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Effective Field Theories Scientific Work submitted by Daniel Jaud on November 2012 Ludwig-Maximilians-University Munich Department for theoretical particle physics academic advisor Prof. Dr. Gerhard Buchalla Effective Field Theories 2 Effective Field Theories Zulassungsarbeit für die erste Staatsprüfung für das Lehramt an Gymnasien eingereicht von Daniel Jaud im November 2012 durchgeführt an der Ludwig-Maximilians-Universität München am Lehrstuhl für theoretische Teilchenphysik unter Betreuung von Prof. Dr. Gerhard Buchalla 3 Effective Field Theories 4 Effective Field Theories Acknowledgement My special thank goes to my supervisor Prof. Dr. Buchalla for giving a brilliant first theoretical lecture on classical mechanics that arouse my interest for theoretical physics without I probably wouldn't have done this work. Also I'm glad for the opportunity that he has given me to write this work as well as the guidance and for taking time to answer all my questions in detail. 5 Effective Field Theories 6 Effective Field Theories Contents 1. Conventions and Notations ........................................................................... 9 2. Introduction.................................................................................................. 10 3. Main Ideas and Introductory Examples .................................................... 11 3.1. Properties and Basic Ideas of an Effective Field Theory ....................... 11 3.2. Examples of Effective Field Theories .................................................... 13 3.2.1. Effective Field Theory of the Blue Sky .......................................... 13 3.2.2. Fermi Theory of Beta Decay ........................................................... 15 4. Techniques and Construction of Effective Lagrangians .......................... 17 4.1. A Short Introduction into the Path Integral Formalism .......................... 17 4.1.1. Feynman's View of Quantum Mechanics........................................ 17 4.1.2. Path Integral Formalism in Various Problems ................................ 22 4.1.3. Fields in Path Integral Formulation................................................. 26 4.2. Dimensional Analysis............................................................................. 27 4.3. Symmetries ............................................................................................. 30 4.3.1. Lorentz Invariance .......................................................................... 30 4.3.2. Gauge Transformations ................................................................... 31 4.3.3. Parity and Charge Conjugation ....................................................... 44 4.4. Constructions of Effective Lagrangians ................................................. 48 4.4.1. General Concepts ............................................................................ 48 4.4.2. The Matching Procedure ................................................................. 52 4.4.3. The Fermi Theory once again ......................................................... 57 5. The Renormalization Procedure ................................................................ 63 5.1. Renormalization ..................................................................................... 63 5.1.1. Concepts of Renormalization .......................................................... 63 5.1.2. Renormalization in the Context of Theory ............................... 70 5.1.3. Renormalization Schemes ............................................................... 77 5.2. The Power Counting Method ................................................................. 78 5.3. Renormalization in an Effective Field Theory ....................................... 86 5.4. Matching at One Loop Level .................................................................. 90 6. Photon-Photon Scattering ........................................................................... 95 6.1. The Euler-Heisenberg-Lagrangian (EHL) .............................................. 95 6.2. Photon-Photon Scattering as a QED Process ......................................... 97 6.3. The Scattering Amplitude in QED ......................................................... 99 7. Abstract ...................................................................................................... 112 8. References ................................................................................................... 113 7 Effective Field Theories 8 Effective Field Theories 1. Conventions and Notations Throughout the entire paper we will use the following convention and notation. Some of them are noted again during the text where it seems important to remember. • We will use the metric in the following form, 1 µ = 0 0 − • We will use natural units, i.e. • We will use in some places the short notation, ℏ==1 ≔ • The S-matrix expansion is given by, ' = () where, • − … … !{ℋ$% … ℋ$% } ! ℋ$% = −ℒ$% Throughout the work we will call ℒ the Lagrangian despite it is in fact the Lagrangian density. 9 Effective Field Theories 2. Introduction This paper covers the topic of effective field theories in the context of relativistic fields. The main object of this paper will be to give an overview of what effective field theories are and which techniques are used to construct an effective Lagrangian under given constraints. The whole work requires the knowledge of a first semester quantum field lecture e.g. quantum electrodynamics (QED) only. Because the theory of interacting fermion fields is well known from QED we will develop many concepts of an effective field theory with the help of scalar field theories because this topic is often not covered in depth in the lectures. The structure of the work will be a short introduction with first examples of effective field theories. Afterwards we will briefly discuss the path integral formulation of quantum mechanics and will give a few examples to illustrate the new concept of describing quantum mechanics. It's to be mentioned that the path integral formulation is just a side topic in this paper and its purpose is to illustrate later chapters and to simplify some calculations, in general the theory of effective field theories could be discussed without knowing anything about path integrals. Afterwards we will learn about special techniques that simplify the construction of an effective Lagrangian. The most important concepts will be dimensional analysis and gauge transformations as well as Lorentz invariance. In the chapter of gauge transformations we will explicitly show how the requirement of gauge invariance will lead to couplings between matter and electromagnetic fields, therefore we will cover in detail the complex scalar field and develop the Feynman rules for complex scalar electrodynamics. After we have learned about the techniques to construct an effective Lagrangian we will explicitly begin constructing some examples and will also show how some Lagrangians can be gained out of a generic theory (e.g. quantum electrodynamics QED) and discussing also an example introducing the so called matching procedure. Leaving the subject of construction behind we will concentrate on the question if a theory is renormalizable. Therefore we will first have a short overview of renormalization in the + theory before we look at renormalization in an effective field theory. Also we will derive a very powerful tool, the power counting method, for checking if the renormalization procedure can be achieved or not. Finally we will use all our gained knowledge to tackle the main calculation of this work. We will show how soft photon scattering is described by the EulerHeisenberg effective Lagrangian (EHL) and also show that all infinities cancel out by considering the corresponding QED process. 10 Effective Field Theories 3. Main Ideas and Introductory Examples 3.1. Properties and Basic Ideas of an Effective Field Theory As an introduction in the topic of effective field theories we will first look at a simple example of electron-electron scattering in quantum electrodynamics (QED). In lowest order perturbation theory we have the corresponding Feynman diagram (fig. 1 (left)) from which the scattering cross section for this process can be calculated. If we are now considering higher orders in perturbation theory we find loop diagrams (fig.1 (right)). If one wants to calculate the corresponding scattering amplitude we would find that the result would be divergent. This is a serious problem because the divergent result tells us that we can tell nothing about what is happening. It took about 20 years to solve this problem. The solution is given by the so called renormalization procedure. The basic idea of renormalization is of redefining free parameters of the theory i.e. the mass, charge or coupling constant such that the divergences are absorbed by the redefinition. Applying the procedure one finds that we now obtain finite answers. Because of the great success of the renormalization procedure it was widely believed that every wise theory describing quantum fields should be renormalizable. Requiring renormalizability, quantum chromodynamics (QCD) and the theory of weak interactions were developed. ,- ,- ,- ,- . ,- ,- ,- fig. 1 Electron scattering in lowest order (left), example for electron scattering at first loop order (right). 11 ,- Effective Field Theories A special characteristic of all mentioned theories is that each contains a finite set of terms only. In QED for example the interaction term is given by 0 . 1 Ψ2 where the theory predicts renormalizability (more on that in the −,Ψ chapter power counting method). Nevertheless there are other terms which are on the one hand Lorentz invariant as well as gauge invariant that don't appear in the Lagrangian. So the natural question arising is why these terms don't appear and what would be the consequences if we would add those terms to the theory. We will later see that the theory wouldn't be renormalizable if we would add arbitrary interaction terms, that could not be compensated by corresponding counter terms, so this would be a serious problem because our theory would again give no predictions (infinities). In the chapter renormalization in an effective field theory we will see that a theory can be renormalizable again if we add up an infinite number of terms, which is on the other hand again a problem because how can anyone calculate an infinite set of terms? Now let's focus on what an effective field theory is made of. As main references we refer to [1.], [2.], [3.], [4.], [14.], [17.] and [20.]. The basic idea of any effective field theory is not to describe a system in every aspect but only up to a given energy scale. That means we are restricting ourselves to the low energy limit, where low is associated to some given energy scale Λ characteristic for the system, and following from Heisenberg's uncertainty principle viewing only low energies is equivalent of saying viewing long distances 1. Because we are now describing our system in a given low energy regime we might expect that we don't need the full Lagrangian of an underlying fundamental theory, for example the QED Lagrangian, but a simpler Lagrangian containing only several interaction terms that will produce the same results as the full theory calculated in the given low energy case. The reason there are in general a finite set of interaction terms is that if we are working in the low energy limit we can order terms in powers of their energy/Λ, so working with energies below Λ higher order terms won't significantly contribute to the solution and can therefore be neglected. The task will be to construct an effective Lagrangian (that's why it's called effective field theory) that describes the same physics in the given low energy limit and also, if required, take symmetry or invariance requirements into account. Thereby we will see that some Lagrangians that describe the system very well won't be (for the beginning) renormalizable. We will describe how to solve this problem, a more detailed derivation will be presented in the chapter renormalization in an effective field theory. Because we are constructing our effective Lagrangian such that we describe the low energy limit, it seems natural that the interaction terms in our Lagrangian are ordered in increasing powers of the inverse energy which is equivalent to saying in powers Δ5∆ ≥ so for low energy, i.e. low momentum, Heisenberg predicts high 9 uncertainties in space. 1 ħ 12 Effective Field Theories of inverse mass (related to Einstein's famous formula : = ; 9 ). So again, if we are only viewing the low energy limit higher orders in energy will contribute little so in the end those terms can be neglected and we again can perform our renormalization procedure. That's the basic concept of an effective field theory. Throughout the whole work we will construct step by step concepts that will help us in our process of constructing effective theories. In the following chapter we will present two introductory examples giving a first impression on how an effective Lagrangian can be constructed. 3.2. Examples of Effective Field Theories 3.2.1. Effective Field Theory of the Blue Sky For this part we refer to the references [1.] and [14.]. First of all we draw your attention to a spinless particle for example the pion or on the other hand a spinless and neutral atom. The particle is described in quantum field theory by the complex scalar field2 Φ and its Lagrangian is given by ℒ = Φ= Φ − >9 Φ= Φ. In general we don't have to describe the particle via the complex scalar field but rather through a real scalar field. One of the main advantage of an complex scalar field is that its transformation properties directly lead to coupling with charged particles so the real scalar field Lagrangian only describes spinless neutral particles which therefore, at a first glance, don't interact with the electromagnetic field3. We can now start to think about which lowest order dimensional terms, that are on the one hand Lorentz invariant and on the other hand "truly" gauge invariant (here we write "truly" and mean that the term is gauge invariant for its own, in the chapter gauge transformations we will see that we can also construct gauge invariant terms that are only gauge invariant if they occur coupled). Thinking (and also playing with the possibilities) we would find after a while the lowest order interaction term given by: Φ= Φ?1 ?1 1 >9 2 The great advantage of a complex scalar field is that by Noether's theorem there will be a conserved quantity, e.g. charge or baryon number. 3 It's to be mentioned that the real scalar field can also interact with the electromagnetic field via its higher order multipol moments. 13 Effective Field Theories The reason why the pre-factor inhabits the inverse mass squared will be discussed in detail in the chapter constructions of effective Lagrangians. At this point there is no reason and no need also to go deeper in detail. So more on that later. The only thing to be said, is that it's there to ensure the right dimension of the interaction term. Looking at the interaction term we immediately recognise that it describes the interaction between the particle and the electromagnetic field, so in general we describe a scattering process. So working in the low energy limit where the photon energy :@ is much lower than the mass of the particle ;A there is no reason to treat the interaction relativistically so we can do a non relativistic expansion, i.e. Φ = B, -$C% , where ; corresponds to the mass. The Lagrangian is therefore given by, ℒ = ℒ) + ℒ$% = B = E% − 59 G B + ℒ$% 2 2;A We immediately recognise that if we treat B and B = as independent fields, by Hℒ I M KL the equations of motion HJ equation. Hℒ − HLIM = 0 we reproduce Schrödinger's The interaction term (also obeying gauge invariance in the low energy limit) is given by, ℒ$% = NB = BO PQ + 9 RQ S3 where N, and 9 are some (till now) unknown constants. Naively we would say now that our scattering amplitude is somewhat proportional to V9 because the electric and magnetic fields are proportional to the time derivative of the vector potential 2 which is proportional via, 2 ~ exp[ 4 So in the derivative we gain a factor proportional to V and therefore the whole scattering amplitude ℳ, ℳ~V9 5 From Fermi's golden rule for transitions we now find that the total cross section has to be proportional via, _V~|ℳ|9 ~V 6 14 Effective Field Theories Looking at this formula we recognise that we have just reproduced the famous Rayleigh formula up to a proportionality constant for non-relativistic soft photon scattering off a neutral particle. It is remarkable that we have found the right answer just under the assumption of gauge invariance and Lorentz invariance, so by now it seems clear that symmetries play an important role in constructing effective Lagrangians. In a later chapter we will deal with photonphoton scattering that doesn't occur in classical electrodynamics but is a consequence of quantum field theory. We will also be especially interested in discussing a low energy effective Lagrangian called the Euler-Heisenberg Lagrangian. 3.2.2. Fermi Theory of Beta Decay As another example we will deal with the Fermi theory of beta decay in the low energy (i.e. low momentum) limit (see [2.]). For now we only outline the main idea. A more formal discussion will follow later when we will discuss a similar calculation in the context of muon decay. One of the great advantages of this example is that we don't have to guess the right interaction terms but we are working with the underlying fundamental theory of weak interactions, thereby we won't construct the Lagrangian explicitly but will sketch the basic idea of working out the result. First of all we take a look at the corresponding Feynman diagram representing the beta decay (fig. 2). As we can see the corresponding interaction particle is the so called Wboson which has roughly a mass >b = 80 GeV. The corresponding boson propagator is proportional to, − c9 1 9 7 − >b 15 Effective Field Theories 5h i- ,- j̅l fig. 2 Beta decay in lowest order. If we now restrict our energies to : ≪ >b and therefore c ≪ >b we can perform a Taylor expansion of the W-boson propagator. The propagator becomes, 1 c 9 + f > 8 >b b So we see that we obtained a fourpoint interaction (fig. 3). Therefore we conclude that in the low energy limit our effective Lagranian involves a fourpoint interaction term. The only task remaining is to look for a suitable interaction term giving the same result as if we were calculating in the underlying theory in the low energy limit. In the chapter construction of effective Lagrangians we will explicitly show how such effective interaction terms can be found. The example here had therefore for this purpose illustrated how an effective Lagrangian can be constructed out of a fundamental underlying theory and also that in effective field theories the heavy interaction terms (here the heavy W-boson propagator) are hidden in the coupling constants between the light fields or in other words the heavy fields involved are integrated out and what remains are just interactions between the light fields of the theory. 