The Effective Potential and the Mass of the Higgs Particle. Det effektive potential og Higgs-partiklens masse

Transcription

1 The Effective Potential and the Mass of the Higgs Particle Det effektive potential og Higgs-partiklens masse Speciale-rapport udarbejdet af Stud. scient. André Fettouhi Fysisk Institut, SDU-Odense Universitet, Campusvej 55, DK-53 Odense M. Denmark 3. juli

2 Copyright c Stud. scient. André Fettouhi at Fysisk Institut, SDU- Odense Universitet, Denmark. All rights reserved. I hope to make this Master thesis freely available on the World Wide Web under the address if not there, then on my little brothers homepage at i

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4 Vejleder: Niels Kjær Nielsen, Fysisk Institut, SDU-Odense Universitet. Startdato: 5. august. Afleveringsdato: 3. juli. Officielle afleveringsdato: 3.juli.

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6 Preface I ve decided to use the preface to tell a little about how this project came about instead of talking about physics right away. The physics of this project will be presented in the introduction. The first sketches of my Master thesis came together in the beginning of March. My plan was to do a thesis on the geometry of the Universe at its very beginning, i.e. around the Planck scale. The idea came to me during a course in Astronomy FY9) I ve had a semester earlier. I asked my advisor Niels Kjær Nielsen, NKN), who was my advisor during my Bachelor project, if he was willing, interested and had the time to be my advisor again. After a three month short introduction to Quantum Field Theory we began to take a closer look at the problem. But around the beginning of July we realized that there wasn t enough material available to make up a Master thesis. The main reason was, that new observations in the last decade had shown that the cosmological constant Λ was nonzero. So the old articles by Hawking and Guth and others lost their appeal to me. My advisor then gave me three articles concerning the effective potential and the lower bounds on the mass of the Higgs particle. The subject appealed to me immediately and during the summer vacation I began the first calculations of the potential. During August the same year I got enrolled as a Master student at Fysisk Institut, SDU-Odense Universitet. So the last year I have calculated the effective potential in different scalar field theories using different methods. In the following introduction I will explain what the purpose of my thesis is and give a very short introduction to Quantum Field Theory. So in order for the reader to understand this thesis he or she must at least have had a one semester course in QFT Quantum Field Theory). The reason why I don t give a more indebted introduction to QFT is, that it s hard to write a good introduction which cannot be more than ten or twelve pages long. A good QFT introduction needs at least a pages. Instead I will sum up some important results from QFT and classical field theory. Then I will give a short list of QFT books which I find that give a good or reasonable iii

7 iv introduction to this field. This about covers the first chapter of my thesis. Chapter two will fully explain what the effective potential is, i.e. definitions and etc. and how it can be calculated and what it has to do with mass of the Higgs particle. There will also be a brief review on what kind of a particle the Higgs particle is, i.e. spin quantum number, mass and spontaneous symmetry breaking. Chapter three and on will hold the explicit calculations I have done during this year. Though not all of them will be included because that would make this thesis about 4 pages long. So the most important and relevant calculations and results will be presented. Many of these calculations are rather or very long and they will be put into an appendix. This document was typeset with L A TEXε.

8 Contents Preface iii Introduction. Classical Mechanics and Field Theory Path Integrals in Quantum Mechanics Quantum Field Theory Perturbation Theory in QFT Notation The Effective Potential 3. Defining the Effective Potential The Effective Action Γ[Φ] Γ[Φ] and V eff Φ) Spontaneous Symmetry Breaking The Higgs-mechanism One-loop Calculations in λφ 4 Theory 3 3. V eff Calculation, the Naive Treatment Tree Level One-loop Level Improving The Effective Potential The Renormalization Group Equations Improving the Potential in Massless φ 4 Theory One-loop Calculations in Scalar QED V eff Calculation, the Naive Treatment Tree-level and One-loop level A Little More SSB Improving the Effective Potential Comments to V eff in Scalar QED v

9 vi CONTENTS 5 Two-loop Calculations in λφ 4 Theory V eff Calculation, the Naive Treatment Calculation of Equation Improvement Once More Improved Potential The Calculation The Leading Logarithms Two-loop Calculations in Scalar QED Structure of the Two-loop Contribution Checking Our Calculations The Gauge Dependence The Ward-Takahashi Identities Applying the Ward Identities The Nielsen Identities Proving BRST Invariance of the Lagrangian The Derivation Discussion of the Identities Removal of ξ The Mass of the Higgs Particle Conclusions Beyond this Thesis A Feynman Rules for λφ 4 Theory 89 A. Derivation of Equation B Feynman Rules for Scalar QED 93 B. Derivation of Equation 4.56) C Derivation of I ϕ) in Equation 5.7) 99 C. One or Two Massless Lines D Last Two Graphs E Renormalization 9 E. Determining δz ) 3 and δz ) E. Determining δz ) and δz ) Bibliography 5

