −∞ dq0 = Z +i∞ −i∞ dq0. (13)

(13) shows we can safely replace q0 by iq0 in the expression of I(n, α).3 This replacement turns out to be of great use, because now we have q2 = −(iq0)2+ q2= q0 2+ q2, which in turn means that I(n, α) contains no poles, since neither q0 2 nor m2 can be negative. We write

I(n, α) = i

Z dnq

(q0 2+ q2+ m2)α ,

where we have removed the −iε factor, since it is no longer necessary. The integral can now safely be evaluated. The actual evaluation can be done relatively easily by switching to spherical coordinates, that is, a coordinate system using n − 1 angles θ1, ..., θn−1 and a radius

r, by using the relation Z dnx = Z dr rn−1 Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθn−1sinn−2θn−1, (14)

a result well-known for n = 1, 2 and 3 dimensions which we will now prove using induction: assuming the relation holds true for n dimensions, we will try to prove it for n + 1 dimensions. We write Z dn+1x = Z dxn+1 Z dnx , (15)

where we have introduced an extra variable xn+1to extend the n-dimensional space to (n + 1)

dimensions. We denote the n-dimensional radius by r0 and the radius in n + 1 dimensions by r. The variable xn+1 is perpendicular to the n other axes and makes an angle θn in the

(n + 1)-dimensional space with radius r, a fact we express mathematically as xn+1= r cos θn.4

From the expression of xn+1 we can see that in the case of θn= 0 the radius r simply equals

xn+1, meaning that the variables of the n axes must all equal zero, in turn meaning that

r0 = 0. This paired with the fact that xn+1 is perpendicular to all of these n axes, means

that r0 = r sin θn. Armed with these expressions and notations, we insert equation (14) into

equation (15) to get Z dn+1x = Z dxn+1 Z dr0r0n−1 Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθn−1sinn−2θn−1, (16) 3

Notice that this is equivalent to going from Minkowski space to Euclidean space: −(q0

)2 + qiqi(Minkowski) → −(iq0)2+ qiqi= (q0)2+ qiqi(Euclidean)

4

If this seems complex, think of the case n = 3 where x3 = z = r cos θ2 , with r now the three-dimensional

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which we now wish to express in terms of r and the angles θ1, ..., θn only. To change the differentials we use Z dxn+1dr0 = Z |J| dr dθn, (17)

where J is the Jacobian determinant, which in this case equals J = ∂xn+1 ∂r ∂r0 ∂θn −∂xn+1 ∂θn ∂r0 ∂r = r cos 2θ n+ r sin2θn= r .

Inserting this expression into (17), we get Z

dxn+1dr0 =

Z

r dr dθn,

all in all giving us for (16) Z dn+1x = Z r (r sin θn)n−1dr dθn Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθn−1sinn−2θn−1 = Z rndr Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθnsinn−1θn,

which is exactly (14) for n → n + 1, meaning from induction that the relation (14) holds for all n.

Using the just proven relation, we can switch from Cartesian to spherical coordinates and write I(n, α) as I(n, α) = i Z dq q n−1 (q2+ m2)α Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθn−1sinn−2θn−1 = iIq(n, α) Iθ(n, α) , (18) where we have written the integral as a product of the angular integral, denoted by Iθ, and

integral over the length of qµ, denoted by Iq. The actual evaluation of I(n, α) can be carried

out using the beta function, defined by

B(x, y) ≡ Z 1

0

tx−1(1 − t)y−1dt , (19)

which has the property

B(x, y) = Γ(x) Γ(y)

Γ(x + y) , (20)

where Γ(z) the gamma function, which has the useful properties

Γ(z + 1) = z Γ(z) , (21)

Γ(n + 1) = n!, (n = 1, 2, ...) (22)

and

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three properties we will use later on. Notice the second property of the gamma function follows from its first property.

For the evaluation of the angular integral Iθ, we can rewrite the beta function by

sub-stituting t = sin2(θ/2) into equation (19). The integral will now run from 0 to π and have differential dt =12sin θ dθ. We get

B(x, y) = 1 2

Z π

0

(sinθ2)2x−2(cosθ2)2y−2sin θ dθ

= 1 2

Z π

0

(sinθ2)2x−1(cosθ2)2y−1 sin θ sinθ2cosθ2 dθ =

Z π

0

(sinθ2)2x−1(cosθ2)2y−1dθ .