16 Effective Field Theories 5 ,- j̅l 5 ,- j̅l fig. 3 Transition from a propagation interaction to a four point interaction in the limit c ≪ >b . 4. Techniques and Construction of Effective Lagrangians 4.1. A Short Introduction into the Path Integral Formalism 4.1.1. Feynman's View of Quantum Mechanics The following part has the intention to give a brief introduction into the path integral formalism which is an alternative way of describing quantum mechanics. For this chapter we have used as reference in particular [6.] and [15.] as well as [13.]. We will begin with a short derivation of the formalism and afterwards give a few examples to obtain a better feeling of the whole story. Before we go into the derivation it is to be said that the path integral approach is absolutely equivalent to the Schrödinger description of quantum mechanics. Nevertheless the path integral formalism opens a new view of how strangely quantum mechanics works. Often it is irrelevant with which formalism we work, but in some problems it is much easier to work in one or the other domains. We now first begin with the derivation, therefore we start with two position kets |⟩ in coordinate space that indicate a particle at an initial |$ ⟩ and 17 Effective Field Theories a final position |n ⟩4 . Further on we know that if the particle is at position $ at initial time o$ and we want to know how the particle envolves during the time from o$ to a later final time on we find, exp E− pOon − o$ SG|$ ⟩9 ℏ So we find the probability amplitude for a transition from place $ to a place n in a time interval on − o$ , rOn , on ; $ , o$ S = tn u exp E− pOon − o$ SG |$ ⟩10 ℏ It is conventional to write r instead of ℳ in the given context. This approach is straightforward and just follows from the Schrödinger formulation of quantum mechanics but now we are taking the nontrivial step from which we’ll derive the new concept. For simplicity we introduce a new notation which simplifies the calculation afterwards, we set |C ⟩ ≔ |oC ⟩ Now we are dividing the time interval into v equal parts : w = The probability amplitude therefore reads as, %x -%y z z r = tn u { exp − pw |$ ⟩11 ℏ |( After each small time interval we can insert a 1 via the completeness relation, 1 = C |C ⟩}C |12 So we can rewrite the amplitude, $ $ r = 9 ⋯ z- }n |, -ℏ |z- ⟩}z- |, -ℏ |z-9 ⟩ ∙ ⋯ ∙ $ ∙ } |, -ℏ |$ ⟩13 In the following we are concentrating on one general factor because the calculation for the other factors is the same. 4 We are assuming that the kets |$ ⟩ are already normalized to 1. 18 Effective Field Theories So we have the general factor }Ch |, -ℏ |C ⟩ y Again we are inserting a 1 except this time via 1 = 5C |5C ⟩}5C | So we obtain, $ $ }Ch |, -ℏ |C ⟩ = 5C }Ch |, -ℏ |5C ⟩}5C |C ⟩ = 5C = exp − pw + Ch − C 5C 14 2ℏ ℏ ℏ where we have used the well known formula, t u5| = 1 exp 5| 15 ℏ √2ℏ for distinctness it is to be said that p = pC , 5C A A general Hamiltonian is given by p = 9C + C . Therefore we can compute the integral (14) via completing the square and using the formula for a gaussian integral, so we find, $ }Ch |, -ℏ |C ⟩ = ; w ; Ch − C 9 exp − C 2ℏw w ℏ 2 16 The whole probability amplitude therefore becomes, r = ⋯ z- = ⋯ z- z z- ; 9 w ; Ch − C 9 { exp − C = 2ℏw ℏ 2 w C() z z- ; 9 w ; Ch − C 9 exp − C 2ℏw ℏ 2 w C() 19 17 Effective Field Theories This is the overall expression. We now take the limit v → ∞ which equals w → 0 and are calling the expression, lim ⋯ z- z→' z ; 9 =: 18 2ℏw if we are looking at the exponential we recognize that in the limit w is going to be a differential time interval o and therefore the sum is going to be an - integral, furthermore we see that in the limit = % = So we obtain the limiting expression, %x ; r = exp o E 9 − G 19 ℏ %y 2 Looking at the integral expression in the exponent we recognize nothing else % C but the definition of the action = % x o 9 9 − so we can write, y rO$ , o$ ; n , on S = exp 20 ℏ The big question arising is how to interpret the given formula. To answer this question we use a graphical representation showing what exactly we are doing in our calculation (fig. 4). The key idea is the following: In our calculation we are connecting the initial and the final spacetime point by all possible intermediate steps. So in the end we obtain a very large number of paths how the particle could move. For each path in spacetime we calculate the action and in the end add up all the corresponding exponentials. So the particle chooses every possible path in spacetime even those that violated physical laws (e.g. the speed of light isn't the upper limit any more). So we summarize the calculation of the path integral in the following three steps: 1. "Draw" all connected paths form the initial to the final state 2. Calculate the action of each path 3. Sum up all paths, where by "sum up" we $ ∑¤¥¥A¤%¦§ exp 5¢oℎ ℏ 20 mean Effective Field Theories x(t) t fig.4Spacetimediagramshowingapossiblepathredpathandtheclassicalpath greenpath. As a further property we are considering the classical limit ℏ → 0. In this limit the exponential oscillates wildly so the net sum will be 0 except those terms where the action reaches becomes stationary under variation. In this case there will be equal contributions from the exponentials and therefore these terms won't cancel each other out. So in the classical limit we see that only the paths that lead to stationary action contribute. This can be written as· = 0from which follows directly the Euler Lagrange equation which is the well known formula for classical motion. So we see that in the classical limit, the path integral formulation delivers the classical result for motion directly. 21 Effective Field Theories 4.1.2. Path Integral Formalism in Various Problems We are now in the position to consider two simple examples. The first example we will deal with is the free particle, we present it in a way discussed in [6.]. We will do the calculation explicitly to show how in principle the path integral is calculated. Afterwards we will consider the example of reflection due to Feynman. Thereby we won't do any calculations but give a graphical representation of what we would do in our calculation that also gives a good impression why the classical path (reflection law) is preferred. So let's start with the free particle = 0. We begin with (17), = ⋯ z- rOn , on ; $ , o$ S = z- z ; 9 w ; Ch − C 9 exp 21 2ℏw ℏ 2 w C() Now we are considering the first integration overz- therefore we recognize that we have to consider two terms of the exponential, w; z(n − z- 9 z- − z-9 9 + G¹22 exp ¸ E 2ℏ w w So we see that by completing the square the integral over z- leads to a Gaussian integral. The solution is, w; z(n − z- 9 z- − z-9 9 + G¹ = z- ,5 ¸ E 2ℏ w w = ℏw 9 ; − z-9 9 »23 exp º 2ℏw ∙ 2 z ; Having solved the first integral we can go on calculating the next one, so we find, z-9 exp º ; ℏw 9 ; − z-9 9 » = z-9 − z- 9 » exp º 2ℏw ; 2ℏw ∙ 2 z 22 Effective Field Theories 9 ℏw 9 ; − z- 9 »24 = exp º ; 2ℏw ∙ 2 z So far we have neglected the overall factor. We see that we have v − 1 C ½ integrals but a factor of9$¼ℏ so we now multiply to each integral a factor C 9$¼ℏ except for the first integral where we multiply by for the first integral, C 9$¼ℏ ,we obtain 9 ; 9 ℏw 9 ; − z-9 9 » = exp º 2ℏw ; 2ℏw ∙ 2 z 9 ; ; − z-9 9 »25 = exp º 2ℏw ∙ 2 2ℏw ∙ 2 z for the second integral we find after multiplying, 9 ; 9 ; ; − z- 9 »26 = exp º 2ℏw ∙ 3 2ℏw ∙ 3 z 2ℏw We see that by multiplying one easily finds a recursion formula which tells us that after [ − 1 steps we obtain, 9 ; ; − z- 9 »27 exp º 2ℏw ∙ [ 2ℏw ∙ [ z With the help of this we see that, 9 ; ; 9 rOn , on ; $ , o$ S = exp º − )($ S »28 O 2ℏw ∙ v 2ℏw ∙ v z(n Through the whole derivation we used z = n and ) = $ therefore we have chosen the expressions z(n and )($ . By looking again at the definition of w we see furthermore that vw = on − o$ , so we can finally write the probability amplitude as, 23 Effective Field Theories rOn , on ; $ , o$ S = E 9 ; ; 9 G exp ¸ Oz(n − )($ S ¹ 2ℏon − o$ 2ℏon − o$ so for the total probability we find, ¾On , on ; $ , o$ S = |r|9 = ; 2ℏOon − o$ S 29 30 The result is very interesting, on the one hand the limit v → ∞ hasn't been taken because K is already limit free. On the other hand the probability for a particle travelling from $ to n is just a function of time difference. As a further problem we'll do the reflection law described by the path integral formalism. We will strictly follow the Richard Feynman approach he presents in his book QED the strange theory of light and matter [5.]. The question now is what is the probability for a particle (e.g. an electron) travelling from point a to point b? For our problem we are considering that the electron can only travel in a straight line and that when hitting a scattering surface it can change direction. Furthermore we are requiring that the particle has to hit the surface first before travelling on. That means that we don't allow the particle to travel from point a to b in a straight line. We now follow the three steps of calculating the path integral, therefore we first consider a discrete number of paths. The first step is to draw all paths that connect point a and b that are possible by the mentioned boundary conditions see (fig. 5). In the figure one sees a set of possible paths which the particle can travel. For simplicity we have labelled the scattering point with the capital letters A to O, so if we are speaking e.g. of path A we mean the path where the particle scatters at point A. By now we have drawn "all" possible paths satisfying the boundary conditions. The next step in the recipe is to calculate the action for each individual path. Again we give a graphical representation, see fig. 6. 24 Effective Field Theories fig. 5 Representation of some possible connections valid by the boundary conditions. Picture out of [5.] fig. 6 Representing the individual paths and their corresponding actions. Picture out of [5.] 25 Effective Field Theories Now if we add up all paths ∑A¤%¦§ exp we see that we will always find ℏ two paths such that the corresponding phase factors cancel each other out except those parts near path H because the action is stationary under small variations of the path. So we conclude that the significant contributions arise from all paths near H. If we look closely we see that for all paths near H we reproduce the classical reflexion law. $ As a next step we will briefly take a look on how to include fields in the path integral formalism. We won't do any example calculations and are quoting the results for completeness only because the theory of path integrals would represent a paper of its own. 4.1.3. Fields in Path Integral Formulation To include fields in the path integral formulation we consider an analogy by first looking on the definition of the probability amplitude ¿n , on Àexp E− pOon − o$ SGÀ $ , o$ Á31 ℏ Now if we are not interested in position space but in field configuration we can simply write by analogy, tΦÂ OÃÄ , on Su exp E− pOon − o$ SG |ΦÅ Ã$ , o$ ⟩32 ℏ So in the path integral formulation we are no longer summing over all paths but summing over all field configurations, therefore we obtain, Å rOΦn , Φ$ S = Φeℏ 33 where the action is now given as, Æ %x = ℒ 34 %y For a real scalar field we find for example the action, 1 = O Φ Φ − ;9 Φ9 S35 2 26 Effective Field Theories We won't discuss further applications here but rather quote that also in the field language the action appears in the exponential. Therefore the action has to be dimensionless. We will use this fact in the chapter dimensional analysis to derive some useful results. 4.2. Dimensional Analysis When we are doing our calculations in SI units it is commonly known that we can express every quantity in terms of mass (;), length (Ç) and time (o). So for example one finds that the velocity has units length per time (= Ç/o). In a quantum field description it proves to be beneficial to work no longer in SI units but rather choose natural units i.e. the most important natural constants ℏ and are set equal 1, so ℏ = = 1. If we are now concentrating especially on relativistic quantum fields, quantities like mass or length are no longer independent but can be related to each other. For an example, we are looking at the length. Using natural constants the unit of length can also be expressed in terms of units of mass via, Ç= ℏ 36 ; An equal example can be found by concentrating on the units of time, o= ℏ 37 ; 9 Therefore we see that every fundamental unit can be expressed in term of unit of mass. In the following we will use the results presented above to do some dimensional considerations. Therefore we will use the dimensional computation rules for dimensional (i.e. mass-) dependent parameters 2, É 5 (for reference see [1.] and [17.]), 5 2É = 2 + É38 2 º » = 2 − É39 É 1 º » = −240 2 We will denote the dimension of a parameter by [...]. 27 Effective Field Theories With these simple rules we look what consequences arise for individual parameters. By doing so, we can now first measure every unit in terms of energy : to some power. So the unit of mass is given by, ; = : h = :141 As seen above the length and time are mass dependent so calculating the dimensions in natural units ℏ = = 1 we obtain, = Ç = 1/; = −; = −142 and also by an analogous calculation, o = −143 Furthermore there can be done some more exotic dimensions, for example the derivative, Ê Ë = º » = 144 as well as an D-dimensional volume element, Ì = Í = −Í45 In the chapter of path integral formulation of quantum mechanics we have seen that the action emerges in the exponential. Because the exponential has to be dimensionless we conclude that the dimension of the action has to be, = 046 Looking upon the definition of the action we find, 0 = = º ℒ» = −4 + ℒ47 by this we obtain, ℒ = 448 So we have shown that the Lagrangian density or short the Lagrangian in a four-d quantum field theory has to have dimension four. This simple result will prove useful in constructing effective Lagrangians because we have found a simple criterion which mass dependence the coupling constants will have to fulfill (more on that in chapter construction of effective Lagrangians). Using equation (48) we will now also derive the dimension of the three most important fields in a quantum field theory the scalar field, the Dirac field and the photon 28 Effective Field Theories field. In doing so we will see the common representation of the Lagrangians for free particles. The scalar field Lagrangian is given by, ℒ = Φ Φ= − ;9 Φ9 49 It is no matter which term is used to determine the dimension of Φ. For completeness we will do the calculations in both cases, so we find for the kinetic term: 4 = Φ Φ = 2Ê Ë + 2Φ = 2 + 2Φ50 for the mass term we find, ⇔ Φ = 151 4 = ;9 Φ9 = 2; + 2Φ = 2 + 2Φ52 ⇔ Φ = 153 So as said before both terms give the same result namely Φ = 1. The fermionic Lagrangian is given by, 0 O. − ;SΨ54 ℒ = Ψ from a similar calculation to the one for the scalar field we find, 3 Ψ = 56 2 Remains the photon field. The corresponding Lagrangian is given by, from which follows, 1 ℒ = − ? ? 57 4 Ê? Ë = 2 ⇔ Ê2 Ë = 158 So we have seen that it's easy to determine the dimension of various field configurations. In the following table we will summarize the most important results of this chapter. 29 Effective Field Theories parameters/fields ; o Φ Ψ ? dimensions 1 -1 -1 1 1 3/2 2 4.3. Symmetries 4.3.1. Lorentz Invariance In the following we will shortly review the concept of Lorentz invariance. Our main reference for this chapter will be [10.] and [15.]. Because the concept is used in so many theories (e.g. all theories involving relativistic kinematics) the methods are well known. The basic idea of building up a Lorentz invariant theory is that the theory should be invariant under Lorentz transformations i.e. the result should be independent of the frame of reference or inertial system. To ensure that we have constructed a Lorentz invariant theory there is one simple rule: The Lagrangian has to be Lorentz invariant, i.e. a Lorentz scalar. To give a deeper insight of the rule we will consider three simple examples. 1. As a first example we consider the field strength tensor in electrodynamics ? . The tensor obviously has two open Lorentz indices μ and j. Our rule now states that we have to contract those to other fields that there are no open Lorentz indices left e.g. we could contract it to itself so we would obtain ? ? . Otherwise we could (in principle) contract each index to a momentum 5 , so we would gain ? 5 5 . 2. As another example we look to an interaction term we will derive in the next chapter. The term is given by c 9 2 2 Φ= Φ. Obviously all Lorentz indices are contracted so the given term is definitely Lorentz invariant. 3. At last we will discuss an example which is not Lorentz invariant. The term is given by Ψ = . Ψ. Obviously the term has the open Lorentz index μso the theory we would describe isn't Lorentz invariant and is therefore dependent in which frame we are doing our calculations. 30 Effective Field Theories This short excursion should be enough to remember the basics of Lorentz invariance. As seen by the given examples it is quite easy to build up Lorentz invariant terms. In the next chapter we will be dealing with another symmetry considering gauge transformations. 