10 List of Figures. Graphical version of eq..8) A 3D-plot of equation.3) Tree-level contribution to the potential The one-loop level contribution to the potential The poles in the k complex plane Diagrams that do not contribute to the potential The photon one-loop level contribution to the potential Two-loop vacuum graphs First three two-loop vacuum graphs The next three two-loop vacuum graphs The next three two-loop vacuum graphs The next three two-loop vacuum graphs The last three two-loop vacuum graphs Graphical version E. The two-point function for the wave-function correction E. Mass correction diagrams E.3 Photon wave-function correction E.4 The λ contribution to Z ) E.5 The e contribution to Z ) E.6 The box diagrams vii

11 viii LIST OF FIGURES

12 Chapter Introduction The motivation for this thesis is based on an article by Lonaiz and Willey [] from 997 where they look at the gauge dependence on the lower bounds of the Higgs mass. Their study goes through the effective potential and its gauge dependence. The gauge dependence of V eff φ, ξ) is controlled by the Nielsen identities [], which are a different kind of Ward identities. These have the following look ) + Cφ, ξ) V eff φ, ξ) =,.) ξ φ here Cϕ, ξ) is a known function calculable in some expansion scheme. We ll derive these identities in a later chapter. Loinaz and Willey [] end up with a gauge dependent λ and they conclude in their little toy model extended Abelian Higgs model) that the lower bounds on the Higgs mass is gauge dependent in their model. We will show that there exists a gauge independent approximation for the effective potential, we can actually remove the gauge dependence through a simple substitution. This will be done to first order in the Abelian Higgs model and then we do a comment on how second order should be approached. The gauge dependence and the problems it brings along have sofar to the author s knowledge) been avoided in the literature. There have been several articles that have checked the Nielsen identities, like Aitchison and Fraser [3] to see if they hold, but no greater study has been made expect for [-]. There are two ways of calculating the effective potential, the first one goes through calculating a bunch of Feynman diagrams and the second one is a differential approach through the renormalization group equations RGE). The second one is actually better, because we widen our approximation scheme, this is why it s called the improvement of the effective potential. We ll show

13 CHAPTER. INTRODUCTION the equivalence between these two, up to second order in λφ 4 theory and the Abelian Higgs model. Before we begin with all the formal definitions of the effective potential, I have chosen to give a very brief summation of classical field theory and Quantum Field Theory. I hope the reader is familiar with QFT if not, there are two rather excellent textbooks: Quantum Field Theory by Itzykson and Zuber [4] a classic) and Gauge Field Theories by Pokorski [5] also nice, but to many errors in the second edition, a shame though). Also I can recommend some lecture notes by Ambjørn and Lyng Petersen from the University of Copenhagen [6]. My thesis is structured the following way. Chapter two gives the formal definitions of the effective potential and an example will be presented to show what it has to do with the Higgs mass. The next two chapters will hold the explicit calculations of the potential in the two previous mentioned models. Here we go to first order and then in the preceding chapters we go to second order. Then comes the heart of this thesis, the gauge dependence of V eff φ, ξ), derivation of the Nielsen identities and the removal of ξ in the Abelian Higgs model. Finally we conclude and look beyond this thesis and model.. Classical Mechanics and Field Theory Since our first school years we are taught Newton s three laws in physics class. The way that we apply these laws, is that we have a given system containing particles/objects. Next we look at what external forces that are acting on that system, i.e. gravity, friction etc. This analysis enables us to write down Newton s second law F = m d x dt,.) so we are left with solving a second-order differential equation. Solving it gives displacement, velocity, acceleration etc. at any time, i.e. the far past or future, of our system. If we now consider relativity general/special) we know that Newton s laws aren t correct. Though they do work as long as we have velocities much smaller than the speed of light c and weak gravitational fields. So if we want the true laws of Newton we need to consider relativity and we find that the laws are given by the following equation d x α ds g µν dx µ dx ν x α ds ds =, α =,,, 3..3)