This result becomes, when substituting x = 1/2 + k/2 and y = 1/2, B(12 +12k,12) = Z π 0 sink(θ2) dθ = Z π 0 sinkθ dθ , (24)

where we performed the last step through a geometric argument shown and explained in figure 6. If we now use the property of equation (20), we get the result

Figure 6: Comparison of the integral of sin(θ) to sin(θ/2), with θ from 0 to π and k = 1. To realize the integrals of equation (24) are equal for any k, notice that half the area of the integral of sin(θ) is the same as that of the integral of sin(θ/2), but with half a base of π/2 instead of π. This means the two areas are equal, since the total area of integration equals h(θ) b, where b denotes the base and h(θ) the height dependent on θ, and the only difference between half the area of the integral of sin(θ) and the full area of the integral of sin(θ/2) is that the latter has a two times as large width of base. Since choosing a different k would change both sink(θ/2) and sink(θ) equally, their relative height would

stay the same, meaning that the areas would remain equal to each other and thus, the integrals would as well.

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B(12 +12k,12) = Z π 0 sinkθ dθ = Γ( 1 2) Γ( 1 2 + 1 2k) Γ(1 +12k) , (25)

which we apply n − 2 times, counting for every integral over a sin θ term, giving us Z 2π 0 dθ1 Z π 0 dθ2sin θ2 · · · Z π 0 dθn−1sinn−2θn−1= 2π (Γ(12))n−2 Γ(1) Γ(32) Γ(32) Γ(2) Γ(2) Γ(52)· · · Γ(n2 − 12) Γ(n2) . Notice that every Γ term, except Γ(1) and Γ(n2), disappears. Using this fact combined with the properties of equations (22) and (23), we get for Iθ(n, α)

Iθ(n, α) =

2π(√π)n−2 Γ(n2) =

2πn/2

Γ(n2) . (26)

For the evaluation of integral Iq(n, α) we rewrite this integral to one of the forms of the

beta function. To do this, we first perform the substitution q = m l in Iq(n, α), giving us

Z dq q n−1 (q2+ m2)α = m Z dl (ml) n−1 ((ml)2+ m2)α = m n−2α Z dl l n−1 (l2+ 1)α. (27)

With the m factor now taken out of the integrand, we will furthermore perform the substitu-tion l2 = z/(1 − z) and n = 2β, where the latter substitution is for convenience’s sake. The integral will now run from 0 to 1 and the differential will be dl = 1/(2z1/2(1 − z)3/2), all in all giving us mn−2α Z dl l n−1 (l2+ 1)α = m2(β−α) 2 Z dz ( z 1−z) β−1/2 (1−zz + 1)αz1/2(1 − z)3/2 = m 2(β−α) 2 Z dz z β−1/2−1/2 (1−z1 )α(1 − z)β+3/2−1/2 = m 2(β−α) 2 Z dz zβ−1(1 − z)α−β−1.

We recognize the beta function with the substitutions x = β and y = α − β as the integral in the last line, so filling this in and using the gamma function to beta function connection (equation (20)), we end up with

Iq(n, α) = m2(β−α) 2 B(β, α − β) = m2(β−α) 2 Γ(β) Γ(α − β) Γ(α) . (28)

Inserting this result and the one of equation (26) into equation (18) and reverting the β substitution by inserting β = n/2, we get

I(n, α) = iIq(n, α) Iθ(n, α) = i m2(β−α) 2 Γ(n2) Γ(α −n2) Γ(α) 2πn/2 Γ(n2) = iπ n/2(m2)n/2−α Γ(α − n 2) Γ(α) . (29)

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It is important to know that the gamma function we have used all this time has simple poles for negative integers,5 meaning that for this expression the condition that α − n/2 and α does not equal a negative integer has to be met. It is this observation that forms the basis of dimensional regularization, since if we momentarily analytically continue the integrals to complex values of n, the given expression will always remain finite. This is the reason for what was mentioned in the beginning of this chapter: by substituting n = 4 +  and making  complex, we avoid the poles again using Cauchy’s theorem. Later on, when we have removed the poles by renormalization (as described in the previous chapter), we take the limit n → 4 to get a finite result.