4.3.2. Gauge Transformations An important requirement when building up an effective Lagrangian can be gauge invariance (see [8.] and [9.]) of the Lagrangian i.e. we can perform a transformation of the fields involved such that the whole Lagrangian is left invariant. In the following we will start with the review of the already known gauge invariant theory, the Maxwell theory of classical electrodynamics. Afterwards we will explicitly deal with gauge invariance of the complex scalar field, thereby we will arrive at a Lagrangian describing complex scalar electrodynamics (SED) for which we will in detail derive the Feynman rules. Having done this important step we will briefly review the QED Lagrangian whose derivation is based on the same requirement of gauge invariance as SED. From all three parts we will work out important results that we will use later to construct effective Lagrangians obeying the requirement of gauge invariance. We will start with the gauge transformations known from classical electrodynamics. As a first example of a gauge invariant quantity we remember the magnetic field. In classical electrodynamics one introduces the vector potential defined via, ÐÑ = Ò × 2Ñ59 É From the discussion of electrodynamics we know that the magnetic field is invariant under the following gauge transformation of the vector potential, 2Ñ → 2ÑÔ = 2Ñ + ÒΛ60 where Λ is some scalar function. This is our first example of a gauge invariant quantity. Because we are interested in fields we will now take a deeper look on special gauge invariant fields. Remembering the electromagnetic field strength tensor ? = 2 − 2 we immediately see that the tensor is invariant under the transformation of the four dimensional vector potential, 2 → 2Ô = 2 + Λ61 This can easily be seen by inserting the transformed potential into the field strength tensor 31 Effective Field Theories Ô ? = 2 + Λ − O2 + ΛS = 2 − 2 = ? 62 So we see that the homogenous Maxwell equations (which are described by the field strength tensor) are already in their gauge invariant formulation. Applying again the requirement of Lorentz invariance we finally find the Lorentz and gauge invariant formulation of the free photon field including the right numerical factor, 1 − ? ? 63 4 Because this is all well known from classical electrodynamics and also from QED, we will end the discussion for basic gauge transformations of the vector potential at this point. Nevertheless it has to be remarked that when speaking of gauge invariance we explicitly mean the invariance under a (in our case only abelian) gauge transformation of the vector potential or the invariance under a field transformation as discussed in the next part. Should we mean something different by gauge transformations in a given context we will explicitly clarify it. Next we will concentrate on the complex scalar field and will study its transformation properties under global and local U(1)-symmetry6. The Lagrangian for a complex scalar field is given by: ℒ = Φ= Φ − ;9 Φ= Φ64 The question now arising is what are the consequences of demanding invariance under U(1) transformations. At first we will consider a global transformation of the field i.e. we are transforming the fields as, Φ → ΦÔ = , $ÕÖ Φ65 Ô Φ= → Φ= = , -$ÕÖ Φ= 66 where c is a constant (later we will see that it refers to a charge the photon field is coupling to) and Λ is independent of spacetime coordinates (the independence is the characteristic of the global transformation). Obviously equation (64) is invariant under such global transformations i.e. U(1)-symmetry group is the group of unitary transformations, in our case multiplication by a phase factor expc×. 6 32 Effective Field Theories ℒΦÔ = , -$ÕÖ Φ= , $ÕÖ Φ − ;9 , -$ÕÖ Φ= , $ÕÖ Φ = = Φ= Φ − ;9 Φ= Φ = ℒΦ67 We now know by Noether's theorem there is a conserved quantity in our case the conserved current Ø , Ø = −OΦ= O ΦS − O Φ= SΦS =: −OΦ= ÙÐÐÑ ΦS68 So we see that from the invariance under global transformations immediately follows a conserved quantity. Now we are in the position to look what happens if we are requiring invariance under local transformations i.e. Λ → Λ and the fields transform like, Φ → ΦÔ = , $ÕÖÚ Φ69 Ô Φ= → Φ= = , -$ÕÖÚ Φ= 70 By inserting the transformations into eq. (64) we obtain, ℒOΦ, ΦS = O, -$ÕÖ Φ= S O, $ÕÖ ΦS − −;9 O, -$ÕÖ Φ= SO, $ÕÖ ΦS = = O−c Λ, -$ÕÖ Φ= + , -$ÕÖ Φ= S ∙ ∙ Oc Λ, $ÕÖ Φ + , $ÕÖ ΦS − ;9 Φ= Φ = = c 9 Λ ΛΦ= Φ − c ΛÊΦ= Φ − Φ= SΦ + + Φ= Φ − ;9 Φ= Φ71 where we have skipped the dependence of Λ in the last line, from now on we will allways skip the dependence but keep in mind that we are talking of Λx. Now looking at the result we see that the Lagrangian is obviously not invariant under such local transformations so we have to modify it such that it becomes ÙÐÐÑ ΦS invariant. To do so we make the following observation: the term −OΦ= represents, the current density, as seen from global transformations. By analogy with electrodynamics we know that a current couples to the vector potential so we interpret Λ as a kind of vector potential and cthe corresponding charge of the particle coupling to the current. As seen in the example of classical electrodynamics the vector potential can be transformed by gauge term. So we 33 Effective Field Theories replace the ordinary derivative by a covariant derivative that will allow us to construct a gauge invariant Lagrangian, so we replace, → Í = − c2 72 where 2 is the corresponding vector potential describing the photon field. We demand the vector potential to transform such: OÍ ΦS → Í Φ′ = Í ′Φ′ = eÅÜÖ D Φ 73 From that we conclude the following transformation property of the vector potential for a U(1) gauge: −c2Ô + cO ΛS = −c2 As a check we calculate, ⇔ 2Ô = 2 + O ΛS74 OÍ ΦS → Í Φ′ = Í ′Φ′ = ∂ − iqA − iq∂ ΛeÅÜÖ Φ = = eÅÜÖ D Φ75 So the transformation is valid. It should be mentioned that 9 ℒ$ = uÍµ Φu = Íµ∗ Φ= Í µ Φ so there is an additional minus sign corresponding to the complex conjugation. So we see that the Lagrangian by including the covariant derivative is left invariant under local gauge transformations obeying the condition that we simultaneously perform a gauge transformation of the vector potential given in eq. (74). The constructed Lagrangian that is gauge invariant under the coupled transformations, Φ → ΦÔ = , $ÕÖÚ Φ76 Ô and Φ= → Φ= = , -$ÕÖÚ Φ= 77 2 → 2Ô = 2 + Λ78 34 Effective Field Theories is given by7, ΦS + c 9 2 2 Φ= Φ ℒ = Φ= ∂ Φ − ;9 Φ= Φ + c2 OΦ= ÙÐÐÐÑ (79) The achieved Lagrangian is gauge invariant under simultaneous transformation of the scalar field and the vector potential. Because the Lagrangian now describes the interaction between spinless charged particles and the electromagnetic field we are forced to add the gauge invariant term for free photons − ? ? . So in the end we constructed the Lagrangian for scalar electrodynamics (SED), ℒãäÌ = Φ= ∂ Φ − ;9 Φ= Φ + c2 OΦ= ÙÐÐÐÑ ΦS − 1 +c9 2 2 Φ= Φ − ? ? 80 4 In the chapter renormalizability we will see that we have to add another interaction term to the Lagrangian but for now we are satisfied with the given solution. We are now in the position to derive the Feynman rules associated with the ΦS interaction terms of the SED Lagrangian. Therefore we treat −c2 OΦ= ÙÐÐÐÑ and −c 9 2 2 Φ= Φ as interactions. Let's start with the free particle solutions, therefore we find for an incoming particle, }0|Φ¢= 5|0⟩ = 5Ô å å = }0| Ê¢5Ô , -$A + æ = 5Ô , $A Ë¢= 5|0⟩ = 2 2V = , -$A 81 where we have used the commutation relation for complex scalar fields Ê¢5, ¢ = 5Ô Ë = 2 2V·5ç − 5çÔ , Êæ5, æ = 5Ô Ë = 2 2V·5ç − 5çÔ and all other commutators are trivial. 7 It's to be said that this Lagrangian is only invariant under coupled gauge transformations. The single terms on their own are not invariant under the given transformations. 35 Effective Field Theories The same calculation can be done for an outgoing particle, }0|¢5Φ= |0⟩ = , $A 82 For the incoming and outgoing antiparticle we find, and }0|Φ= æ= 5|0⟩ = , -$A 83 }0|æ5Φ|0⟩ = , $A 84 It has to be said that these are the only terms that give contribution ≠ 0. All other terms, for example }0|¢= 5Φ= |0⟩ = 0 because the commutator acts trivially in this case. The same can be done with the photon field. We can just quote the results because this is well know from QED, and }0|2 ¢é= [|0⟩ = wé [, -$ 85 }0|¢é [2 |0⟩ = wé [, $ 86 Also we already know from basic quantum field theory how the propagators for the scalar and photon field look like. For the scalar propagator in momentum space we find, 87 59 − ;9 and for the photon propagator in momentum space we have, − 88 [9 Now we come to the interesting part of calculating the vertex functions. For this purpose we will use the perturbation theory method an Wick contractions. We'll begin with the first interaction term −c9 2 2 Φ= Φ . We are considering scattering of an incoming spinless particle off an incoming photon to first order in perturbation theory. The initial state is given as 36 Effective Field Theories ¢= [¢= 5|0⟩ and the final state is given as }0|¢[ Ô ¢5Ô . In first order perturbation theory we have to calculate8, c 9 }0| !ê¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5ë|0⟩89 Now we are using Wick's theorem that tells us that every possible contraction yielding none vanishing entries has to be taken into account. Furthermore we know that the incoming and outgoing particle only couples to the scalar field and that the incoming and outgoing photon only couples to the photon field. Therefore we find two possible contractions: c 9 }0| !ê¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5ë|0⟩ = = c 9 }0| ¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5|0⟩ + +c 9 }0| ¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5|0⟩ 90 Because there is no difference between coupling to the first photon field or the second, both possible contractions give the same result so we can combine the calculation to, c 9 }0| !ê¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5ë|0⟩ = = 2c 9 }0| ¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5|0⟩ 91 8 The zero-th order isn't to be taken into account because it's just representing free -$í propagation solutions, also note that ì = ∑ … !{ℋî … ℋî } ! where ℋî = −ℒî . So in our case ℋî = −Oc2 OΦ= ÙÐÐÐÑ ΦS + c 9 2 2 Φ= ΦS. 37 Effective Field Theories using the results of the free propagating solutions we find, 2c 9 }0| ¢[ Ô ¢5Ô 2 2 Φ= Φ¢= [¢= 5|0⟩ = = 2c 9 w [ Ô w [ exp−5 + [ − 5Ô − [ Ô = = 2c 9 w [ Ô w [2 · 5 + [ − 5Ô − [ Ô = = 2c 9 1 w [ Ô w [2 · 5 + [ − 5Ô − [ Ô 92 So we see because 2 · 5 + [ − 5Ô − [ Ô is just the overall momentum conservation9 and w $ [ for = j, ï are the free propagating photon polarization vectors that the vertex factor has to be, 2c 9 93 From this result the interaction in terms of Feynman diagrams has for example the following representation (fig. 7). [ [′ 2c 9 5 5′ fig. 7 Feynman diagram for scalar particle-photon scattering in scalar electrodynamics. 9 In future cases when calculating the scattering amplitude we will just drop the overall momentum conservation. 38 Effective Field Theories For the antiparticle there can be done the same derivation. Thereby we would see that we would gain the same vertex factor that's why we won't explicitly do the calculation. After having dealt with the first interaction term we now concentrate on the second term. The interesting specialty is that it contains a derivative so naively we expect the vertex factor to explicitly depend on the momentum. As a first step we look what c2 OΦ= ÙÐÐÐÑ ΦS explicitly looks like, c2 OΦ= ÙÐÐÐÑ ΦS = c2 ÊO Φ= SΦ − Φ= ΦË = = c2 ¸E[ ¢= [, $ − [ æ[, -$ G Φ¹ − −c2 ¸Φ= E−[ ¢[, -$ + [ æ= [, $ G¹94 ð where we have used the short notation = 9¼ð 9ñ Now again we want to calculate the vertex factor using first order perturbation theory. It's important to notice that we now have only one photon field so we can either have an outgoing or an incoming photon but not both (corresponding to the interaction term). We are restricting our calculation to the outgoing photon case. The incoming case is then analog in calculation and in result also. So again we have to calculate, ΦS¢= [ Ô ¢= 5ë|0⟩95 }0| !ê¢5Ô c2 OΦ= ÙÐÐÐÑ Now by Wick’s theorem there is only one single possible contraction namely, }0| ¢5Ô c2 ÊO Φ= SΦ − Φ= ΦË¢= [′¢= 5|0⟩ 96 Here we used the notation that the contraction above the equation and under the equation are to be taken separately. This integral can easily be handled, the only difference is now an additional factor of [ so we first had to calculate, 39 Effective Field Theories }0| O−[ ¢[, -$ + [ æ= [, $ S¢= 5|0⟩ = −5 exp−5 97 and also, }0|¢5Ô O [ ¢= [, $ − [ æ[, -$ S|0⟩ = 5Ô exp5Ô 98 By doing the calculation one has to be careful not to forget the additional minus sign in the interaction. This is the reason why in the solution occurs only plus signs. So we finally obtain for the integral eq. (95). }0| !ê¢5Ô c2 OΦ= ÙÐÐÐÑ ΦS¢= [′¢= 5ë|0⟩ = = −cw [ Ô 5 + 5Ô 2 · 5 + [ Ô − 5Ô 99 So once again we can read the vertex factor term. This time we obtain for the vertex factor, cO5 + 5Ô S100 In the language of Feynman diagrams we have (fig. 8) The same results can easily be obtained for the case of an incoming photon and also for the cases of antiparticles. The only difference is that antiparticles have opposite momentum flow characterized by an additional minus sign. 40 Effective Field Theories 5 c5 + 5Ô 5′ [′ fig. 8 Feynman diagram for particle + photon →particle in scalar electrodynamics. We sum up the Feynman rules for scalar electrodynamics (not including the counter-terms necessary for renormalization) 1. External lines • For each incoming scalar particle draw a dashed line with an arrow pointing to the vertex and label it with incoming momentum 5 flowing in the diagram (write a numerical factor 1 in the amplitude) • For each outgoing particle draw a dashed line with an arrow pointing away from the vertex and label it with the outgoing momentum 5′ (write a numerical factor 1 in the amplitude) • For each incoming antiparticle draw a dashed line with an arrow pointing away from the vertex and label it with the incoming momentum −5 (write a numerical factor 1 in the amplitude) • For each outgoing antiparticle draw a dashed line with an arrow pointing to the vertex and label it with the outgoing momentum −5′ (write a numerical factor 1 in the amplitude) • Each photon with momentum [ attached to a vertex gives a factor : w [ 41 Effective Field Theories 2. Internal lines • For a scalar propagator write a factor: • For a photon propagator write a factor: $ A -C $òKó - 3. Vertices • For a vertex joining two scalar lines and one photon line write a factor: c5 + 5Ô • For a vertex joining two scalar lines and also two photon lines write a factor: 2c 9 As mentioned before there is one additional interaction term due to renormalization. The term is given by: − ô 9 OΦ= ΦS 101 2! 2! This corresponds to the well known Φ theory of real scalar fields. Because this theory is well known and can be found in any book on quantum field theory we won't do any calculations at this point but just quote the last Feynman rule for 9 õ scalar electrodynamics including the − OΦ= ΦS term: • For a vertex joining four scalar lines write a factor10: −ô õ 9 Again we look to the corresponding Feynman diagram for the − OΦ= ΦS interaction (fig. 9) The factor 1/4 is canceled because there are four equal Wick contractions cancelling the factor. 10 42 Effective Field Theories 5 5 ′ −ô 59 59 ′ fig. 9 Feynman diagram showing particle-particle scattering in scalar electrodynamics õ including interaction term − Φ= Φ9 . In our whole derivation we have considered a complex scalar field. For a real scalar field we can neither perform a general11 global nor a local gauge transformation of the field because in that case the fields would become complex which contradicts the assumption of real fields. The only possibility is the case when we set c = 0. In that case we would reach interaction of a neutral spinless particle with the electromagnetic field. Like in the hydrogen atom the electromagnetic field isn't interacting with the particle itself but for example with the corresponding dipol moment. We summarize our solutions of this chapter. The field strength tensor is invariant under a gauge transformation of the photon field. By gauge invariance we understand the invariance of the corresponding term in the Lagrangian under a gauge transformation of each field individually. The requirement of invariance under global and local U(1)-transformations of the Lagrangian led to a conserved current as well as coupled gauge transformations of fields which led to interactions between matter and the electromagnetic field. 11 Meaning with some arbitrary phase. 43 Effective Field Theories 4.3.3. Parity and Charge Conjugation So far we have dealt with continuous symmetries so it is time to also discuss some discrete symmetries (see [12.]). The most important discrete symmetries for particle physics are parity (space reflection), charge conjugation (i.e. exchange of particle and antiparticle) and time reversal. Though time reversal isn't unimportant and it is a general fact that every quantum field theory is CPT (i.e. charge conjugation, parity and time reversal) invariant, we won't discuss it in this context but just focus on the other two symmetries. We will start with parity transformations via the parity operator ¾ defined by, ¾: → 102 and therefore follows for the momentum, ¾:5ç = ç → −5ç103 o So we see that parity corresponds to space reflection. As is commonly known, every operator can be expressed as a transformation of unitary operators denoted ìö (unitary operators obey the relation ìö= = ìö- ). So we have the transformation of a general operator c under parity, ¾c = ìö cìö- 104 We now want to know how creation and annihilation operators transform under parity, therefore we start by the property of our unitary operator, ìö |5ç⟩ = ÷|−5ç⟩105 where ÷ is a constant . To simplify our life we will introduce the short notation |5ç⟩ =: |5⟩. From the following calculation, 1 = }5|5⟩ = t5uìö= ìö u5 = ÷ ∗ ÷}−5|−5⟩ = |÷|9 106 We conclude that ÷ = , $ø , ù ∈ ℝ . For our purpose it's enough to consider the cases where ÷ ∈ {1, −1, , −}. Which of the four possibilities is taken depends on the viewed particle so we will write ÷$ where corresponds on the considered particle and therefore considered creation or annihilation operators. Therefore we conclude that our creation and annihilation operator transforms like, ìö ¢5 = ÷¤ ¢−5ìö 107 ìö ¢= 5 = ÷¤∗ ¢= −5ìö 108 44 Effective Field Theories Rewriting it in terms of our parity transformation rule eq. (104), ìö ¢5ìö- = ÷¤ ¢−5109 ìö ¢= 5ìö- = ÷¤∗ ¢= −5110 Having achieved this so far we now want to look what follows for the transformation of a real scalar field under parity so we want to calculate, ìö Φìö- = = 5 ì O¢5ç, -$A + ¢= 5ç, $A Sìö- = 2 2VA ö 5 Ê÷ ¢−5, -$A + ÷¤∗ ¢= −5, $A Ë111 2 2VA ¤ Now by changing the variables via 5çÔ = −5ç and under usage of the invariance of the measure i.e. 5/2 2VA = 5′/2 2VAÔ we find, ìö Φìö- 5Ô = Ê÷ ¢5ç′, -$AÔü + ÷¤∗ ¢= 5ç′, $AÔü Ë = ÷Φü 2 2VAå ¤ 112 where we have introduced the notation 5Ô = 5)Ô , 5çÔ and ü = ) , −ç. Also we have used that for a real scalar field ÷¤ = ÷¤∗ =: ÷ . So we see that a real scalar field transform under parity like a scalar up to a phase factor. If the phase factor ÷ equals +1 we are talking of a pure scalar, if on the other hand the phase factor ÷ equals -1 we speak of a pseudo scalar. These are the general steps to obtain the parity transformed field. In the following we list up how the most important fields transform under parity. • • • ìö Φìö- = ÷Φü ìö Ψìö- = . ) Ψü ìö 2 ìö- = 2 ü scalar field fermionic field 12 photon field With these three important results we can look at two examples. As a first example we look how the + theory transforms under parity so we calculate, 1 1 ô ìö º O + +S − ;9 + 9 − + » ìö- = 2 2 4! 9 ÷9 ÷ ô÷ = O +ü +üS − ;9 + 9 ü − + ü = 2 2 4! 12 Where . ) is the first Dirac gamma matrix and ÷ = 1. 45 Effective Field Theories 1 1 ô = O +ü +üS − ;9 + 9 ü − + ü113 2 2 4! where we have used that ÷ 9 = 1 and the short hand notation of ü = ) , −çý . Also we used the trick of rewriting for instance ìö + 9 ìö- = = ìö +ìö- ìö ΦU- = η9 Φ9 . So we see that the Lagrangian is even (i.e. ¾ℒ = +ℒü) under parity transformations so the solution tells us that it is all the same where the reaction takes place which tells us our theory is parity invariant. As a further example we look at part of a term corresponding to weak interaction, 1 0 . 1 ± . Ψ114 Ø = Ψ 2 Doing the parity transformation we find, 1 1 0 . 1 ± . Ψxìö- = Ψ 0 ü. ) . 1 ± . γ) Ψü = ìö Ψ 2 2 1 0 ü. 1 ∓ . Ψü =Ψ 2 115 So our theory violates parity invariance. Summarized we conclude if our theory is parity invariant the Lagrangian itself must be invariant too. Next we look at charge conjugation. As the name suggests, charge conjugation describes the interchange of charged particles and corresponding antiparticles. Thereby charge does not necessarily have anything to do with electric charge but can also be something more abstract quantity as for instance color charge in strong interactions. In the language of creation and annihilation operators for particle and antiparticle we therefore have under charge conjugation in analogy to parity transformations, ¢5 - = æ5116 ¢ = 5 - = æ = 5117 As an example we look at how the complex scalar field transforms under charge conjugation. The complex scalar field denoted ΦC to distinguish from the real scalar field above is given by, Φ = 5 Ê¢5, -$A + æ = 5, $A Ë118 2 2VA So we find for the charge conjugated complex scalar field: 46 Effective Field Theories ΦC - = = 5 Ê¢5, -$A + æ = 5, $A Ë - = 2 2VA 5 = 119 Êæ5, -$A + ¢= 5, $A Ë = ΦC 2 2VA We see that the charge conjugated field corresponds to the hermitian conjugated field. Once again by applying these simple rules we can distinguish between charge conjugated invariant and non-invariant Lagrangians i.e. we can calculate if the theory is invariant under the exchange of particles by their antiparticles or vice versa. This can be a crucial build up point of effective field theories obeying this result. For example we look at the complex Φ theory with Lagrangian: ô 9 ℒC = Φ= Φ − ;9 Φ= Φ − OΦ= ΦS 120 4 We are interested in the properties of the Lagrangian under charge conjugation therefore we calculate, ô 9 ℒC - = Φ Φ= − ;9 ΦΦ= − OΦΦ= S = 4 ô 9 = Φ= Φ − ;9 Φ= Φ − OΦ= ΦS 121 4 So obviously the Lagrangian is invariant under charge conjugation. As a conclusion of the discussion of charge conjugation we again show how the three fundamental fields transform under that kind of transformation (because the calculations are somewhat lengthy and always proceed exactly the same, we skip them here), • • • complex scalar field13 fermionic field photon field ΦC - = Φ= Ψ - = . 9 Ψ ∗ 2 - = −2 It is interesting that the last transformation of the photon field could also have been achieved classically by obeying the invariance of charge conjugation of the term Ø 2 and knowing that the current Ø transforms under charge conjugation like, Ø → −Ø 122 13 As mentioned before the real scalar field describes only neutral particles. Charge conjugation would have no effect and has therefore not to be obeyed. 47 Effective Field Theories To let the term Ø 2 be invariant under charge conjugation 2 it has to therefore transform as 2 → −2 123 which is equivalent to the quantum field theoretical result. So we by now have developed many constraints that can be used to construct effective Lagrangians under given boundary conditions. In the next section we will begin to explicitly construct effective Lagrangians and also show how such constraints can be included. 4.4. Constructions of Effective Lagrangians 4.4.1. General Concepts By now we have developed the main concepts of symmetries and dimensional analysis. Both of these concepts now prove quite useful when we begin to construct general effective Lagrangians obeying given boundary conditions. The first question when beginning to construct an effective Lagrangian is what kind of particles do we describe? This will immediately provide the free Lagrangian. So if we are considering for example the scattering of fermions by scalar particles we know that our free Lagrangians are provided by the real scalar Lagrangian, 1 ;9 9 ℒ§¤¥¤é = Φ9 − Φ 124 2 2 and the fermionic Lagrangian, 0 O. − ;SΨ125 ℒnléC$ = Ψ Having found the basic terms ℒ) (here the real scalar and free fermion Lagrangians) we can now begin to build up our effective theory. For this chapter we refer to [1.] - [4.] as well as [17.]. In general we can add up all kinds of interaction terms which on the one hand obey boundary conditions that we might demand and on the other hand describe the physics up to the given order in energy scale Λ 14. This scale is given by the properties of the system and also 14 This is a crucial point because an effective field theory describes physics in the low energy regime. By ordering in powers of energy/Λ we can immediately tell which terms will be relevant up to a given order in our calculation. 48 Effective Field Theories the range of physics we want to describe. So we find in general our effective Lagrangian: ℒlnnl%$ = ℒlnn = ℒ) + ℒ$% = ℒ) + $ $ 126 $ Thereby $ are operators which are constructed out of light15 field combinations and $ are the corresponding coupling constants also called Wilson coefficients. Because our Lagrangian is describing the low energy limit, all heavy16 degrees of freedom (i.e. interactions that occur only in the high energy limit) are absorbed in the coupling constant. We now illustrate more clearly what we mean by ordering in energies. From dimensional analysis we know how we can calculate the dimension of the operators and of the coupling constants, so we know if, then it follows immediately, that $ = $ 127 $ = 4 − $ ⇔ $ ~ 1 Λy - 128 Thereby Λ is some characteristic energy scale17 and we want to describe the physics below that scale. We see that if we are considering interactions on an energy scale below Λ, i.e. : ≪ Λ, our interaction terms will give smaller contribution to the result when ä § the order of $ is larger because they will be suppressed as Ö , > 0. In general we can divide the operators $ into three categories relevant ($ < 4) marginal ($ = 4) irrelevant ($ > 4) The categories are quite self-explanatory: For relevant operators :/Λy - it is not small and has therefore to be included in the calculations. 15 Here light means that the energy scale of the fields involved is much smaller than a given energy scale provided by the problem. 16 Heavy means those terms involving energy scales above our estimated given energy scale. 17 One can think of Λ also to be some characteristic heavy mass scale so Λ has the same dimension as the mass. 49 Effective Field Theories For marginal operators it's in general not clear if the terms have to be considered. On the other hand we cannot simply throw them away, so they are kind of special. The effects of irrelevant operators are suppressed by powers of :/Λ. So we see that working in the low energy limit has some nice advantage namely the ä dimension of our operators (if our result should be correct up to order f Ö) is limited to four and one can easily check that there are not that many possibilities to achieve those combinations of operator fields. So in general it is a question of which accuracy in :/Λ we want to achieve. At this point it's a good time to present an example of what we have just worked out. We are considering the interaction between light scalar particles of mass ; and heavy fermionic particles Ψ of mass > as well as photons2 embedded in the field strength tensor ? . Also we set our characteristic energy scale equal to the heavy mass, i.e. we are considering working in an energy limit less than >. The interaction Lagrangian that describes our system is given by, 0 Ψ + 9 ?µ ? µ + …129 ℒ$% = ) + + 9 Ψ Now we use first the method of dimensional analysis to calculate the dimensions of the coupling constants, so we find, 4 = ) = ) + 3 = ) + 3 ⇔ ) = 1130 Therefore the coupling constant is given by, ) = ¢>131 where > is the corresponding heavy energy mass scale we consider and ¢ is a dimensionless quantity. The same procedure can be done for the other interaction terms so we can rewrite the interaction Lagrangian as, 0 Ψ + ℒ$% = ¢> + æ + Ψ d 9 ?µ ? µ + …132 M9 So considering energies with : ≪ > we see that the last interaction term is irrelevant (gives no significant contribution) and has therefore not to be considered in our effective Lagrangian, which becomes, 0 Ψ + f ℒlnn = ¢> + æ + Ψ 50 1 133 M9 Effective Field Theories This simple example was meant to show how by energy arguments the important part of an effective Lagrangian can be sorted out describing the low energy limit. So far we have not considered symmetry arguments that can also occur given a specific problem. Therefore we will also look at an example. We want to construct an effective Lagrangian describing the interaction between spinless particles with the electromagnetic field that on one hand should be gauge invariant and on the other hand should obey Lorentz symmetry. Once again we write down all possible terms in order of dimension. The interaction Lagrangian is given by, ℒ$% = c) Aµ + 2µ 2µ 9 + 9 2µ µ + 9 ?µ ? µ + … 134 Of the requirement for Lorentz invariance we immediately see that the term c) Aµ Φ vanishes. The requirement of gauge invariance18 eliminates two other terms namely 2µ 2µ Φ9 and9 2µ Φ µ Φ. Therefore only the term Φ9 ?µ ? µ of lowest dimensional order remains where has the mass dependence = ¢/>9 . So finally we obtain our effective Lagrangian as, ℒlnn = ℒ) + ¢ 9 ?µ ? µ 135 >9 Remembering the introductory examples of this work we see that we have reproduced exactly the Lagrangian describing Rayleigh scattering in an ä effective field theory up to order f . By now we have constructed the effective Lagrangian that shall describe our low energy system but looking at the Lagrangian we see that there is still one unknown. In our case the constant ¢. The question now is how to obtain the constant. In general there are two possibilities: 1. As a first possibility we could calculate the interactions between the involved particles in the unknown constants and fix the constants such, that our calculations provide the same results as measurements in experiments. 2. As a second possibility we could calculate the interactions in an underlying fundamental theory as well as in our effective theory and match both solutions, and therefore the unknown constants in the low energy case. This procedure is called the "matching procedure" which we will deal with in the next chapter. 18 Once again we mean purely gauge invariant and not coupled gauge invariant. 51 Effective Field Theories As a summary we again present all important steps involved in constructing an effective Lagrangian: 1. Determine the set of fields describing the interaction of the system in the low energy case. 2. Write the effective Lagrangian as a sum of operators and corresponding (unknown) coupling constants ∑$ $ f$ . 3. Determinate the dimensions of the coupling constants in powers of the characteristic energy scale Λ and order the operators in powers of Ö. 4. Look if the system and therefore the Lagrangian obeys symmetries, e.g. gauge invariance or parity invariance. 5. Keep all interaction terms obeying the given symmetries up to a given order Ö to which accuracy we want to describe our system. 6. High energy dynamics are purely absorbed into the coupling constants. Though by now it seems at a first glance a bit strange how the high energy dynamics can be absorbed into the couplings we will see an example in the next chapter providing a good insight. At the end of this chapter we present a short list of some renormalizable and Lorentz invariant terms of dimension less than four: 0 Ψ; ∂µ ∂µ ;Ψ 0 γµ Dµ Ψ;Ψ 0 Ψ; Fµν F µν f ≤ 4;Ψ 4.4.2. The Matching Procedure We will now study a simple example of how the unknown constants in an effective Lagrangian can be found by the so called matching procedure (for reference see in particular [1.] - [4.]). At first we consider a simple example, presented in [2.], of two real scalar fields and Φ of mass ; and > where ; ≪ >. The Lagrangian density is given by 1 1 1 1 ô 9 9 ℒ = O S − ;9 9 + O ΦS − >9 Φ9 − 9 Φ136 2 2 2 2 2 Clearly the interaction term couples one heavy scalar field to two light scalar fields, so the Lagrangian is describing the interactions of three scalar fields. Working out the Feynman rules (the calculations are trivial so we just quote the result) we see that by calculating the vertex factor there will be two equal contractions so the factor 9 in the interaction term is cancelled and we obtain the Feynman rule for the vertex factor of the given interaction −ô137 52 Effective Field Theories The other Feynman rules follow from our discussion on scalar electrodynamics. So in general we can have the following Feynman diagrams in first order perturbation theory (fig. 10), 5 5 A 5 59 59 B 5 5 5 5 59 5 C fig. 10 Lowest order scattering diagrams. Here we show in this illustration that dashed lines correspond to light scalar fields and that whole lines correspond to heavy scalar fields. So if we are now considering soft light scalar scattering with energies 5$ ≪ > A we can Taylor expand the heavy scalar propagator around y = 0 to leading order, thereby 5$ means all possible external momenta involved in the process: 53 5 Effective Field Theories 1 1 5$ = − + f 138 >9 > 5$9 − >9 We see that in this limit the propagator reduces to a point interaction, in terms of Feynman diagrams (fig. 11), 5 5 5 5 59 5 59 5 fig. 11 Translation from fundamental to effective diagram. Therefore in this limit (which is the typical limit of any effective field theory) the interaction could also have been described by the + theory, so we have found our effective Lagrangian for light scalar scattering in this context: 1 1 ¢ ℒlnn = 9 − ;9 9 − 139 2 2 4! >9 where ¢ is (until now) an unknown coupling constant. From dimensional analysis we know that ¢~;9 to get the right dimension of the Lagrangian, so we set ¢ = ;9 where is an dimensionless coupling constant and finally obtain for the effective Lagrangian, 1 1 ;9 ℒlnn = 9 − ;9 9 − 140 2 2 4! >9 By now we are left with the task of finding the value of . The procedure to achieve this is called the matching procedure i.e. we calculate the scattering amplitude in the fundamental theory (here our light-heavy scalar interaction) in the limit where 5$ ≪ > and also with the effective Lagrangian. In the soft scattering regime both scattering amplitudes have to be equal so the unknown 54 Effective Field Theories constant can be found by comparing both amplitudes and matching the unknown constant such that they are equal. In the introduction to this paper we have seen a similar example, Fermi's theory of weak decay. The same procedure can be applied in this context to find the low energy effective Lagrangian. So now we come back to our example and match the variable by first calculating the individual scattering amplitudes. õ We start with the ordinary theory involving the interaction term − 9 9 Φ. The lowest order scattering Feynman diagrams are shown in (fig.11). Using the Feynman rules we obtain the corresponding scattering amplitudes. ℳ = −ô ô9 −ô 141 = − 5 + 59 9 − >9 − >9 ℳ = −ô ô9 −ô = − 143 5 − 5 9 − >9 − >9 ℳ = −ô ô9 −ô = − 142 5 − 5 9 − >9 o − >9 where we made use of the Mandelstam variables , o and . The whole scattering amplitude in the limit 5$ ≪ >, from which follows , o, ≪ >9 , is given by ℳ ≔ ℳ + ℳ + ℳ = 3ô9 5$ + f 144 9 > > In the last step we again used the Taylor expansion of the heavy propagator to first order. Now we also calculate the scattering amplitude in the effective theory. In this case we obtain, ℳlnn = − ;9 145 >9 In the regime 5$ ≪ > both amplitudes must be equal so we find, ℳ = ℳlnn ⇔ = − 55 3ô9 146 ;9 Effective Field Theories Now we have matched the prefactor and we find the effective Lagrangian for soft light particle scattering, 1 1 3ô9 ô ℒlnn = 9 − ;9 9 + + f G147 E 4! >9 2 2 > Thereby we have neglected higher orders in 1/> § > 2. If we would now want to describe our interaction in more detail we have to take higher orders of perturbation into account. The corresponding Feynman diagrams of the fundamental theory and of the effective theory are shown in (fig. 12), fig. 12 First order corrections. õ We see that the first order corrections arise out of loop diagrams and are of order . In the next chapter we will go into depth with the calculations of loop diagrams and so called renormalization, therefore we will just quote the results at this point and will do the calculations in a following chapter when we have developed all the important tools. The Lagrangian is then given by, 1 ¢ ô9 1 æ ô9 3ô9 ô9 + ℒ lnn = E1 + 9 G 9 − E;9 + 9 G 9 + E1 G 2 > 2 > 4! >9 >9 148 56 Effective Field Theories where ¢ , æ and can again be matched by the corresponding diagrams from (fig. 12). Our Lagrangian describes now the physics of the system up to orders õ f so we have generated a more accurate description of the low energy physics contained. It is quite interesting that in second order we produce not only corrections to the coupling constants but also corrections to the mass term. This will be a characteristic of the renormalization procedure that we will tackle in the next chapter. This part was intended to give a brief introduction into the matching procedure. In a later chapter we will be dealing with the photon-photon scattering. In finding an effective Lagrangian we will also use the matching procedure. As a summary it is to be said that matching is always a good tool as long as the underlying fundamental theory is known. If we have no knowledge of the underlying theory we'll have to fix the variables by experiment. So we leave some work for our experimental friends out there. 