For more general integrals we can employ a different trick of analytical continuation. Using the product rule, we can write for a function f (q)

Z dnq ∂ ∂qµ q µf (q) = ∂qµ ∂qµ Z dnqf (q) + Z dnq qµ ∂ ∂qµf (q)  =∂q 1 ∂q1 + ∂q2 ∂q2 + · · · + ∂qn ∂qn Z dnqf (q) + Z dnq qµ ∂ ∂qµf (q)  = n Z dnqf (q) + Z dnq qµ  ∂ ∂qµf (q)  . (30)

At the left-hand side we recognize, by Gauss’ theorem, a surface integral, which may be dropped if f (q) vanishes sufficiently fast as q tends to infinity. For some functions this may be the case, for some not. The trick of analytical continuation for these general f (q) integrals will now be to drop the surface integrals, regardless of the fact if f (q) actually does or does not meet the vanishing requirement. It can be proven that this assumption does not cause inconsistencies for the integrals and functions we deal with in QFT, but this proof is highly difficult and so, in this bachelor thesis, it will not be further discussed. Having dropped the left-hand side of the equation, we can rewrite equation (30) to

n Z dnq f (q) = − Z dnq qµ ∂ ∂qµf (q)  . (31)

This can be used to express divergent integrals into finite ones for complex values of n. To see how this equation analytically continues functions, let us insert the function f (q) =

5

The Γ function being an extension of the factorial for all possible values, this result is due to the definition of factorials [11]. Taking the factorial for an integer n, we know that n! = n (n − 1)!, which with some rearranging gives us (n − 1)! = n!/n. If we are now interested in the factorial of −1, we see from the found relation we will be dividing by a zero and thus, the factorial will diverge. Since all smaller, negative integer factorials will depend on (−1)!, we conclude every factorial for a negative integer is divergent.

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(q2+ m2)−α into it and see what happens. Using q2 = ηµνqµqν, we get n Z dnq 1 (q2+ m2)α = − Z dnq qµ ∂ ∂qµ  1 (ηµνqµqν + m2)α  = α Z dnq qµ 2ηµνq ν (ηµµ(ηµνqµqν + m2)α+1 = 2α Z dnq q 2 (q2+ m2)α+1 .

From this, it is not immediately clear how we have analytically continued the inserted function. To make this more transparent, we use q2 = (q2+ m2) − m2 on the right-hand side and get

2α Z dnq q 2 (q2+ m2)α+1 = 2α Z dnqq 2+ m2− m2 (q2+ m2)α+1 = 2α Z dnq 1 (q2+ m2)α− m2 (q2+ m2)α+1  , (32) which we can insert into the previous equation and rearrange to obtain

Z dnq 1 (q2+ m2)α = α m2 α −12n Z dnq 1 (q2+ m2)α+1 . (33)

We can now see better how this method shows an analytical continuation. Whereas the original integral (the left-hand side) was only finite for n < 2α, the right-hand side is finite for n < 2α + 2, since if these requirements are not met, the integration will give a logarithmic or even faster diverging term. Notice however that with this method, we have obtained one simple pole for the case n = 2α, marking the only price we pay after every application of this method: we obtain one extra simple pole. The benefits outweigh the disadvantages however, given we have an entire new region to explore. Now we have a method of analytical continuation, we can keep on repeating the procedure of equation (31) to continue the integral to higher and higher values of n until we have reached a point where the integrals are finite near the number of space-time dimensions we are interested in. The extra simple poles that arise can be dealt with using renormalization and are actually the same simple poles that arise in equation (29), just cast in a different form. We will see this later when we perform the  dimensional regularization substitution into equation (29). There is one important side note to all this however: simple poles arise only if the function f (q2) vanishes for large values of the power of q2. If this is not the case, we will actually get higher order poles every time we perform

the described procedure. We can illustrate this if we insert f (q) = (q2+ m2)−αln(q2+ m2) into equation (31) and rearrange to get, similar to equation (33),

Z dnq ln(q2+ m2) (q2+ m2)α = 1 2n m2 (α − 12n)2 Z dnq (q2+ m2)α+1 + α m2 α −12n Z dnq ln(q2+ m2) (q2+ m2)α+1 . (34)

Notice the double pole at α = n/2 that has appeared in addition to the simple pole at that value.

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The question could arise if this method, with the surface integral vanishing approximation mentioned previously, is actually consistent with equation (29). It is, but for extra assurance, we will check it for equation (33). We recognize I(n, α) on the left-hand side of equation (33) and I(n, α + 1) on the right-hand side. Using the Γ(1 + z) = z Γ(z) property of the gamma function (equation (21)) and the result of equation (29), which we fill in for I(n, α + 1) on the right-hand side of equation (33), we see if we indeed obtain I(n, α). We get