4.4.3. The Fermi Theory once again In our matching procedure example we have seen how to derive the effective Lagrangian and the corresponding effective coupling by comparison of all possible diagrams (to lowest order) in perturbation theory. Now we will walk the other line, i.e. we will construct an effective Lagrangian without explicitly calculating the effective coupling constant but take the constant as an input from experiment. Thereby we will also show, that the low energy limit gives suitable answers by explicitly calculating the weak interaction fine structure constant. Our main reference for this chapter will be [16.] and [19.]. Consider a weak interaction process by exchange of a i - boson. This can be expressed in form of Feynman graphs in the ordinary and in the effective regime (fig. 13) From the theory of weak interaction we know that the interaction Lagrangian for the exchange of a i - boson is given by, ℒ where, Ø = l¤ = i- Ø 149 1 1 Ψ. 1 − . Ψ150 2 √2 57 Effective Field Theories The boson propagator is given by, − 5 5 9 >b 151 9 59 − >b − fig. 13 Weak interaction in fundamental and effective theory. So we can write down the decay amplitude, 9 1 ℳ = Ψ. 1 − . Ψ 2 2 5 5 9 1 >b Ψ. 1 − . Ψ152 9 9 2 5 − >b − 9 Now we consider the low energy limit where 59 ≪ >b . Therefore we can expand the propagator as, 5 5 9 1 >b = − 9 + f E G153 9 9 5 − >b >b >b − So we obtain an effective decay amplitude up to order f " , ℳlnn = − ! 9 1 1 1 − . Ψ Ψ. 1 − . Ψ154 9 Ψ. 2 2 2>b We see now that we could have achieved the same result by an effective Lagrangian which has the structure of a four point interaction, ℒlnn = − #$ √2 OΨ. 1 − . ΨSOΨ. 1 − . ΨS155 58 Effective Field Theories where #$ is the so called Fermi constant and %& √9 = ò . '! In the following as a kind of typical application we will calculate the decay rate of the muon by using the effective Lagrangian, thereby we will in the end calculate the theoretical value of the Fermi constant by using the measured data of the muon mass and lifetime. For the decay amplitude we obtain, ℳ = Therefore, #$ √2 |ℳ|9 = Ou5 . ) 1 − . vp SOu59 .) 1 − . up S156 #$9 Ou5 . ) 1 − . vp SOu59 .) 1 − . up S ∙ 2 ∙ Ov5 . * 1 − . up S u5 .* 1 − . up9 157 We now average over spins and use the relations, 55 = . 5 + ; 158 §A$§ and, +5+5 = . 5 − ; 159 §A$§ also we set the mass of the neutrinos to zero, so we obtain the averaged amplitude as: |ℳ|9 = #$9 !ÊO. ø 5,ø + ;l S. ) 1 − . . , 5,, . * 1 − . Ë ∙ 4 ∙ !ÊO. H 5 ,H + ; S.* 1 − . . 59, .) 1 − . Ë160 We calculate the first trace explicit, the second is then equivalent, except exchange of indices !ÊO. ø 5,ø + ;l S. ) 1 − . . , 5,, . * 1 − . Ë = = 25,ø 5,, O!Ê. ø . ) . , . * Ë − !Ê. . ø . ) . , . * ËS161 59 Effective Field Theories Using the identities, !Ê. ø . ) . , . * Ë = 4Oø) ,* − ø, )* + ø* ), S162 !Ê. . ø . ) . , . * Ë = −4w ø),* 163 we can simplify our formulation and obtain: 85,ø 5,, Oø) ,* − ø, )* + ø* ), + w ø),* S164 So we obtain for the whole amplitude, |ℳ|9 = 16#$9 5,ø 5,, 5H 59 Oø) ,* − ø, )* + ø* ), + w ø),* S ∙ ∙ OH* ) − H *) + H) * − wH*) S165 we perform all contractions and use the identity w ø),* wH*) = −2 ·ø ·H − , −·Hø · S, , |ℳ|9 = 64#$9 O5, 5 SO5 , 59 S166 Now we are in the position to calculate the decay rate for the process in the muon rest frame, 5$ 1 5 9 |ℳ| Γ = 2 · − 59 − 5 − 5 { 167 2 2:$ 2; Further we (|. | =: 5 note that :9 = |.Q | =: 59 and $(9 also : = |. | =: 5 We want the reader to pay extra attention to change of notation. Such we find for the differential decay rate Γ, Γ = 4#$9 O5 5 SO5 , 59 S·O; − 59 − : − 5 S ∙ ; 2 , ∙ · .Q + ./ + . 59 5 5 168 59 : 5 solving the first integral for 5 we obtain, 60 Effective Field Theories 4#$9 59 5 |. |S 5 5 − 5 − : − + . O5 SO5 S·O; , , 9 Q / 9 ; 2 59 : |.Q + ./ | 169 Changing to spherical coordinates such, 59 = 599 59 sin0 0+170 and also defining, |.Q + ./ | = 599 + 59 + 259 5 cos0 =: 171 from which follows, 59 5 sin0 =− 172 0 Therefore performing the substitution we can rewrite the integral as, Γ = − 9¼ ' 4#$9 5 5 5 + ∙ O5 SO5 S , 9 9 ; 2 , ) ) 12 ∙ 1 ·O; − 59 − : − S where we have defined, 5 173 5 : ∓ = 599 + 59 ∓ 259 5 174 The integration gives 1 if the relation - < ; − 59 − : < h is valid. The relation is an indicator of the kinematic properties. So we find three important inequalities considering the cases where : = 0 or 59 = 0. First we find, 59 < ; − 59 ↔ 59 < second, äð ≈Að 5 < ; − : 5667 5 < ; 175 2 ; 176 2 and by squaring the right hand side inequality again using the approximation : ≈ 5 61 Effective Field Theories 9 9 ↔ 59 + 5 > O; − 59 − 5 S < h ; 177 2 Combining all three relations we find a bound for the momentum 59 , ; ; − 5 < 59 < 178 2 2 This seems a good point to talk of kinematics anyway. From overall four momentum conservation we find, 5 − 59 = 5 + 5 /. .9 ↔ ;9 − 2; 59 = ;l9 + 25, 5 179 Because the electron mass is much smaller than the muon mass ;l ≪ ; we can set the electron mass to zero without making a big error, as a consequence : = 5 Having discussed all relevant consequences above we can now return to our differential decay rate expression and therefore find, Γ = − ' 9¼ 2#$9 9 9 ; 5 5 − 2; 5 + ∙ 9 9 9 ; 2 ) ) 12 ∙ 1 ·O; − 59 − : − S 5 180 59 The integration is now trivial and by using (eq. (175) & eq. (177)) we find a restriction for the integration domain namely, CK 9 2#$9 5 9 9 ; Γ = − 5 5 − 2; 5 = 9 9 ; 2 CK -Að 9 59 =− 9 ; 9 2;9 2#$9 Ω5 E 5 − 5 G181 ; 2 2 3 From (176) we also have a restriction to the other integration domain. The surface integration is trivial and gives a value of 4. We therefore obtain, CK ; 9 2;9 8#$9 9 Γ=− 5 E 5 − 5 G = ; 2 ) 2 3 =− ;9 ; #$9 8#$9 E− = 182 G ; 2 96 192 62 Effective Field Theories The lifetime is therefore, := 1 192 = 183 Γ ; #$9 We want the reader to pay extra attention and remember that this result is only valid for small momentum transfer compared to the i boson mass, therefore the formula is just valid in the low energy regime. Putting in the experimental data of the muon mass ; = 105,7>, and also the mean lifetime : = 2,2 ∙ 10-9 we find an experimental value for the Fermi constant, #$ = 1,166 ∙ 10- /#,9 = So the corresponding value for is, √2 9 184 8 >b = 0,653185 and so we find for the weak fine structure constant, ù l¤ 9 1 = ≈ 186 4 29,5 5. The Renormalization Procedure 5.1. Renormalization 5.1.1. Concepts of Renormalization When dealing in general with higher orders in perturbation theory19 we encounter loop processes such as shown in (fig. 14). Unfortunately when calculating the corresponding scattering amplitude we can encounter integrations yielding an infinite contribution to the scattering amplitude. So the question is how to get rid of these infinities. The procedure to achieve this is the so called renormalization procedure which we will deal with in the following section. The main references for chapter 5. are [15.], [16.] and [18.]. For chapter 5.1.3. and 5.2. we also refer to [1.] - [4.]. In the example that we will calculate below we will see that there will be a loop process even in first order perturbation theory when considering the + theory. 19 63 Effective Field Theories fig. 14 One loop diagram in QED. The basic idea of renormalization is that we redefine the masses, fields and coupling constants such that they cancel out the occurring infinities in our calculation. Therefore we are no longer working with the bare quantities (that are given by non interacting systems) but rather with physical quantities, i.e. in experiment we don't measure for example the bare mass but rather a renormalized or better called a physical mass that includes all kinds of radiative corrections and therefore differs from the bare mass. By now we have just presented the main idea of renormalization. In the next step we will show how this is handled mathematically. As mentioned before we are redefining measurable quantities such that they cancel the infinities in our calculations. To illustrate this mathematically we deal with the + theory. In the following we will distinguish between the bare mass and coupling, and the renormalized mass and coupling. The bare Lagrangian is given by, 1 ;)9 9 ô) ℒ = +) 9 − + − + 187 2 2 ) 4! ) Because in measurements we don't really measure the bare quantities. We reexpress them through their renormalized quantities which are measurable. So we find: ;) = ;<C ;188 ô) = <õ λ189 64 Effective Field Theories +) = <> +190 where ;) , ô) and +) are the bare quantities, ;, ô and + are the renormalized quantities and the <$ are the corresponding proportionality factors which are still not known. It has to be mentioned that not only mass, charge and coupling are renormalized but also the fields of the system . We can now replace the bare quantities in the Lagrangian and obtain, 1 <C <? ;9 9 ô + − <>9 <õ + = ℒ = <> +9 − 2 2 4! 1 1 ô = +9 − ;9 + 9 − + + ℒý 191 2 2 4! where ℒý is the so called counter-term Lagrangian and is given by, 1 ;9 9 1 ℒý = <> − 1+9 − <C <? − 1 + − O<>9 <õ − 1Sô + 2 2 4! The counter-terms factors are given by, 192 9 ,<>9 <õ − 1ô =: ·õ <> − 1 =: ·> ,<C <? − 1;9 = : ·C 193 Looking at the renormalized Lagrangian we see that it contains the original (with the renormalized Lagrangian) qunatities plus some additional terms summed up in ℒý that give rise to either a corretion to mass, coupling constant20. The additional terms provide a new set of Feynman rules which have H to be added to the old ones. Treating !@ ϕ as a perturbation we find the following vertex factor, −·õ 194 So we obtain a new type of Feynman diagram (fig. 15 left) for this kind of interaction (to distinguish between the original interaction and the new interaction corresponding to the counter-term it is convenient to draw the vertex as a cross), In scalar field theory the mass of the particle is given by the square route of the prefactor of the Φ9 term. So in our case we have ;9 + ·;9 Φ9 so physical mass is given by √;9 + ·;9 . 20 65 Effective Field Theories fig. 15 Feynman diagrams corresponding to the new interaction term due to renormalization. The same procedure can be done for the other two new terms. The terms correspond to a two point interaction and the corresponding vertex factor can again be calculated via, − }0|¢5! B− ·> ·C 9 = +9 + + C ¢ 5|0⟩ = 2 2 ·> ·C = }0|¢5 E ++ − ++G ¢= 5|0⟩ + 2 2 ·> ·C + }0|¢5 E ++ − ++G ¢= 5|0⟩ 2 2 195 so we find for the vertex, O·> 59 − ·C S196 66 Effective Field Theories Our theory now also describes the two point interaction shown in (fig. 15 right). We will now focus on how the counter-terms will help us to cancel out the infinities in our calculation, therefore we consider a simple loop diagram shown in (fig. 16). Because we are now working with the renormalized theory, we have also to take the counter-terms interaction into account when calculating the scattering amplitude in (fig. 16). We only showed an additional contribution due to counter-terms. Why we will need exactly this kind of contribution will be shown in the later calculation). Nevertheless in our counter-terms there are still unknown parameters. The idea is now to choose the free parameters such that they cancel the infinities produced by the loop integration. To make the procedure more clear we will calculate the loop diagram shown in (fig. 16 right) and explicitly comment on the steps done in the calculation. fig. 16 Loop diagram in first order perturbation theory (left) and two point interaction diagram due to renormalization (right). Before continuing with our calculation we first discuss the concept of exact propagators and vertex corrections due to loop diagrams. In general when we are doing our renormalization procedure we can obtain two different kinds of corrections. The first correction is due to the propagator and the other correction is due to the vertex functions. For the correction to the propagator we introduce the concept of a one particle irreducible diagram. A one particle irreducible diagram follows one simple rule: A one particle irreducible diagram doesn't decompose in two separate diagrams when cutting an internal line of the diagram in two. 67 Effective Field Theories To illustrate this concept we look at (fig. 17). When cutting an internal line of the left hand diagram the diagram doesn't decompose into separate parts so it is obviously irreducible. On the other hand when considering the right hand diagram we see that when we cut the internal electron line into two we obtain two separate diagrams so this type of diagram is reducible. fig. 17 One particle irreducible diagram (left) and reducible diagram (right). So we see that in general we can build up every Feynman diagram in terms of one particle irreducible diagrams which are connected by propagators. So by now we are in a position to present a graphical definition of the exact propagator D as, = 1PI 1PI + + 1PI + 1PI 1PI 1PI fig. 18 exact propagator in terms of one particle irreducible diagrams (1PI ≜ −Σ) so we find that, D = Í + Í−ΣÍ + Í−ΣÍ−ΣÍ + … = 68 Effective Field Theories = Í + Í−ΣD199 Therefore by multiplying −Í - from the left and −D - from the right we find that the exact propagator is, −Í - = −D - + −Σ ⇔ D - = Í - − Σ = 59 − ;9 − Σ ⇔ D = 59 200 − ;9 − Σ 59 201 − ;9 So we have derived the form of the exact propagator. The big question is if there are any constraints on Σ that we need to take into account. The answer is yes. Therefore we first of all calculate the free propagator and find, where ; is now the physical mass. So we see from the propagator of the free field theory that the mass is defined as the position of the pole i.e. 59 = ;9 . By the condition of preserving the pole when 59 = ;9 we extract two renormalization conditions for the exact propagator namely, 59 − ;9 − Σ59 G|A (C = 0 and, ⇔ Σ59 G|A (C = 0202 Σ59 G|A (C = 0203 59 So the main task will be to calculate the contribution Σ corresponding to the one particle irreducible diagram. In doing so we again come back to our example above and will explicitly calculate Σ for two different kind of loop contributions. Now let's view the correction to the vertex (vertex corrections = +. .). In contrast to the propagator correction there is no obvious relation defining the magnitude of the counter term (remember for the propagator we defined the counter term via the conservation of the pole). One definition is that the counter term at zero momentum is chosen such that it reproduces the ordinary four point coupling −ô without considering any loop process. In formulas, ℳ = −ô + +. . = 4;9 , o = 0, = 0 − ·õ = −ô204 69 Effective Field Theories where we have used the values of Mandelstam variables at zero momentum. A graphical illustration is given by (fig. 19) via the amputated diagrams. =−ô at = 4;9 , o==0 amputated fig. 19 Amputated diagrams at zero momentum. 5.1.2. Renormalization in the Context of H Theory In this chapter we will explicitly calculate two different kinds of loop corrections corresponding to the renormalized graphs shown in (fig. 15). Thereby we will develop an important trick due to R. Feynman that will help us to simplify and to reduce unknown loop integrals to those we already know. We will start with the first loop diagram. By viewing again (fig. 16 left) we see that the scattering amplitude follows from the first order term, − ô }0|¢5!{ΦΦΦΦ}¢= 5|0⟩205 4! so there are four equal possibilities to contract ¢5 with an + field. After having contracted the first pair there are three other equal possibilities to contract ¢= 5 with a remaining + field. In the end there remain only two other fields that have to be contracted together so we find 4 ∙ 3 ∙ 1 possibilities of contraction. Therefore the one particle irreducible amplitude part without the corresponding counter-terms is given by, − [ ô [ 4 ∙ 3ô = − 206 9 9 9 2 [ − ; 2 [ − ;9 4! 2 The task is now to evaluate this very integral. For this purpose we will use the method of dimensional regularization (there are also other methods to regularize 70 Effective Field Theories the integrals for example the cutoff method, nevertheless dimensional regularization is the most profitable method because it can be applied in various contexts) i.e. we won't calculate the integral in four dimensions but in 4 + 2w or equivalent in 4 − 2w dimensions and afterwards take the limit w → 0. In doing so we will explicitly see from which terms the infinities occur, but more on that step later. So let's get started. First of all we use a Wick rotation to simplify our integral therefore we make the substitution, [) = [I ) 207 Therefore we obtain for the integral, ï9 $' -9 [ 1 [I ) -9 [ 1 9 = ï = -9 -9 2 [)9 − [Ðç 9 − ;9 −[I )9 − [Ðç 9 − ;9 -$' 2 = −1ï9 -9 [I 1 9 208 -9 I 2 [ + ;9 where we have introduced [I 9 = [I )9 + [Ðç 9 and also the parameter ï9 to ensure that the dimension of the integral is the same for every w. So now we can change into spherical coordinates via, −1ï9 -9 [I 1 9 = -9 2 [I + ;9 ' = −ï9 Ω-9 ) [I [I -9 209 2-9 [I 9 + ;9 In order to calculate we first derive how the surface integral in dimensions can be calculated. Therefore we use, So we find in dimensions, , - = √ ℝ exp− 9 − 99 − … − 9 9 … = 9 210 On the other hand we can again change to spherical coordinates and find, exp− 9 − 99 − … − 9 9 … = 71 Effective Field Theories = Ω - , -é 211 by doing the substitution 9 = o follows, 1 1 n - , -é = o 9 - , -% o = Γ 212 2 2 2 where we used in the last step the definition of the gamma function. Therefore by comparing eq. (210) and eq. (211) we find, 2 9 Ω = 213 Γ 2 So by now we have achieved the first integral in eq. (209) leaving the second one so we find ' ) 1 ' 9 9-9 -9 = 9 214 9 + ¢ 2 ) +¢ by substituting æ ≔ 9 + ¢ it follows that, 1 ' æ − ¢ æ 2 ¤ æ - ¤ 1 ' ¢ = ææ - E1 − G 2 ¤ æ - 215 now by another substitution ∶= K and the integral representation of the Euler beta function, É, L = oo - 1 − oM- = we find, ) ' ) -9 1 = ¢ 9 +¢ 2 - ΓΓL 216 Γ + L Γ−1 + w217 →) where we have used the fact that Γ1 = 1 and Γ2 − w NOP Γ2 = 1 . Below, when dealing with the second loop example we will quote a general formula for this kind of integrals. 72 Effective Field Theories So finally we obtain for our whole expression, − ô [ ô ï9 9- Γ−1 + w = − 2 [ 9 − ;9 2 ;-9h9 2 2-9 9 ô ï Γ−1 + w =− = 9- 2 4 ;9 - h ô ;9 ;9 E =− G 2 16 9 4μ9 - Γ−1 + w218 This is the overall result for the one particle irreducible diagram but obviously when taking the limit w → 0 i.e. the four dimensional limit our result includes divergences corresponding to the gamma function. The question is how to cancel those divergences. Therefore we remember the second diagram shown in (fig. 16) that we haven't yet included. Taking the second diagram also into account we find for the one particle irreducible amplitude, ô ;9 ;9 −Σ = − E G 2 16 9 4μ9 - Γ−1 + w + O·> 59 − ·C S219 where ·C and ·> are still unknown parameters. For small values of w we can expand the Gamma function as well as the mass term ; via the general formulas, Γ− + w = where, and the special case, −1 1 E + Ψ + 1 + fwG220 w 1 1 Ψ + 1 = 1 + + … + − .