α m2 α −12n Z dnq (q2+ m2)α+1 = α m2 α − 12nI(n, α + 1) = α m2 α −12niπ n/2(m2)n/2−α−1Γ(α − 1 2n + 1) Γ(α + 1) = α m 2 α − 12niπ n/2(m2)n/2−α−1(α − 1 2n) Γ(α − 1 2n) α Γ(α) = iπn/2(m2)n/2−αΓ(α − 1 2n) Γ(α) = I(n, α) ,

with which we have confirmed that the relation of equation (33) holds for equation (29). To get to the heart of the matter, we now perform the method of regularization we described in the beginning of this chapter: we substitute n = 4 + , keeping track of poles at  = 0 and later on, once the poles have been removed, take the limit  → 0 (or equivalently, n → 4). Now that we have an evaluated integral (equation (29)) to use this method on, we can see how this method is put into practice. We take the case α = 2 and insert the substitution n = 4 +  to obtain I(n, 2) = iπ2(πm2)/2 Γ(− 1 2) Γ(2) = iπ 2(πm2)/2Γ(−1 2) , (35)

where we used Γ(n + 1) = n! (equation (22)). At this point we are only interested in seeing exactly where the poles arise, therefore, we perform a series expansion of the functions to make the position of the poles more explicit. The Laurent expansion for the gamma function Γ(−12) around  equals Γ(−12) = −2  − γE−  2 π2 12 + 1 2γE  + O(2) .

We stopped writing the expansion after the 2 terms, since these will disappear in the limit  → 0. For (πm2)/2 we will perform a Taylor expansion using xε= exp(ε ln x). We obtain

(πm2)/2= 1 +  2ln(πm

2) + O(2) , (36)

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equation (35), we end up with I(n, 2) = iπ2h1 + 2ln(πm 2) + O(2)ih 2  − γE−  2 π2 12 + 1 2γE  + O(2)i = iπ2 h2  − γE− ln(πm 2) + O()i = −2iπ2 h1  + 1 2γE+ 1 2ln(πm 2) + O()i.

It seems this result is correct as is, but another adjustment needs to be made. As mentioned in the previous chapter, because we are using natural units, everything is in units of mass. This includes this integral, which is supposed to have the dimension [mass]. Looking at the

obtained expression however, it can be seen that the right-hand side is unitless. We correct this by inserting an arbitrary reference mass µ to get

I(n, 2) = iπ2µπm

2

µ2

/2

Γ(−12) . (37)

Notice that we simply multiplied by a factor (µ/µ) and so have the same expression as equation (35). In the expansion however, this expression becomes

I(n, 2) = −2iπ2µ h1  + 1 2γE+ 1 2ln πm2 µ2  + O() i , (38)

which is now dimensionally correct because the µ mass term moved to the front, giving us the required [mass] dimension, with the argument of the logarithm being dimensionless as well. It seems worrying that we have to introduce vague mass terms just to keep everything dimensionally correct. This is a typical example of something that is a ‘feature, not a bug’: later, when we introduce the renormalization group, this arbitrary mass will play an important role.

It can be now be clearly seen that for this single loop integral a single pole arises, just as with the general integral analytic continuation procedure of equation (31). As expected, more loops in a Feynman diagram will lead to more poles. It would then seem this is due to poles arising from lower-order diagrams entering into higher-order calculations and getting integrated over. In that case, integration over poles encountered at one-loop order would enter into the two-loop calculation and through integration in this calculation, become double poles. They would afterwards enter into the three-loop calculation and get integrated over again, leading to triple poles and so on, up to the order considered. In the previous chapter, however, we already showed how at the one-loop order we had accounted for all simple poles using appropriate counterterms in the Lagrangian. This would mean that, once we would begin integration for the two-loop order diagram, the simple poles would already have been removed and can therefore not be the reason more poles arise at higher-order. The actual

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reason the higher-order poles arise is because of something mentioned in the previous chapter: we encounter logarithms of the momenta at the one-loop level, as well as combinations of functions which will subsequently enter into the integrand of the two-order loop and when integrated over, will lead to a double pole. Specifically, the analytic continuation of integrals with logarithmic functions or combinations of functions, needed to perform evaluation at the number of space-time dimensions we are interested in, is what causes the higher-order poles to arise. This was shown at equation (34) for a logarithmic function giving rise to a double pole. We can therefore see from induction that this procedure will give m-order poles for a diagram with m loops (as was also mentioned in the previous chapter), even though we remove all lower-order poles through renormalization when evaluating a loop diagram that is one order higher.

4.2 Feynman parametrization and methods of integral evaluation

The evaluation of the integral mentioned in the previous subsection of this chapter was quite lengthy. There exist other, easier techniques for evaluating the integrals we often encounter in QFT. For example, for an integral with the function fµ(q) = qµ/(q2+ m2)αwe can perform

the action q → −q and see that Z +∞ −∞ dnq qµ (q2+ m2)α = − Z −∞ +∞ dnq (−qµ) ((−q)2+ m2)α = − Z +∞ −∞ dnq qµ (q2+ m2)α, (39)

and since for any object A for which −A = A it can be concluded that A = 0, we can conclude that for this integral

Z

dnq qµ

(q2+ m2)α = 0 . (40)

Using a similar argument, there is a trick to evaluate an integral with an integrand with fµν(q) = qµqν/(q2 + m2)α. Since terms calculated in QFT must be Lorentz-invariant, the

integral with this integrand must be proportional to a Lorentz-invariant tensor. Additionally, since it does not matter whether qµqν or qνqµ is written in the integral, the tensor should be

symmetric under interchange of the µ and ν indices as well. The only tensor that has these two properties is the Minkowski metric ηµν, so this must mean that

Z

dnq qµqν

(q2+ m2)α = a ηµν. (41)

To find a, we contract with ηµν and use ηµµ= n and qµqµ= q2 to get

a = 1 n Z dnq q 2 (q2+ m2)α , (42)

we rewrite the right-hand side Z dnq q 2 (q2+ m2)α = Z dnqq 2+ m2− m2 (q2+ m2)α = Z dnq  1 (q2+ m2)α−1 − m2 (q2+ m2)α  , (43)

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we recognize the integral I(n, α−1) and I(n, α) that we encountered in the previous subsection and so we can finally write for equation (41),

Z

dnq qµqν

(q2+ m2)α = aI(n, α − 1) − m

2I(n, α) . (44)

These methods of integrations are called symmetric integration.

These are useful tricks, but we will have to need other tools to evaluate more realistic integrals. The ’typical integral’ we evaluated in the previous section served as an example and was not necessarily realistic. Integrals corresponding to realistic Feynman diagrams involve integrands with product of propagators of different momenta, making the evaluation harder. Feynman figured out a handy technique, called Feynman parametrization, for evaluating these types of integrals. Feynman observed that

1 AB = Z 1 0 dx [Ax + B(1 − x)]2 , (45)

something we can verify fairly easy. First we rewrite, 1

AB =

(A − B)2 AB(A − B)2 =

B2− AB + A2− AB

(AB − B2)(A2− AB)

= 1 AB − B2 − 1 A2− AB = 1 A − B 1 B − 1 A  = 1 A − B Z B A du u2 , (46)

after which u = Ax + B(1 − x) is inserted. The differential will then become du = (A − B)dx and x will go over the interval 0 to 1. All in all, this changes the integral to give

1 A − B Z B A du u2 = A − B A − B Z 1 0 dx [Ax + B(1 − x)]2 = Z 1 0 dx [Ax + B(1 − x)]2 , (47)

as wanted. The same trick exists for the three term case, 1 ABC = 2 Z 1 0 dx Z 1 0 dy [Ax + By + (1 + x − y)C]3, (48)

which can be extended to n-terms, giving 1 A1A2· · · An = (n − 1)! Z 1 0 dx1 Z 1 0 dx2· · · Z 1 0 dxn δ(1 − x1− x2− · · · − xn) [A1x1+ A2x2+ · · · + Anxn]n , (49)

where δ(x) is the Dirac delta function. This trick can be of great use in the more difficult in-tegrals encountered in QFT. Take for example the following integral that is often encountered in QFT, I(k2, m21, m22) = 1 (2π)n Z dnq ((q + 12k)2+ m2 1)((q − 12k)2+ m22) . (50)

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If we write A = (q +12k)2+ m21 and B = (q −12k)2+ m22, we can insert the identity of the two-term Feynman parametrization (equation (45)) into this equation. The integral in equation (50) is only convergent for n < 4 and so, for n < 4, we can flip the order of integration,

I(k2, m21, m22) = 1 (2π)n Z 1 0 dx Z dnq [Ax + B(1 − x)]2. (51)

It is important to note that this can only be done if the integral is convergent, since otherwise we could be calculating a different outcome than the original integral would have given. The swapped integration might give a finite answer, while the original did not.