ä 221 2 Γw = 1 − .ä + fw222 w where .ä = 0,5772 … is the so called Euler constant. On the other hand we can C - expand ¼ for small values of w, ;9 E G 4μ9 - = 1 − ln E ;9 G w + fw 9 223 4μ9 Therefore we can expand our loop integral for small values of w up to order fw and find, 73 Effective Field Theories − =− ô ;9 ;9 E G 2 16 9 4μ9 - Γ−1 + w = ;9 ô ;9 1 1 − ln E G w + fw 9 E + 1 − .ä + fwG = 9 9 4μ w 2 16 =− ô ;9 1 ;9 + 1 − . + ln4 − ln E G + fw224 ä 2 16 9 w μ9 So by dimensional regularization we immediately see how the divergences enter into our result as a pole for w → 0. - õ C C 9¼ ¼ But now comes the big step. We can cancel the divergences of the first part by defining ·C = − 9 Γ−1 + w and ·> = 0 (to define ·> = 0 corresponds to the fact that the one particle irreducible amplitude is independent of 59 ). In addition the boundary conditions derived in the last chapter also hold so we see that, −Σ = 0225 and therefore provide no contribution to the exact propagator (and also satisfying the renormalization condition mentioned above), or in other words in first order perturbation theory the exact propagator is equal to the "normal" propagator and the occurring loop diagram gives zero contribution to the scattering amplitude and can therefore be neglected. Now let's consider the second and more interesting loop diagram due to Φ theory (fig. 20). This loop calculation will serve as a standard example to calculate similar loop diagrams in other theories. fig. 20 Corresponding one loop diagrams in second order perturbation theory. 74 Effective Field Theories We see at once that we have three "different" kind of processes depending on which channel it occurs (−, o − or − channel). The scattering amplitude of the very left diagram corresponding to the o channel follows from, − ô 9 L}0u¢5 ¢5 !{Φ xΦ L}¢= 5 ¢= 59 u0⟩ = 4! −ô9 4 ∙ 3 ∙ 4 ∙ 3 ∙ 2 [ = ∙ 4!9 2 c − [9 − ;9 [ 9 − ;9 where c 9 = o = 5 − 5 . ∙ 2 · 5 + 59 − 5 − 5 226 So the task now is to solve the integral over the open four momentum [. Therefore we will use a wonderful and also powerful trick due to R. Feynman using the identity, 1 Γù + N ø- 1 − ,= 227 ¢ + æ1 − øh, ¢ø æ, ΓùΓN ) Using the identity we can rewrite the integral part of (226) as, [ 1 = 9 9 c − [ − ; + [ 9 − ;9 1 − 9 2 ) = [ 1 9 228 [ − 2c[ + c 9 − ;9 9 2 ) By completing the square we obtain, [ 1 229 9 9 2 ) [ − c − c 9 + c 9 − ;9 9 We now make the substitution = [ − c and also ¢9 ≔ c 9 9 − c 9 + ;9 and obtain the final expression, 1 1 = 230 9 9 9 9 − ¢ 2 ) 2 − ¢9 9 ) Now we again apply dimensional regularization to calculate the second integral and use the general formula, 75 Effective Field Theories [ 9 Ì [ = 2Ì [ 9 − ¢9 K 4 OΓ2 + − wΓæ − − 2 + wS 9 -Kh9 −¢ = 9 E G231 16 9 ¢ ΓæΓ2 − w where Í = 4 − 2w. So we obtain for the integral expression (229), ï9 ) 4 Γw232 16 9 ¢9 Expanding up to first order in wwe again obtain, = ) 1 4μ9 E + ln G − .ä G = E ¢9 16 9 w 1 c9 ;9 − − . + ln4 − ln ln + O − 1SG E G E ä 16 9 w μ9 c9 ) 233 Thus we obtain the scattering amplitude for the o − channel diagram of (fig. 20) by, ℳ% = − ô9 1 o ;9 E − . + ln4 − ln − ln + O − 1SG ä 32 9 w μ9 o ) 234 The corresponding − and − channel scattering amplitudes are equivalent to the o − channel by interchanging the corresponding channel, i.e. ô9 1 ;9 ℳ§ = − E − .ä + ln4 − ln 9 − ln + O − 1SG 32 9 w μ ) ℳ1 = − 235 ô9 1 ;9 E − . + ln4 − ln − ln + O − 1SG ä 32 9 w μ9 ) 236 76 Effective Field Theories So the whole scattering amplitude for the loops is given by, ℳ = ℳ§ + ℳ% + ℳ1 − ·õ 237 from the corresponding renormalization conditions we obtain, =− ·õ = ℳ§(C + ℳ%() + ℳ1() = ô9 3 4;9 E − 3. + 3 ln4 − ln E G ä 32 9 w μ9 1 ;9 − ¸ln E + − 1G + 2 ln E 9 G¹G μ 4 ) 238 The result presents us the lowest order correction to the coupling constant. At this point we will end the discussion of renormalization. In the next chapter we will take a brief look at different renormalization schemes that can be used by convention. 5.1.3. Renormalization Schemes By now we have done two different loop calculations but in both results we saw that we have a finite and an infinite part. Therefore we can split the results into a meaningless divergent part21 and a finite part. Of course there are several ways to split up the obtained term and the choice is somehow ambiguous. A given choice defines a scheme. In general there are three different kind of schemes which we will briefly discuss, also we will use the convention Í = 4 − 2w. (µ) scheme In the (µ) scheme one defines the divergent part via the term, where, 1 + ¢239 ŵ 1 1 = − .ä + ln4240 ŵ w and ¢ is the constant containing the finite terms of the amplitude. 21 The part is meaningless because it gets cancelled by the corresponding counter term. 77 Effective Field Theories (MS) scheme In the (MS) scheme we define the divergent part only by 1/w. RRRR) scheme (MS RRRR) scheme is defined by 1/ŵ. The divergent part in the (MS There is not much to say about all of this except that when dealing with renormalization one should always pay attention which scheme is used to avoid confusions. 5.2. The Power Counting Method So far we have seen how a theory can, step by step be renormalized but we haven't considered the question if a theory is renormalizable in general at all or if there are any constraints that distinguish between renormalizable and nonrenormalizable. Luckily there is a very simple method, the so called power counting method by which one can easily tell if a theory is in general, renormalizable or not. The key idea is to count the "superficial degree of divergence" in a given process. In the following we will develop the concept of power counting and will apply it on the one hand explicitly to the + theory and afterwards derive the general formalism valid in every theory. After having derived the general formalism we will apply it also to QED as well as SED. First of all we recall that the renormalization procedure was necessary because of infinities in our theory. Thereby it's a well known fact that a loop integral corresponding to a one particle irreducible diagram can either converge or diverge depending on the number of propagators within a loop. A general loop process looks like, Ç5~ [ 1 æ ∈ ℝ241 [K It seems natural to introduce the degree of divergence Í via order of the numerator minus order of the denominator. Í = ,S[;,¢o − ,S[,;¢o 242 78 Effective Field Theories By introducing the superficial degree of divergence we know that the corresponding loop process converges or diverges like, depending on the value of Í. Ç5~ Ì- 243 Because any process can be represented by Feynman graphs, it seems natural to deduce the superficial degree of divergence right out of the Feynman diagram. In the following we will show how in the three above mentioned theories the degree of divergence can explicitly be counted. Therefore we will start with the general + theory in four spacetime dimensions. In the + theory different particles are coupled in a point. If we are considering a loop process we clearly find, Í = 4T − 2U244 Where T is the number of loops and U is the number of internal lines. The factor four corresponds to the four dimensional integration [ for each loop, the factor two corresponds to the propagator which is proportional to . So far we have included only loops and internal lines, but it would be more beneficial if we would reexpress the degree of divergence via the external lines which can be counted much easier. So if we introduce the number of external lines : and the number of vertices we find in our given theory the general connection, = 2U + :245 This result is easily explained because every vertex couples particles. The particles can on the one hand belong to external lines and on the other hand correspond to propagators. Because internal lines always connect two vertices the number of internal lines is multiplied by a factor of two. The last general result can be obtained when looking directly at given Feynman diagrams. In (fig. 21) we see two types of Feynman diagrams corresponding to = 4. 79 Effective Field Theories fig. 21 Example processes in the + theory. First of all by looking at the left diagram we see that (245) is valid because 4 ∙ 2 = 8 = 2 ∙ 2 + 4. Looking at the right diagram we also see that (245) holds because 4 ∙ 1 = 4 = 2 + 2 . We are now looking for a relation between the loops, vertices and internal lines. Looking at (fig. 21) we conclude, T = U − + 1246 The additional term of one reflects overall momentum conservation. This holds for any value of (as can easily be verified looking at other processes in different magnitude of ). So putting together (244), (245) and (246) we find, Í = − 4 + 4 − :247 where we have used the notation Í corresponding to the + theory. That's the final result for the degree of divergent for any integer number . It is interesting that the degree of divergence is just a function of the number of vertices and the external lines. We also conclude that if we leave fixed depending on our theory can either become more divergent or less divergent. In order to get more familiar with the concept we will discuss the solution in the context of = 4 and also = 6. For = 4 i.e. in a + theory we obtain, Í = 4 − :248 So the superficial degree of divergence is a function of the number of external lines only. Furthermore we see that : = 2 and : = 4 are the only two divergent 80 Effective Field Theories contributions which are shown in (fig. 21)22. We see that there exist only two general possibilities where divergences occur so we have a finite set of divergent graphs and therefore we know that our theory is renormalizable if the divergences can be cancelled by corresponding terms in the Lagrangian. That's what we meant above in the phrase finite set of processes . If instead of = 4 we consider = 6, we obtain for the degree of divergence, Í9 = 2 + 4 − :249 We see immediately an interesting property, namely that the degree of divergence is now dependent on two quantities, the number of vertices and the number of external lines. Therefore we have an infinite set of possibilities of constructing Feynman graphs with Í9 ≥ 0 and therefore the theory is not renormalizable. In the chapter renormalization in an effective field theory we will explicitly discuss a graphical interpretation why there is an infinite set of non-renormalizable terms, but more on that later. So far we have discussed only these two examples but if we were looking at other values of we would work out the general rule, that every theory with ≤ 4 is renormalizable and all others i.e. for > 4 are non-renormalizable. if we are now looking directly at the interactions terms for ∈ {3,4,5,6} we have, ;ô + ,ô9 + , 1 1 ô + , 9 ô + 9 ; ; If we look closely we can convert the statement that theories are nonrenormalizable for > 4 into the statement that a theory is non-renormalizable if the mass dimension of the coupling of the interaction term is smaller than zero. This result is valid in general, though the proof isn't trivial. That's why we won't present it here but again just quote the important result: A theory is non-renormalizable if the mass dimension of the coupling is smaller than zero For example, we consider the two possible interaction terms 0Ψ 1. Yukawa interaction ôΦΨ = 2. Φ Φ? ? By dimensional analysis we know that the dimension of ô is equal to zero so the first interaction term would describe a renormalizable theory. The dimension of is equal to minus two so we immediately know that a theory including the given interaction term is non-renormalizable. 22 There is no graph corresponding to : = 3 because of symmetry under Φ → −Φ. 81 Effective Field Theories We now turn to develop the superficial degree of divergence in a general formalism though we know by now how to distinguish between renormalizable and non-renormalizable theories, nevertheless it is an interesting concept worth looking at. Therefore we introduce the following notations: U¤ = ;æ,So,¢ÇÇ,SoL5,¢ :¤ = ;æ,S,o,¢ÇÇ,SoL5¢ ¤ = ;æ,S+,o,SoL5,¢ Also we know that every propagator Δ¤ [ of type ¢has the asymptotic behaviour, Δ¤ [~[ K = [ 9§V-9 250 where we have introduced ¤ where for a scalar field § = 0. For a fermionic field n = and for a vector field e.g. the photon field A = 0. 9 So far we haven't discussed the influence of derivatives in the vertex function. For each derivative in a vertex the superficial degree of divergence increases by one. So if we have ¤ derivatives of type ¢ per vertex the superficial degree of divergence is given by, Í = 4T + U¤ 2¤ − 2 + ¤ ¤ 251 ¤ ¤ Once again we find (in analogy to our introductory example) that the number of loops T is given by, T = 1 + U¤ − ¤ 252 ¤ ¤ Therefore we can replace the number of loops T in the degree of divergence and obtain, Í = 4 + U¤ 2¤ + 2 + ¤ ¤ − 4 253 ¤ ¤ Now we are left to find another relation for to replace U¤ . Therefore we introduce the number of particles of type ¢ connected to a vertex of type æ via K¤ . Having introduced this new quantity we find (again in analogy to our previous example), 2U¤ + :¤ = K K¤ 254 K 82 Effective Field Theories So we have found an expression for U¤ , which we can use in (253) to finally obtain an expression for the superficial degree of divergence depending only on the number of external lines and the number of vertices, Í = 4 − :¤ ¤ + 1 + K K¤ ¤ + 1 + ¤ ¤ − 4 ¤ ¤ ¤ K 255 This is now the general result to obtain the degree of divergence. Because the formula isn't that trivial we present two examples to get more familiar with it. First of all we are looking at QED. In any QED process the number of derivatives ¤ in a vertex function is zero. On the other hand the number of vertices is independent of the type of particles so we can rewrite, K K¤ ¤ + 1 = K K¤ ¤ + 1 256 ¤ K ¤ K Looking at the second term we can now explicitly insert the rules of QED, 1 K¤ ¤ + 1 = 2 + 1 + 10 + 1 = 4257 2 ¤ because in QED the only possibilities are that we couple two fermions with n = 9 and one photon with A = 0. So we can rewrite the degree of divergence in QED as, ÍWäÌ = 4 − :¤ ¤ + 1 + 4K − 4¤ 258 ¤ K ¤ Because the sum over æ and ¢ are equal and therefore the vertex expressions cancel each other out we finaly obtain, 3 ÍWäÌ = 4 − :¤ ¤ + 1 = 4 − :n − :@ 259 2 ¤ where we introduced the short notation for the number of external lines :n that are fermions and also for the number of external photons :@ . Shown in (fig. 22) are all three possible one particle irreducible graphs of degree of divergence bigger or equal to zero that can occur in QED. Though not all diagrams have degree of divergence of zero it can be seen by directly doing the calculation that they are only logarithmically divergent in reality. 83 Effective Field Theories 1PI Í=0 1PI 1PI Í=1 Í=2 fig. 22 One particle irreducible Feynman graphs in QED with degree of divergence Í ≥ 0. Again we see that there is only a finite set of possibilities with ÍWäÌ ≥ 0 and also each of the terms in (fig. 22) corresponds to a term in the QED Lagrangian. We conclude that QED is renormalizable. As a last example we will look at the case of SED. SED has two different kind of interaction terms, one of which is containing a derivative. We will first look at each sum in the degree of divergence independently and add them up in the end. Looking at the first sum we see, :¤ ¤ + 1 = :§ + :@ 260 ¤ where :§ is the number of external scalar particle lines. Next we look at the second term, K K¤ ¤ + 1 = 2 + 1 + 2 + 2 = 3 + 4 ¤ K 261 Here we used the notation that the number in the index corresponds to an npoint interaction. So let's view the last term, ¤ ¤ − 4 = 1 − 4 + −4 = −4 − 3 262 ¤ So by putting everything together we find, 84 Effective Field Theories ÍãäÌ = 4 − :§ − :@ 263 Again we are representing all one particle irreducible graphs (fig. 23) which have the superficial degree of divergence equal or bigger than zero. 1PI 1PI 1PI 1PI 1PI Í=0 Í=0 Í=0 Í = 2 Í=1 1PI Í=2 fig. 23 One particle irreducible Feynman graphs in SED with degree of divergence Í ≥ 0. Considering the solution we conclude that SED is also renormalizable, which we could easily have told before because of dimensional reasons. Nevertheless the power counting method is quite a useful tool to describe the degree of divergence in a one particle irreducible diagram. In both cases (QED and SED) we have just shown the one particle irreducible graphs that give non-zero contribution to the amplitude. All other imaginable graphs are either zero because of Lorentz invariance of the current or by gauge invariance. In the next chapter we will first give a graphical reasoning why the + 9 theory, which is an example of a theory of mass dimension higher than four, is nonrenormalizable before we answer the question how the renormalization procedure applies in an effective field theory. 85 Effective Field Theories 5.3. Renormalization in an Effective Field Theory So far we have only dealt with renormalizable theories i.e. those in which the individual terms had dimensions smaller than four (as long as we are working in a four-dimensional spacetime). In the following we will restrict ourselves to a four dimensional spacetime. The generalization to different dimensions can be simply done but won't be discussed here. We look at the example of + 9 theory to illustrate why terms of mass dimension smaller than four are renormalizable (the general prove is quite difficult and won't be discussed here). The Lagrangian is given by, 1 1 ô ℒ = +9 − ;9 + 9 − + 9 264 2 2 6! We can now start to think what kind of Feynman graphs we can build in this theory (fig. 24). fig. 24 Possible Feynman diagrams due to Φ9 theory including required counter-terms. We see that in order to carry out the renormalization procedure in our Lagrangian there also must be terms of order four and order eight. So we complement our Lagrangian by the given terms in the hope that our new Lagrangian can then be renormalized (though we already know from the previous chapter that this won't be the case). We obtain: 86 Effective Field Theories 1 1 ô μ ℒ = +9 − ;9 + 9 − + − + 9 − + ' 265 2 2 4! 6! 