To continue our evaluation of I(k2, m21, m22), we take a closer look at the denominator of equation (51). We fill in the full A and B expressions again and write the denominator out,

((q +12k)2+ m21)x + ((q −12k)2+ m22)(1 − x)

= (q2+ q k + 14k2+ m21)x + (q2− q k + 14k2+ m22)(1 − x)

= q2+ 2q(x −12)k +14k2+ m22+ (m21− m22)x . (52)

If we introduce Kµ = (x − 12)kµ and M2 = 41k2+ m22+ (m21− m2

2)x we can write equation

(52) as q2+ 2 q · K + M2. If we now also introduce variable Q = q + K, we see that we can write the denominator as

q2+ 2 q · K + M2 = (q + K)2− K2+ M2= Q2+ M2− K2. (53)

We also get differential dQ = dq since k has no q dependence and so, equation (51) becomes I(k2, m21, m22) = 1 (2π)n Z 1 0 dx Z dnQ (Q2+ M2− K2)2. (54)

We recognize I(n, 2) but with M2− K2 instead of m2, so using what we obtained at equation

(29), filling in α = 2 and m2 = M2− K2, we get

I(k2, m21, m22) = iπ n/2 (2π)n Γ(2 − n2) Γ(2) Z 1 0 dx (M2− K2)2−n/2 = i(4π)−n/2Γ(2 − n2) Z 1 0 dx (M2− K2)2−n/2 = i(4π)−2Γ(2 − n2) Z 1 0 dx ((4π)−1M2− K2)2−n/2, (55)

where we used Γ(n + 1) = n! (equation (22)) and kept (M2− K2)2−n/2 inside the integral

since M and K have a dependence on the variable x. To make the existence of poles clearer, we reintroduce  = n − 4 and expand the result as we did with I(n, 2) in equation (38) (after

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we already introduced the arbritrary reference mass µ), where now (πm2)/2 in I(n, 2) has been replaced by ((4π)−1(M2− K2))/2 to get

I(k2, m21, m22) = −2iµ  16π2 h1  + 1 2γE+ 1 2 Z 1 0 dx ln M2− K2 4πµ2  + O() i = −iµ  8π2 h1  + 1 2γE+ 1 2 Z 1 0 dx ln k2x(1 − x) + m2 2+ (m21− m22)x 4πµ2  + O() i , (56) where in the last step we wrote out M2−K2in the argument of the logarithm. Aside from the

 poles, we can now see additional poles from the logarithm term when its argument becomes negative. Since x goes from 0 to 1, this will only happen if −k2 > (m

1+ m2)2. Defining a new

variable s = −k2, we can find this criterion by examining the point the argument becomes negative, that is,

−sx(1 − x) + m22+ (m21− m22)x < 0 , (57) where the 4πµ2 term in the denominator was not included since that term will never be negative. Taking the s term to the other side in equation (57), we rewrite to get

s > m 2 2+ (m21− m22)x x(1 − x) = m22  1 x(1 − x)− 1 1 − x  + m 2 1 1 − x = m22  1 − x x(1 − x)  + m 2 1 1 − x = m 2 2 x + m21 1 − x. (58)

Because we have defined s as −k2, we are trying to find the smallest lower bound on s, since this will give the smallest upper bound for k2 for which the argument of the logarithm becomes negative. The argument of the logarithm will then become negative throughout the interval of x from zero to one, as the k2 term will disappear at the start of the interval when x begins infinitesimally close to zero, then return after x will quickly become bigger than this infinitesimal distance, turning the argument negative given k2 satisfies the criterion. To find

the x for which we have the largest upper bound on s for which the criterion of equation (57) is satisfied, we first take the derivative of equation (58),

ds dx > − m22 x2 + m21 (1 − x)2,

and let it equal to zero to find the critical points of the criterion, −m 2 2 x2 + m21 (1 − x)2 = 0 ,

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which we rewrite and take the square root of to find m2(1 − x) = ± m1x .

Since x lies in the interval (0,1) and the masses are not negative, the negative solution of this equation will never be attained. We are therefore only left with one critical point,

xc=

m2

m2+ m1

.

To find the largest s in the criterion, we insert this result into equation (58) to obtain scrit> m22(m2+ m1) m2 +m 2 1(m1+ m2) m1 = m2(m2+ m1) + m1(m1+ m2) = m22+ m21+ 2m1m2 = (m2+ m1)2, (59)

where we can read off from the last step of equation (59) that the smallest k2 for which the argument of the logarithm becomes negative is for k2 = −s < −(m1+ m2)2. If this criterion

is met, then the logarithm, throughout the interval of x, will run into poles. We can find these poles by again writing −k2 = s and equaling the argument of the logarithm to zero,

m22+ (m21− m22) x − s x(1 − x) = 0 ,

which we rewrite in the form of a quadratic equation, x2+ x m2 1− m22− s s  +m 2 2 s = 0 , on which we use the quadratic formula to obtain

x±= 1 2 " s − m21+ m22 s ± r m2 1− m22− s s 2 − 4m 2 2 s # = 1 2s " s−m21+m22± r  m2 1− m22− s 2 − 4m2 2 # ,

where of course 0 < x±< 1. Because of the existence of these zeros of the argument, we see

that the integral I(k2, m2

1, m22) will develop a branch cut in the complex s-plane.

Now that we have a clear image of renormalization and regularization as it is used in QFT, there is a still a much more prominent role to be discussed for a parameter we encountered early on: the arbitrary mass parameter µ.

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5

The renormalization group

It seems at this point, all of renormalization has been covered: we have ways to both expose and remove the divergences which troubled our calculations. However, of many parameters we are still not sure of their explicit dependence on µ. Once we have fitted our theory to experimental results, the renormalized result will be reasonably independent of µ for µ-values near the relevant physical scale. Our theory would then work as intended near the energy scale of the data we used to fit the theory to. In the loop corrections of our perturbation expansion, however, terms of the form ln2(µ/µ0) will enter, where µ0 the original energy scale

of the events we used the experimental data from to fit our theory to [7]. If we now would wish to make predictions and perform calculations at much different energy scales than the one of the experimental data we used to fit our theory to, we would get inaccurate results since we would not account for this change in the coupling constant used in the expansion.

It seems we could avoid this complication by refitting our theory to the experimental data of an event that takes place at the energy scale we would like to make predictions at, since the logarithmic terms would then disappear. While this is an option, the problem remains that if we would like to make predictions about events that take place at energy scales far above the ones regularly encountered on earth, it would become a costly business to perform measurements, or in extreme cases, it may be impossible to perform the required measurements with the current state of technology humanity is at. We would then be limited to what can be measured, not imagined. To circumvent this, we can utilize the parameter µ in a different way.

First, let us consider an n-point Green’s function, denoted by Γ(n)(p

1, ..., pn; g0), which is

composed of the sum of all connected Feynman diagrams with n external lines, as shown in figure 7, where pn stands for the momentum of the particle depicted by the nth external line

and g0 the coupling constant of the theory at the original energy scale µ0. Like any expression

in QFT, this Green’s function will get renormalized through a Z-factor to account for the divergences at loop-order,

Γ(n)(p1, ..., pn; g0) = ZRn/2Γ(n)R (pi; g, µ) ,

where the ΓRthe ’renormalized’ n-point Green’s function and the ZRn/2comes from a different

definition of the n-point Green’s function: Γ(n) can also be defined as

Γ(n)(p1, ..., pn; g0) ≡ hφ(x1)φ(x2) · · · φ(xn)i , (60)

the n-point correlation function, where φ is a field. This shows that, since we redefined fields through φ0 =pZφφ (equation (9)), we have to renormalize every field present in equation

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Figure 7: The n-point Green’s function. The corresponding Feynman diagrams only have their n external lines in common and therefore, the middle is dependent on what Feynman diagrams is being summed over. To stay general, this part is thus shown as a blue circle purely for schematic reasons, not to represent the actual form of the resultant sum. Furthermore, in this picture, the n external lines are represented by the black lines connected to the contributions of the summed Feynman diagrams.

(60), giving us the factor √ZR

n

= ZRn/2 for the renormalized n-point Green’s function. At this point, it is important to note for the next step that for n = 2, the n-point Green’s function equals the propagator. Recall that the physical electron mass was defined as the pole in the fermion propagator, showing that the n-point Green’s function is actually physical. Since it is physical, this in turn must mean that the Green’s function must be independent of any parametrization we have chosen for our theory, in particular, of the arbitrary mass scale µ. We can therefore conclude that

d dµΓ (n)(p i; g0) = d dµ Z n/2 R Γ (n) R (pi; g, µ) = 0 ,

where we can write the full differential more explicitly first through the product rule, d dµ ZRΓ (n) R (pi; g, µ) = Γ (n) R (pi; g, µ) d dµZ n/2 R + Z n/2 R d dµΓ (n) R (pi; g, µ) , (61)

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after which we can use the chain rule to write the dZRn/2 term at the right-hand side as d dµZ n/2 R = Z n/2 R 1 ZRn/2 ∂ZRn/2 ∂µ = ZRn/2∂ ln(Z n/2 R ) ∂ZRn/2 ∂ZRn/2 ∂µ = ZRn/2∂ ln(Z n/2 R ) ∂µ = n 2Z n/2 R ∂ lnZR ∂µ , (62)

and we can do the same thing for the dΓ(n)R (pi; g, µ) term at the right-hand side,

d dµΓ (n) R (pi; g, µ) = ∂g(µ) ∂µ ∂ ∂g + ∂ ∂µ  Γ(n)R (pi; g, µ) , (63)

since the coupling constant g, after renormalization, has a µ dependence. After this, we can insert equations (62) and (63) into equation (61) to obtain