8! Given this new Lagrangian we again can start to think which kinds of interactions we can construct. Of special interest is the + ' interaction term. Some possible interactions are shown in (fig. 25). fig. 25 Possible Feynman diagrams in Φ' theory including required counter-terms. So once again we have bad luck because we now also have to include + 9 terms in our Lagrangian in order to cancel out the infinities emerging from the last graph of (fig. 25). Now we can easily see that if we would now include these new terms we could build up an theory that would need another higher term. This would go on and on up to infinity and therefore it wouldn't make sense to apply the renormalization procedure because we would get an infinite set of unknown parameters. This is exactly the content of eq. (247) claiming that Í9 is smaller than zero, and the theory therefore non-renormalizable, for number of external lines equal to eight. So we see that in general theories that involve terms of dimension greater than four it can't be (so far) renormalized. The theory is therefore called non-renormalizable. We now take a brief review of scalar electrodynamics that we developed in the chapter symmetries. The Lagrangian was given by, 87 Effective Field Theories 1 ℒãäÌ = Φ= ∂ Φ − ;9 Φ= Φ + c2 Ø + c 9 2 2 Φ= Φ − ? ? 4 266 For our theory to be renormalizable we have to add an interaction term in the form of, ô 9 − OΦ= ΦS 267 4 such we can compensate the following interactions shown in (fig. 26) by a four point interaction. The arguments follow the same logic as the previous example that shows the non-renormalizability. fig. 26 Graphical representation for including additional interaction term to SEDLagrangian. So the complete Lagrangian for renormalizable scalar electrodynamics can be written as, ℒãäÌ,éléC = Φ= ∂ Φ − ;9 Φ= Φ + c2 Ø + c 9 2 2 Φ= Φ + λ 1 9 − OΦ= ΦS − ? ? 268 4 4 Now we concentrate on what consequences we can obtain for our effective field theories. Once again any effective Lagrangian can be written as, ℒlnn = ℒ) + $ f$ 269 88 Effective Field Theories where the operators f$ are ordered by their dimension . We can alternatively write the effective Lagrangian as follows, denoting by ℒ the term in the Lagrangian of dimension : ℒlnn = ℒX + ℒ + ℒ9 + …270 In the chapter construction of effective Lagrangians we saw that every term of dimension is proportional to some energy scale Λ (which is proportional to some heavy mass) via Λ- . So we directly see that the effective Lagrangian at ä Y a given energy : is ordered in terms of Ö . The first idea that could transpire (in order to obtain an renormalizable effective theory) is to just use the ℒX term of our effective Lagrangian. Thereby we would neglect higher orders in 1/Λ . Our result would therefore be renormalizable under the circumstance that we would make an error in accuracy of order :/Λ. If we are considering energies : ≪ Λ this ansatz would be very good because our error would be small. Nevertheless, if our energy increases and draws near Λ our error would increase. How can we now solve the problem of increasing energy and increasing error? We have no choice but to take the next dimensional term also into account, so we are calculating with ℒX + ℒ . As seen in the introductory example of this chapter we now would also need the next dimensional term and so on in order to renormalize the theory. Our theory would then again include an infinite set of terms. But now comes the crucial point. The next dimensional term (in our case at the moment the term ℒ9 is of ä 9 order Ö so if we are restricting our accuracy up to that order we don't have to include the term ℒ9 and therefore all higher terms. So we can calculate with ℒX + ℒ and even renormalize the results with the compromise that we are ä 9 only considering terms up to order Ö . So in general we can include all terms ä §h up to order Ö and therefore use the terms ℒX + … + ℒh§ . So we make ä §h an error of order Ö in our calculation. In return we gain a Lagrangian of finite terms that is renormalizable up that order in accuracy. In the end our theory gains predictive power again, so a theory of possibly infinitely many terms (as our effective field theory) can be renormalized by making an error in prediction up to a given order. "A non-renormalizable theory is just as good as a renormalizable theory for computations, provided one is satisfied with a finite accuracy." (A. Manohar, [1.], page 12, chapter 3 Renormalizable Theories vs. Effective Theories) 89 Effective Field Theories In order to present an example we consider an effective Lagrangian that is equal to the + theory. The Lagrangian is given by, 1 1 ô ℒ = +9 − ;9 + 9 − + 271 2 2 5! As shown in (fig. 27) for renormalization we would need the term corresponding to a + 9 interaction. If we are considering an accuracy up to order ä 9 Ö we don't need to consider + 9 because it can't occur up to our considered order. So the interaction as shown in (fig. 27) doesn't exist (in our order of accuracy) and has therefore not been taken into account. Therefore we can apply our renormalization procedure and obtain a finite set of variables that can be calculated (instead of an infinite set which occurs when considering terms to every order in the expansion). fig. 27 One loop diagram and corresponding counter-term. 5.4. Matching at One Loop Level In the following chapter we will review our first example for the matching procedure and will calculate the effective Lagrangian up to the first loop order considering again the toy model presented in [2.]. As a reminder the Lagrangian of the ordinary theory was given by, 1 1 1 1 ô 9 9 ℒ = O S − ;9 9 + O ΦS − >9 Φ9 − 9 Φ272 2 2 2 2 2 and we already obtained the effective Lagrangian at tree level (review chapter 3.4.2. The matching procedure for detailed calculations) as, 90 Effective Field Theories 1 1 ô9 ℒlnn = 9 − ;9 9 + 273 8>9 2 2 Next we consider the matching at one loop level. Therefore we will need the techniques developed in the context of renormalization. We expect corrections to the effective Lagrangian such as, 1 1 1 3ô9 ℒlnn = 9 − ;9 + ¢9 − E− 9 + æG 274 > 2 2 4! We first consider the lowest order loop diagrams leading to corrections to the 9 and terms. These calculations we will do explicitly. The remaining correction due to æ we will just quote because these are obtained in the same way. To lowest order in the full theory we have two loop diagrams (fig. 28 left) which will correspond to the diagrams of the effective action shown in (fig. 28 right). fig. 28 Lowest order one loop diagrams in the full theory (left) and in the effective theory (right). In order to match we calculate the two loop diagrams of (fig. 1 left) starting with the very left and requiring that > ≫ 5 where 5 is the corresponding momentum transfer. So for the first loop diagram we find, ℳ = −ô9 [ 1 1 275 9 9 9 2 5 − [ − > [ − ;9 by making use of Feynman's trick we obtain, ℳ = −ô9 1 9 276 − ¢9 9 2 ) where = [ − 5 and ¢9 = >9 − ;9 + 59 − 1 − ;9 . 91 Effective Field Theories Now by using the technique of dimensional regularization and the general formula for those kinds of integrals, which has been quoted in the context of renormalization, we obtain (in the limit > ≫ 5), − 1 μ9 E + ln4 − . + ln E G ä >9 16 9 w ) ;9 ;9 − ln E1 − 9 G + 9 G = > > ℳ = −ô9 μ9 ;9 ;9 ô9 1 ln E 9 GG277 = E + ln E 9 G + 1 + 9 16 9 ŵ > > − ;9 > Now for the second diagram we find, ℳ9 = −ô9 [ 278 9 9 9 2 [ − ;9 2 c −> in the limit of > ≫ c and again applying the techniques of dimensional regularization we find, ℳ9 = − ô9 ;9 9 4 μ 9 Γ−1 + w = 32 9 >9 ; ô9 ;9 1 ;9 =− E + 1 − ln E 9 GG279 32 9 >9 ŵ μ Considering the first diagram in the effective theory that we have already calculated in the context of renormalization in the + theory. As a result we found, ℳlnn, = 1 ;9 9 ; + 1 − ln E E GG280 32 9 ŵ μ9 where in our case is obtained from the effective Lagrangian at tree level and has therefore the value, =− 3ô9 281 >9 At last we consider the remaining diagram. The corresponding amplitude is very simple, 92 Effective Field Theories ℳlnn,9 = −¢282 In order that both theories match up to first loop order we insist that, that means ℳ + ℳ9 = ℳlnn, + ℳlnn,9 283 ô9 1 μ9 ;9 ;9 + ln + 1 + ln E E G E GG 16 9 ŵ >9 >9 − ;9 >9 ;9 ô9 ;9 1 E + 1 − ln − E GG = 32 9 >9 ŵ μ9 =− 3ô9 ;9 1 ;9 E + 1 − ln GG − ¢284 E 32 9 >9 ŵ μ9 working in the MS scheme we can drop the 1/ŵ terms and "ban" them in the counterterm parts. Therefore our equation reads, ô9 ;9 ;9 μ9 ;9 ô9 ;9 ln E1 − ln + 1 + − E E Eln E G GG GG = 16 9 >9 >9 − ;9 >9 32 9 >9 μ9 =− 3ô9 ;9 ;9 E1 − ln E GG − ¢285 32 9 >9 μ9 expanding in leading orders of ;9 />9 we find, ô9 μ9 ;9 ;9 ô9 ;9 ;9 + 1 + ln − E1 − ln Eln E G E GG E GG = 16 9 >9 >9 >9 32 9 >9 μ9 =− 3ô9 ;9 ;9 E1 − ln E GG − ¢286 32 9 >9 μ9 By now that's a nice result but leaves the question of how "big" the value μ might become. It seems natural to bound μ by the heavier mass > and for simplicity we can set them both equal, i.e. μ = >. Our equality therefore simplifies further and becomes, ô9 1 ;9 3;9 ;9 ô9 3 ;9 3;9 ;9 − + ln = + ln E1 E GG E− E GG − ¢ 16 9 2 >9 2>9 >9 16 9 2 >9 2>9 >9 287 By comparison we see that the only remaining variable that needs to be matched takes the value, 93 Effective Field Theories ¢=− ô9 ;9 + E1 G288 16 9 >9 õ C For a general μ we would have obtained ¢ = − 9¼ 1 + 1 + ln , so in general we see that ¢ = ¢μ. So in the limit where the mass > becomes so heavy it satisfies > ≫ c, ;, we find, ¢ NOOOOP − ≫Õ,C ô9 289 16 9 and therefore our effective Lagrangian becomes, 1 1 ô9 1 3ô9 9 ℒlnn = 9 − E;9 − − + æG 290 G E− 16 9 >9 2 2 4! It remains to match the value of æ, which can be done by considering the diagrams of (fig. 29) and again match in the regime of > ≫ c, ;. fig. 29 Higher orders leading to æ correction term. Because the calculations are now similar we will skip them and just quote the result, æ = æI ô 1 291 16 9 > where æI is a constant that we don't quote here. 94 Effective Field Theories 6. Photon-Photon Scattering 6.1. The Euler-Heisenberg-Lagrangian (EHL) Before starting the formal discussion of the Euler-Heisenberg effective Lagrangian as a low energy theory of QED, ee first of all take a look back at classical electrodynamics. The very heart of electrodynamics are the four Maxwell equations, +:Ðç = 4[ Ðç = 0 +É 1 Ðç = 0 o:Ðç + % É 1 4 Ðç − % :Ðç = \ç oÉ From these one can derive the wave equations for source- free fields, 292 1 9 :Ðç − ∆:Ðç = 0293 9 % 1 9 Ðç − ∆É Ðç = 0294 É 9 % A central feature of these equations is the linearity, i.e. there are no interaction term. So classically there can be no light-by-light scattering. Now let's turn our view to QED. The QED Lagrangian involves an interaction part namely, ℒ$% = cΨ. Ψ2 295 so it couples two fermions to one photon. Because of that one easily finds the lowest order gauge, Lorentz and charge conjugation and parity invariant process in which photon-photon scattering is possible (fig. 30). We will deal later with this scattering more explicitly. For now we want to focus on the idea due to Euler and Heisenberg back in the 1930's. Their idea was to introduce a low energy effective Lagrangian with the assumption that the Lagrangian should be gauge invariant (under U(1)) and also should not violate parity. 95 Effective Field Theories The free photon field is described by the Lagrangian, 1 ℒnéll = − ? ? 296 4 The lowest order Lorentz-, parity, charge conjugation and gauge invariant interactions that can be added are, 9 ℒä] = O? ? S + 9 ? ? ø ?ø, ?, 297 fig. 30 Photon-photon scattering in lowest order QED (left) and in EHET (right). We see that the interaction term is describing the four point interaction of photons (fig. 30) in the so called Euler-Heisenberg effective theory (EHET). Furthermore we can derive some properties of the interaction Lagrangian also called the Euler-Heisenberg effective Lagrangian (EHL). By dimensional analysis we see that the mass dependence of the coupling constants have to be, ℒ$% = 4 = 8 + Ê ,9 Ë so we know that, where Êæ ,9 Ë = 0. ↔ Ê ,9 Ë = −4298 ,9 = 96 æ ,9 299 ; Effective Field Theories The constants can be calculated by matching. They are given by: =− ù9 7ù 9 = 300 9 36; 90; For reference see: [1.] - [4.] as well as [20.]. Another feature is obtained by looking at the photon self energy in the EHET. The corresponding one loop diagram is given in (fig. 31). fig. 31 One loop diagram for photon self energy in the Euler-Heisenberg effective theory. The contribution from this diagram gives zero value because the massless loop is scale free, in other words there are no radiative corrections to the photon propagator in the given theory, i.e. the propagator remains only the free propagator. In the next chapter we will take a deeper look at the QED content of this effective field theory. Thereby we will perform some calculations in the QED regime and show how the infinities come into the process as well as some short calculation of the structure of the scattering amplitude in the EHET. 6.2. Photon-Photon Scattering as a QED Process In the last chapter we already showed one diagram representing photon-photon scattering in QED. Now we will take a more in depth look at all possible lowest order diagrams corresponding to the given process. There are six diagrams contributing to the scattering amplitude, nevertheless there are only three different diagrams. The other three follow by simply changing the direction of the fermion propagators. The first three diagrams are represented in (fig. 32). 97 Effective Field Theories [ [ [ 5 5+[ [9 5 + [ 5 + [ + [9 5 5+[ [ [9 5 + [ 5 + [ + [9 [ [ [ 5 − [ [9 [ 5−[ 5 − [ + [9 [ fig. 32 Lowest order box-diagrams for photon-photon scattering in QED. By power counting arguments we see that we have the degree of divergence, Í = 4 − 4 = 0. So we estimate at most a logarithmic divergence. Actual calculations will show that by considering all six possible diagrams the loop integration turns out to be finite, but more on that later. In the next section we will deal with these loop diagrams in QED. 98 Effective Field Theories 6.3. The Scattering Amplitude in QED In order to obtain the low energy effective action one must first of all calculate the scattering amplitude in the fundamental underlying theory. In the following we will explicitly do some calculations in order to show how the infinities come into the process. Thereby we won't explicitly calculate the whole scattering amplitude but just give an impression of the general procedure. From (fig. 32) we already know the Feynman graphs in QED and we can now easily write down the corresponding scattering amplitudes for the three shown graphs and also for the graphs with the internal momentum flow in the other direction. We therefore find, 5 1 1 1 ∗ 3 ℳ = −, !. w . w ∙ 2 . ø 5ø − ; ., O5, + [,, S − ; 1 1 ∙ . * w*∗ 4 H . ) w) 2 ^ . O5H + [H, + [H,9 S − ; . O5^ + [^, S − ; 301 1 1 5 ℳ9 = −, !. w 1 ø . * w*∗ 4 , ∙ 2 . 5ø − ; . O5, + [,, S − ; ∙ . w∗ 3 ℳ = −, ∙ . * w*∗ 4 1 . H O5H + [H,9 + [H, S − ; . ) w) 2 1 . ^ O5^ + [^, S − ; 302 5 1 1 !. w∗ 3 ø . w 1 , ∙ 2 . 5ø − ; . O5, − [,, S − ; . H O5H 1 − [H, + [H,9 S − ; 99 . ) w) 2 . ^ 5^ 1 − [^, − ; 303 Effective Field Theories ℳ = −, ∙ . ) w) 2 ., O5, ., O5, ∙ . w∗ 3 ., O5, . * w*∗ 4 . H O5H 1 − [H, S − ; 1 5 !. w 1 ø ∙ 2 . O5ø − [ø, S − ; 1 ∗ 4 ∙ . * w* . w∗ 3 . H O5H 1 − [H, S − ; 1 5 !. w∗ 3 ø ∙ 2 . O5ø + [ø, S − ; + [,, − [,, S − ; ∙ . 1 w1 1 ∙ 1 305 −; . ^ 5^ 1 ∙ 1 304 −; . ^ 5^ − [,, − [,, S − ; ℳ9 = −, ∙ . ) w) 2 1 − [,, − [,, S − ; ℳ = −, ∙ . ) w) 2 5 1 !. w 1 ø ∙ 2 . O5ø − [ø, S − ; . * w*∗ 4 . H O5H 1 + [H, S − ; ∙ 1 306 −; . ^ 5^ In the following we will concentrate on calculating parts of the first scattering amplitude. The others will simply follow by interchanging the corresponding external momenta. We can rewrite the scattering amplitude as follows, ℳ = −, ∙ 5 !. . ø 5ø + ;. . , O5, + [,, S + ; ∙ 2 ∙ . * O. H O5H + [H, + [H,9 S + ;S. ) . ^ O5^ + [^, S + ; ∙ 1 1 1 1 ∙ 59 − ;9 5 + [ 9 − ;9 5 + [ + [9 9 − ;9 5 + [ 9 − ;9 ∙ w 1w∗ 3w*∗ 4w) 2307 100 Effective Field Theories At first glance one would now try to calculate the occurring four, six and eight gamma traces. Nevertheless later calculations will show that one only has to calculate a four gamma matrix trace. For completeness we will calculate the six and eight gamma traces in the following short digression because one can obtain a nice recursion formula for calculating traces with an even number of gamma matrices. Afterwards we will return to the scattering amplitude. For simplicity let us denote, !. ¤ . K … . =: !¢æ … 308 So we find using the Clifford algebra {. , . } = 2 !¢æ,S = 2ln !¢æ − !¢æS,309 switching again the S term we obtain, !¢æ,S = 2ln !¢æ − 2n !¢æ, + !¢æS,310 so we can switch the "S" five times and use the cyclicity of the trace to find, !¢æ,S = ln !¢æ − n !¢æ, + n !¢æ, − −Kn !¢, + ¤n !æ,311 By now we can use the well known identity for the trace of four gamma matrices in Í dimensions, !¢æ = 4O¤ K − ¤ K + ¤K S312 and therefore find the final result, !¢æ,S = 4ln ¤ K − ¤ K + ¤K − −4n ¤l K − ¤ Kl + ¤K l + 4n ¤l K − ¤ Kl + ¤K l − −4Kn ¤l − ¤ l + ¤ l + 4¤n Kl − K l + K l 313 101 Effective Field Theories The same can now be done for the trace of eight gamma matrices, !¢æ,Sℎ = ¦$ !¢æ,S − n$ !¢æ,ℎ + l$ !¢æSℎ − −$ !¢æ,Sℎ + $ !¢æ,Sℎ − K$ !¢,Sℎ + ¤$ !æ,Sℎ 314 We can unite the solutions for six and eight gamma matrices into a general result, !¢ ¢9 … ¢9h9 = ¤í ¤í !¢ ¢9 … ¢9 − −¤í ¤í !¢ ¢9 … ¢9- ¢9h + … + ¤ ¤í !¢9 … ¢9h 315 The big question now is how many different traces we obtain in our scattering amplitude expression. From combinatorics we find that we have 4 4 4 + + = 8 possible traces where we have used the fact that the trace 4 2 0 over an odd number of gamma matrices vanishes. The eight possibilities are, !Ê. . ø . . , . * . H . ) . ^ Ë, ; !. . . * . ) , ;9 !Ê. . ø . . , . * . ) Ë, ;9 !Ê. . ø . . * . H . ) Ë, ;9 !. . ø . . * . ) . ^ , ;9 !Ê. . . , . * . H . ) Ë, ;9 !Ê. . . , . * . ) . ^ Ë, ;9 !. . . * . H . ) . ^ 316 In order to obtain the divergent contribution of the amplitude we have to consider those terms that give maximal contribution of inner momenta in the numerator, i.e. the term involving the eight gamma traces. So we find for the part of the first scattering amplitude containing the divergences, ℳ ,' ≔ −, 5 w 1w∗ 3w*∗ 4w) 2 ∙ 2 5ø O5, + [,, SO5H + [H, + [H,9 SO5^ + [^, S!Ê. . ø . . , . * . H . ) . ^ Ë ∙ ∙ 59 1 1 1 1 9 9 9 9 9 − ; 5 + [ − ; 5 + [ + [9 − ; 5 + [ 9 − ;9 102 317 Effective Field Theories We are especially interested in the term proportional to 5ø 5, 5H 5^ in the numerator, i.e. 5ø O5, + [,, SO5H + [H, + [H,9 SO5^ + [^, S → 5ø 5, 5H 5^ Thus we find, ℳ ∙ 59 ,' 318 5 = −, w 1w∗ 3w*∗ 4w) 2 ∙ 2 ∙ 5ø 5, 5H 5^ !