ZRn/2n 2 ∂ lnZR ∂µ + ∂g(µ) ∂µ ∂ ∂g + ∂ ∂µ  Γ(n)R (pi; g, µ) = 0 ,

and since the Z-factor will not equal 0, we can divide by ZRn/2 and multiply by µ to finally obtain n 2 ∂ lnZR ∂µ + ∂g(µ) ∂µ ∂ ∂g + µ ∂ ∂µ  Γ(n)R (pi; g, µ) = 0 ,

which is called the renormalization group equation. This equation is typically written as n 2γ R(g, µ) + βR(g, µ) ∂ ∂g + µ ∂ ∂µ  Γ(n)R (pi; g, µ) = 0 ,

where we have defined

g = g(g0, µ) , βR(g, µ) = µ ∂ ∂µg(µ) g0 fixed , γR(g, µ) = µ ∂ ∂µlnZR(g0, µ) g0 fixed .

Of particular interest to our problem is the βR(g, µ) (beta) function, which encodes how the coupling constant of our theory changes over the change of the energy scale µ at a given loop-order (this dependence on loop-loop-order comes from the dependence of the Z-factor redefinition). The reason our perturbation expansion became inaccurate was because we did not account for this change, instead assuming the coupling constant to be independent of µ. By varying

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the coupling constant over the different µ according to the changes encoded in the beta function, the ln2(µ/µ0) term in our loop-order perturbation expansion can be accounted for

to get accurate, finite corrections at any energy scale. This means that after a single fit of our theory to experimental data at any energy scale, our theory will work as desired, only having to account for the changes in the coupling constant dictated by the beta function. This change in the coupling constant over the energy scale is known as ’running of the coupling’. Another benefit of the renormalization group equation is that its solution can be used to write the renormalized n-point Green’s function dependence on µ in a more exact form than the expansion it would originally appear in, making the renormalized Greens function much easier to handle numerically. This can be done by writing the renormalization group equation as  d dµ + n 2γ R(g, µ)Γ(n) R (pi; g, µ) = 0 , (64)

which is the familiar equation with the full differential of Γ(n)over µ not written out with the chain rule. Solving this equation would give the exact form of the µ dependence.

Returning to the beta function, we could obtain this function by using that the n-point Green’s function has no µ dependence for any parametrization, meaning we could, for example, insert the one that follows from our dimensional regularization at one-loop order into γR(g, µ),

calculate the differentiation of the Green’s function in the renormalization group equation and solve for βR(g, µ). For QED at one-loop order, we would find

βQEDR (α, µ) = 2α(µ)

2

3π ,

where α the fine-structure constant, the coupling constant of QED (since α = e2/4π in natural units, this could have been written in terms of the electron charge parameter e as well) [9]. Since βR= µ ∂ ∂µg(µ), we can write µdα(µ) dµ = 2α(µ)2 3π , (65)

which we can solve to find the change of the QED coupling constant over different energy scales. We rewrite equation (65),

3πdα α2 = 2

dµ µ , and integrate the left-hand side

3π Z α(µ) α(µ0) dα0α0−2= −3π[α−1]µµ0 = 3π 1 α(µ0) − 1 α(µ)  , and the right-hand side,

2 Z µ µ0 dµ0µ0−1= 2[ln µ]µµ0 = 2 lnµ µ0 ,

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which we reinsert and rearrange to obtain α(µ) = α(µ0) 1 −2α(µ0) 3π ln µ µ0 .

Plotting this relation in Mathematica 11.3, we obtain figure 8 in which we see the change of the coupling constant of QED as a function of the energy scale µ. Only for events at a low energy scale comparable to the electron mass do we find the familiar value α ≈ 1371 .

Notice in figure 8 the QED coupling constant continues to increase even for energy scales much higher than µ0. This suggests a far more serious problem than we initially faced when

we considered renormalization: the indefinite growth points to an energy scale where the perturbation theory of QED definitively breaks down. This would happen as follows: the loop corrections in our perturbation expansion will grow larger and larger, to the point they become of the order of the tree-level calculation. The closer the correction gets to that point, the more the one-loop order correction will overpower the original tree-level calculation and with that, will decrease the accuracy of the correction.

Figure 8: The running of the QED coupling, where µ0 the electron mass, µ has been plotted from

0.510 MeV·c−1 to 100·103MeV·c−1 and α(µ

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