Ê. . ø . . , . * . H . ) . ^ Ë ∙ 1 1 1 1 9 9 9 9 9 − ; 5 + [ − ; 5 + [ + [9 − ; 5 + [ 9 − ;9 319 In order to calculate the integral expression we now use a result due to R. Feynman namely, 1 ∏$( ¢$ ∙ ) = − 1! -∑í2 ya y ) - ) - 9 ⋯ 1 ) -∑í2ð ya y -9 ∙ 320 ∑$( ¢$ $ + ¢ O1 − ∑$( $ S In the chapter on renormalization we have used this expression for the case = 2 in order to write, 1 1 = 9 321 ¢æ ) O¢ + æ1 − S For our integral now we consider the case where = 4 , i.e. 1 = 6 ¢æ ) ) - L ) --M b 1 O¢ + æL + b + 1 − − L − bS 103 322 Effective Field Theories Applying this result to ℳ ,' and leaving for the moment the factors, −, w 1w∗ 3w*∗ 4w) 2!Ê. . ø . . , . * . H . ) . ^ Ë323 we find, ∙ ℳ ,$,A = 59 5 3! 2 ) ) - L ) --M b ∙ 5ø 5, 5H 5^ + 25−[ + [ − [ L + [9 b + [ + 2[ [9 b − ;9 By completing the square we can rewrite the denominator as, 5 + ¢9 − æ325 where, and, 324 ¢ = −[, + [ − [ L + [,9 b + [, 326 æ = −[ + [ − [ L + [9 b + [ 9 − 2[ [9 + ;9 327 Therefore we find, ℳ ,' = −6 ∙ ) ) - L ) --M b , w 1w∗ 3w*∗ 4w) 2 ∙ 5ø 5, 5H 5^ 5 !Ê. . ø . . , . * . H . ) . ^ Ë 328 2 5 + ¢9 − æ In order to find an expression for the occurring integral we start with, 1 329 5 + ¢9 − æ A where we have introduced the short notation, ≔ A 5 2 104 Effective Field Theories The measure of (328) is invariant under translation such we can change 5 → 5 + c and therefore obtain, 1 1 = 330 9 9 A 5 + ¢ − æ A 5 + c + ¢ − æ we now differentiate both sides by /c ø and find, 5 + c + ¢ø 331 9 h A 5 + c + ¢ − æ 0 = −2 Again we now differentiate 330 by /c , and use the product rule for differentiation as well as the identity = õ õ = to obtain, 5 + c + ¢ø 5 + c + ¢, ø, −2 + 4 + 1 9 h 9 h9 A 5 + c + ¢ − æ A 5 + c + ¢ − æ = 0332 We proceed with the differentiation of (331) by /c H and find, 4 + 1Êø, 5 + c + ¢H + øH 5 + c + ¢, + ,H 5 + c + ¢ø Ë 5 + c + ¢9 − æh9 A − 8 + 1 5 + c + ¢ø 5 + c + ¢, 5 + c + ¢H + 2 = 0 5 + c + ¢9 − æh A at last we differentiate 332 by /c ^ , Êø, H^ + øH ,^ + ,H ø^ Ë − 5 + c + ¢9 − æh9 A 0 = 4 + 1 Êø, 5 + c + ¢H 5 + c + ¢^ Ë − 5 + c + ¢9 − æh A −8 + 1 + 2 ÊøH 5 + c + ¢, 5 + c + ¢^ Ë − 5 + c + ¢9 − æh A −8 + 1 + 2 105 333 Effective Field Theories ÊH, 5 + c + ¢ø 5 + c + ¢^ Ë − 5 + c + ¢9 − æh A −8 + 1 + 2 Êø^ 5 + c + ¢H 5 + c + ¢, Ë − 5 + c + ¢9 − æh A −8 + 1 + 2 Ê^, 5 + c + ¢H 5 + c + ¢ø Ë − 5 + c + ¢9 − æh A −8 + 1 + 2 ÊH^ 5 + c + ¢ø 5 + c + ¢, Ë + 5 + c + ¢9 − æh A −8 + 1 + 2 +16 + 3! Ê5 + c + ¢ø 5 + c + ¢, 5 + c + ¢H 5 + c + ¢^ Ë 5 + c + ¢9 − æh − 1! A 334 We are now in the position to let c = 0 again and by using (331) we can rewrite our equality 333 to, 5 + ¢ø 5 + ¢, 5 + ¢H 5 + ¢^ = 5 + ¢9 − æh A = Êø^ ,H + ,^ øH + H^ ø, Ë 1 335 5 + ¢9 − æh9 4 + 3 A Rewriting the left hand side of the equality 334we encounter, 5ø 5, 5H 5^ ¢ø 5 + ¢, 5 + ¢H ¢^ + + 9 h 5 + ¢9 − æh A 5 + ¢ − æ A ¢ø 5 + ¢, 5 + ¢^ ¢H ¢ø 5 + ¢^ 5 + ¢H ¢, + 336 9 h 5 + ¢9 + æh A A 5 + ¢ − æ + where we used the fact that all integrals that are proportional to 5 + ¢ø or 5 + ¢ø 5 + ¢, 5 + ¢H vanish (odd function integrated over full Minkowski space). 106 Effective Field Theories Again we can use 331 to simplify the expression and with 334obtain, 5ø 5, 5H 5^ Êø^ ,H + ,^ øH + H^ ø, Ë 1 = − 9 h 5 + ¢9 − æh9 4 + 3 A A 5 + ¢ − æ − ,H ¢ø ¢^ ,^ ¢ø ¢H 1 1 − 9 h 2 + 3 A 5 + ¢ − æ 2 + 3 A 5 + ¢9 − æh − ^H ¢ø ¢, 1 337 2 + 3 A 5 + ¢9 − æh What a result! We managed to express the integration on the left hand side of 335 via a integration that is simple to perform. In fact we can reduce the equation a little further by substituting 5 + ¢ = c. For simplicity we simply set c = 5 and find the final result, 5ø 5, 5H 5^ Êø^ ,H + ,^ øH + H^ ø, Ë 1 = − 9 h 59 − æh9 4 + 3 A A 5 + ¢ − æ − ,H ¢ø ¢^ ,^ ¢ø ¢H 1 1 9 − 2 + 3 A 5 − æh 2 + 3 A 59 − æh − ^H ¢ø ¢, 1 9 338 2 + 3 A 5 − æh We can now calculate the eight-gamma scattering amplitude that was again given by, ℳ ,' = −6 A ) ∙ ) - L ) --M b!Ê. . ø . . , . * . H . ) . ^ Ë ∙ 5ø 5, 5H 5^ w 1w∗ 3w*∗ 4w) 2339 5 + ¢9 − æ using our result 336 for = 0we find, 107 Effective Field Theories ℳ ,' = −6 ) ) - L ) --M b!Ê. . ø . . , . * . H . ) . ^ Ë Êø^ ,H + ,^ øH + H^ ø, Ë 1 − w 1w∗ 3w*∗ 4w) 2 59 − æ9 12 A 1 ,H ¢ø ¢^ 1 ,^ ¢ø ¢H − 9 − 9 6 A 5 − æ 6 A 5 − æ 1 ^H ¢ø ¢, − 9 340 6 A 5 − æ Now every term is finite except those terms involving two metric tensors that give divergent contributions. In the following we will concentrate on those terms only. Contracting the metrics with the gamma matrices in the trace and using the zyclisity as well as the D-dimensional identities, * * * . . . = −21 − w. , . . . . = 4 − 2w. . , . . . * . ) . = −2. ) . * . + 2w. . * . ) and under usage of 232 we can rewrite the divergent part as follows, ℳ ,',$ = −6 ) ) - w 1w∗ 3w*∗ 4w) 2 1 = − 2 ) ) μ9 - L ) --M b!Ê. . ø . . , . * . H . ) . ^ Ë Êø^ ,H + ,^ øH + H^ ø, Ë 1 = 59 − æ9 12 A L ) --M bw 1w∗ 3w*∗ 4w) 2 Ì 5 1 81 − w9 !. . . * . ) − 8Í) * + Ì 9 2 5 − æ9 +4w!. . . ) . * + 8wÍ) * − 4w 9 !. . . * . ) = 108 Effective Field Theories 1 = − 2 ) ) - L ) --M bw 1w∗ 3w*∗ 4w) 2 4 9 μ Γw81 − w9 !. . . * . ) − 8Í) * + 16 9 æ +4w!. . . ) . * + 8wÍ) * − 4w 9 !. . . * . ) We see that the only divergent parts occur from the terms, 341 Γw8!. . . * . ) − 32) * = = 32Γw *) + ) * − 2* ) 342 where we put Í = 4. ¼ The other interesting part is . For w → 0 we find, K 4 = 1 + ln4w − lnæw + fw 9 343 æ Considering the low energy limit where [$ ≪ ; we can expand the terms as, + ln E−[ 4 4 = 1 + ln 9 w + æ ; y ≪C L b 1 9 2 + [ − [ + [9 + [ − 9 [ [9 G w NOOP ; ; ; ; ; 4 1 + ln 9 w344 ; Such the log-divergent part can be reduced to the terms in the integration as follows, 1 2 ) ) ¸1 + ln E - L ) --M bw 1w∗ 3w*∗ 4w) 2 ∙ 16 9 4μ9 G w¹ 32Γw *) + ) * − 2* ) ;9 109 345 Effective Field Theories But now comes the crucial step. Considering all six amplitudes in the given low energy limit. The only difference in the divergent parts, as calculated explicitly for the first amplitude, are the exchanges of gamma matrices within the trace. The six divergent parts, according to the six scattering amplitudes, are given by, 4 º1 + ln 9 w» 32Γw *) + ) * − 2* ) 346 ; 4 º1 + ln 9 w» 32Γw* ) + ) * − 2 *) 347 ; 4 º1 + ln 9 w» 32Γw *) + ) * − 2* ) 348 ; 4 º1 + ln 9 w» 32Γw) * + *) − 2* ) 349 ; 4 º1 + ln 9 w» 32Γw) * + * ) − 2 *) 350 ; 4 º1 + ln 9 w» 32Γw* ) + *) − 2) * 351 ; To obtain the overall scattering amplitude one has to add up all six individual parts. Applying this one sees that by adding up (344-349), that correspond to the divergent parts, cancel each other as the metrics add up to zero because *) + ) * − 2* ) + * ) + ) * − 2 *) + *) + ) * − 2* ) + ) * + *) − 2* ) + ) * + * ) − 2 *) + * ) + *) − 2) * = = 2 *) − *) + 2* ) − * ) + +2 *) − *) + 2) * − ) * + +2) * − ) * + 2* ) − * ) = 0 352 This means that all divergences drop out and the only terms in the whole scattering amplitude that remain are convergent. From these convergent terms 110 Effective Field Theories one can calculate the coefficients for the Euler-Heisenberg Lagrangian. This is, however, beyond the scope of this work. 111 Effective Field Theories 7. Abstract We briefly summarize topics discussed in this work. We started with some introductory examples, followed by a short introduction to the path integral formalism including two detailed examples. Afterwards we concentrated on what techniques could be used to build an effective Lagrangian. In detail we thereby discussed symmetries as well as dimensional analysis which proved a powerful tool to specify the mass dependence of the coupling constants of a theory. We used the matching procedure to obtain the coupling constants in a low energy effective theory such that the results matched the underlying theory. In this context we explicitly calculated the low energy Lagrangian for the muon decay and introduced the Fermi constant, which can be measured in an experiment and from which follows the mass of the W-boson. Afterwards we took a deeper look at the renormalization procedure in the context of Φ theory and developed important calculus tools to deal with loop diagrams. With the help of this very chapter we also calculated the effective Lagrangian of a simple scalar theory up to tree and first loop order. In the final chapter we studied the photon-photon scattering in QED and showed that the infinities cancel out by considering all possible Feynman diagrams. 112 Effective Field Theories 8. References [1.] [2.] [3.] [4.] [5.] [6.] [7.] [8.] [9.] [10.] [11.] [12.] [13.] [14.] [15.] [16.] [17.] [18.] [19.] [20.] A. Manohar, Effective field theories, (arXiv:hep-ph/9606222v1, 4 June 1996) A. Pich, Effective field theory, (arXiv:hep-ph/9806303v1, June 1998) S. Hartmann, Effective field theories, reductionism and scientific explanation, (http://stephanhartmann.org/Hartmann_EFT.pdf) H. Georgi, Effective field theory and electroweak radiative corrections, (Annual Review of Nuclear and Particle Science, vol. 43., 1993) R. Feynman, QED the strange theory of light and matter , (Princeton University Press 1985) R. Feynman and A. Hibbs, Quantum mechanics and path integrals , (Dover 2010) R. Feynman, Quantenelektrodynamik, eine Vorlesungsmitschrift , (R. Oldenbourg Verlag, 1997) F. Mandl and G. Shaw, Quantum field theory, (John Wiley & Sons, 2010) M. Srednicki, Quantum field theory, (Cambridge University Press, UK, 2007) M. Kaku, Quantum field theory, (Oxford University Press, 1993) S. Weinberg, The quantum theory of fields; Volume I Foundations, (Cambridge University Press, UK, 2002) S. Weinberg, The quantum theory of fields; Volume II Applications, (Cambridge University Press, UK, 2010) M. Maggiore, A modern introduction to quantum field theory, (Oxford University Press, UK, 2005) A. Zee, Quantum field theory in a nutshell, (Princeton University Press, USA, 2010) M. Peskin and D. Schroeder, An introduction to quantum field theory, (Westview Press, 1995) E. Rebhan, Theoretische Physik: Relativistische Quantenmechanik, Quantenfeldtheorie und Elementarteilchentheorie, (Spektrum akademischer Verlag, Germany, 2010) D. Kaplan, effective field theories, (arXiv:nucl-th/9506035v1 30 Jun 1995) Matthias Jamin, QCD and Renormalisation Group Methods, (Herbstschule für Hochenergyphysik, Maria Laach, 2006) David Griffiths, Introduction to elementary particles, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008) A. Grozin, Introduction to effective field theories, (arXiv:0908.4392v1, August 2009) 113 ( (µ) scheme ............................................... 77 (MS) scheme ............................................ 78 A Action ............. 20, 21, 24, 26, 27, 28, 91, 99 Amputated diagrams ............................... 70 Annihilation operator .............................. 44 Asymptotic behaviour ............................. 82 B Bare quantities .................................. 64, 65 Blue Sky ................................................... 13 Boson........................15, 16, 57, 58, 63, 112 Box-diagrams ........................................... 98 C Charge Conjugation ................................. 44 Classical electrodynamics .15, 31, 32, 33, 95 Commutator ............................................ 36 Complex scalar field ..10, 13, 31, 32, 43, 46, 47 Conserved current ............................. 33, 43 Contractions .....................36, 37, 42, 52, 60 Counter-term..................................... 65, 90 Coupling constants ....16, 28, 49, 50, 52, 57, 64, 96, 112 CPT........................................................... 44 Cutoff method ......................................... 71 D D-dimensional ................................. 28, 108 Dimensional analysis .10, 27, 48, 49, 50, 54, 81, 96, 112 Dirac .................................................. 28, 45 Divergences ............11, 73, 74, 81, 102, 110 E Effective field theories.... 10, 11, 16, 47, 88, 113 Effective Lagrangian ............... 12, 54, 57, 96 EHET ................................................... 96, 97 EHL ............................................... 10, 95, 96 Einstein .................................................... 13 Euler beta function .................................. 72 Euler constant .......................................... 73 Exact propagator................................ 69, 74 External momenta ................................. 100 F Fermi constant ........................... 59, 63, 112 Fermi Theory ...................................... 15, 57 Fermionic ......................... 29, 45, 47, 48, 82 Feynman ..10, 11, 15, 17, 22, 24, 31, 35, 38, 40, 41, 42, 43, 52, 53, 54, 55, 56, 57, 65, 66, 68, 70, 75, 79, 81, 84, 85, 86, 87, 91, 99, 103, 112, 113 Feynman diagram 11, 15, 38, 41, 42, 43, 65, 68, 79 Fields ........................................................ 26 First order perturbation theory .. 37, 39, 53, 63, 67, 74 Fourpoint ................................................. 16 Free particle ....................................... 22, 35 Fundamental theory .. 12, 15, 51, 54, 56, 57 G Gamma function ................................ 72, 73 Gauge invariance 10, 14, 15, 31, 32, 43, 51, 52, 85 Gauge transformations ... 10, 13, 31, 32, 34, 35, 43 Gauge Transformations ........................... 31 Gaussian integral ............................... 19, 22 Global transformation ............................. 32 Global transformations ...................... 32, 33 H Hamiltonian ............................................. 19 Heisenberg ......... 10, 12, 15, 95, 96, 97, 111 Effective Field Theories I N Incoming particle ............................... 35, 41 Infinities ... 12, 63, 64, 67, 71, 78, 87, 97, 99, 112 Interaction11, 12, 13, 14, 15, 16, 17, 30, 35, 36, 38, 39, 40, 42, 43, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 65, 66, 67, 81, 84, 87, 88, 90, 95, 96 Irrelevant ..................................... 17, 49, 50 Natural units .................................. 9, 27, 28 Neutral particles ................................ 13, 47 Noether's theorem ............................ 13, 33 O One loop diagram ........................ 64, 90, 97 One particle irreducible diagram 67, 69, 73, 78, 85 Operators ................... 44, 46, 49, 50, 52, 89 Outgoing particle ......................... 36, 37, 41 K Kets .................................................... 17, 18 P L Parity........................................................ 44 Path integral . 10, 17, 20, 21, 22, 24, 26, 28, 112, 113 Photon field .. 29, 32, 34, 36, 37, 39, 43, 45, 47, 82, 96 Photon-Photon Scattering ................. 95, 97 Power Counting ....................................... 78 Power counting method......... 10, 12, 78, 85 Probability amplitude ............ 18, 19, 23, 26 Propagator .... 15, 16, 36, 42, 53, 54, 55, 58, 67, 68, 69, 74, 79, 82, 97 Lagrangian . 9, 10, 12, 13, 14, 15, 16, 28, 29, 31, 32, 33, 34, 35, 43, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 64, 65, 84, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 111, 112 Length ................................................ 27, 28 Local transformations........................ 33, 43 Lorentz indices......................................... 30 Lorentz Invariance ................................... 30 Low energy .... 12, 14, 15, 16, 48, 49, 50, 51, 52, 57, 58, 63, 95, 99, 109, 110, 112 Q M QCD .................................................. 11, 113 QED .... 10, 11, 12, 24, 31, 32, 36, 64, 78, 83, 84, 85, 95, 96, 97, 98, 99, 112, 113 Quantum field theory 13, 15, 28, 36, 42, 44, 113 Quantum Mechanics ................................ 17 Mandelstam variables ....................... 55, 70 Manohar .......................................... 89, 113 Marginal ............................................ 49, 50 Mass .. 11, 13, 14, 15, 27, 28, 29, 49, 50, 51, 52, 57, 59, 62, 63, 64, 65, 69, 73, 81, 89, 93, 94, 96, 112 Matching ........................................... 52, 90 Matching Procedure ................................ 52 Maxwell ....................................... 31, 32, 95 Metric ........................................................ 9 Minkowski ............................................. 106 Muon ........................15, 59, 60, 62, 63, 112 R Radiative corrections ................. 64, 97, 113 Rayleigh ............................................. 15, 51 Relativistic kinematics.............................. 30 Relevant ....................................... 48, 49, 62 Renormalization.. 10, 11, 12, 41, 42, 56, 57, 63, 64, 66, 67, 69, 74, 77, 78, 81, 85, 86, 87, 90, 91, 92, 103, 112 115 Effective Field Theories Time 5, 14, 18, 19, 20, 24, 27, 28, 40, 44, 50 Time interval ...................................... 18, 20 Trace ........................ 59, 101, 102, 108, 110 Two point ........................................... 66, 67 Renormalization Schemes ....................... 77 S Scalar field10, 13, 26, 28, 29, 35, 37, 43, 45, 46, 47, 52, 65, 82 Scattering 10, 11, 14, 15, 24, 36, 38, 43, 48, 51, 53, 54, 55, 56, 57, 63, 67, 70, 74, 75, 76, 77, 92, 95, 96, 97, 98, 99, 100, 101, 102, 107, 110, 112 Scattering amplitudes.................. 54, 55, 99 Scattering process ................................... 14 Second order perturbation theory .......... 74 SED .............................31, 35, 78, 84, 85, 88 SI units ..................................................... 27 S-matrix ..................................................... 9 Spherical coordinates ........................ 61, 71 Superficial degree of divergence 78, 79, 80, 82, 83 Symmetries.............................................. 30 U U(1) transformations ............................... 32 U(1)-symmetry ......................................... 32 V Vector potential ......... 14, 31, 32, 33, 34, 35 Vertex 36, 38, 39, 40, 41, 42, 52, 65, 66, 67, 69, 79, 82, 83 W Weak fine structure constant .................. 63 Weak interactions .................................... 15 Wick rotation ........................................... 71 Wick’s theorem ........................................ 39 Wilson coefficients................................... 49 T Taylor expansion ............................... 16, 55 116 Effective Field Theories 117 Effective Field Theories Erklärung zur Hausarbeit gemäß §30 (Abs. 6) LPO I (2002) Hiermit erkläre ich, dass die vorliegende wissenschaftliche Hausarbeit von mir selbstständig verfasst wurde und dass keine anderen als die angegebenen Hilfsmittel benutzt wurden. Die Stellen der Arbeit, die anderen Werken dem Wortlaut oder Sinn nach entnommen sind, wurden von mir in jedem einzelnen Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht. Dies gilt ebenso für die in der Arbeit enthaltene Graphiken und bildliche Darstellungen. ______________________ _____________________ Ort, Datum Unterschrift 118

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