14 .. CLASSICAL MECHANICS AND FIELD THEORY 3 Here g µν is the metric field tensor, the reader who is familiar with differential geometry will note that the equation above is just the equation for a geodesic. Geodesics on a surface are lines where the curvature along them is minimum. In a plane all lines are geodesics. Equation.) is the general relativistic version of Newton s first law of inertia) and second law of a particle in a gravitational field. If we want electric or magnetic fields too, the zero on the dx righthand side of the equation has to be substituted with KF ν µν, here F ds µν is the electromagnetic field tensor and K is a characteristic constant of the particle, actually K = q where m = mass and q = charge. The beauty of m this formulation is that the physics of a particle in a gravitational field is nothing more than a geometrical problem on a surface. Sofar we have taken a differential approach to physics, one might then think, is there another way? The answer to that question is, Yes. We can derive an integral version of Newton s laws. The idea here is instead of looking at the particles in our system, we look at what fields that are present. If we consider merely one particle in a gravitational field and take two points P and P in our space-time. We now want to know what path our particle will choose in order to get from P to P. The distance our particle travels can be written as the following integral the Action) S = P P ds..4) There is an infinite number of paths our particle can choose in order to get from P to P, so we need a criteria which will single out the actual path, i.e. the physical track. An obvious criteria would be, the path where S is extremum. Here we are still left with two choices, minimum or maximum. Since we are used to that everything proceeds through a minimization principle we rule out the maximum. So we define the physical path by S = P P ds is minimum..5) Now if we use that ds = g µν dx µ dx ν, then we can write ds ds = g µν x, x, x, x 3 )dx µ dx ν = dt g µν x, x, x, x 3 ) dxµ dx ν dt dt = Lt, q, q, q 3, q, q, q 3 )dt, in the last equation we set x, x, x, x 3 ) = t, q, q, q 3 ) and we defined the function L, which is a known function of 7 variables, since we are in

15 4 CHAPTER. INTRODUCTION space-time and we claim to know the metric tensor field, i.e. g µν. L is the Lagrangian and S is the Action, so our new constraint is S = t t Lt, q, q, q 3, q, q, q 3 )dt is minimum..6) Using Hamilton s principle, i.e. δs = one can derive the Lagrange equations which again is another way of looking at Newtonian mechanics, this version is more general than the normal Newton equations. We will derive the QFT version of these equations later Euler-Lagrange). Above we now have a path integral version of the law inertia, it simply states that the track physical path) satisfies that S is minimum. With these simple considerations we have made sofar, we move on to Quantum Mechanics. Here we have the normal differential approach, i.e. the Schrødinger equation. We will show that there is a path integral version here to, invented by R. Feynman back in 94 ies. From there we can immediately jump into QFT.. Path Integrals in Quantum Mechanics As we enter the quantum regime, our geometrical picture breaks down because of the Heisenberg uncertainty relation and we must substitute it with a wave picture. Though the classical wave equation can t be used to represent electrons, so another equation must be found. Using the ideas of wave packets one derives the Schrødindinger equation for a free particle. For a particle in a time-independent potential e.g. Coulomb) we have the time-dependent Schrødinger given by ψx, y, z, t) i h t = ] [ h m + V x, y, z) ψx, y, z, t) = ĤSψx, y, z, t),.7) Ĥ S is the Schrødinger Hamilton operator, one can easily see that the above equation breaks relativity. Time and space are not equal. So a new equation must be found if relativity must be uphold. Since the Schrødinger equation comes from quantization of the classical energy expression E = p + V one could perform quantization of the relativistic energy expression Erel = p rel c + m c 4 by letting E i h and t m p i h then we get the Klein-Gordon equation for a free particle h ψx, y, z, t) t = [ h c + m c 4] ψx, y, z, t) = ĤKGψx, y, z, t),.8)

16 .. PATH INTEGRALS IN QUANTUM MECHANICS 5 Ĥ KG is the Klein-Gordon Hamilton operator but this equation still doesn t solve all our problems. Since it s a quantization of a classical expression it can t describe spin just like the Schrødinger equation, though we can define a spin operator like the angular moment operator but it just pops up because we have no classical version of this internal degree freedom, that particles with spin have. So we conclude that the KG-equation can only describe particles with spin, which we call scalar particles. Since the KG-equation is a second order differential equation one could purpose a first order differential equation instead, this is the Dirac equation which has the following look ψx, y, z, t) i h t = [ c α p + βmc ] ψx, y, z, t) = ĤDψx, y, z, t),.9) here α and β aren t numbers but 4 4 matrices and are given by the following α i = β = ) σi, i =,, 3,.) σ i ) I..) I The σ i s are the Pauli-spin matrices and I is the unit matrix. The Dirac equation can be derived by requiring that ĤD is hermitian and that ψx, y, z, t) also satisfies an equation like the KG-equation. The beauty of the Dirac equation is that spin comes out naturally in order to secure that the Dirac Hamilton operator commutes with the total angular moment. I don t want to go deeper into the Dirac equation and its solutions, it s beside the point. I instead want to talk about pictures in Quantum Mechanics. These pictures are used to describe the dynamics of the system we are looking at. Suppose we have a known Hamilton operator and an initial wave function ψ r, ). The initial wave function develops into ψ r, t) according to the timedependent Schrødinger equation. There isn t a unique way of looking at the dynamics of a system, in general we use three pictures, there is the Schrødinger picture which we have used above, the Heisenberg picture and finally the interaction picture Dirac picture). These pictures are of course equivalent since they describe the same physics of a system. In the Schrødinger picture we are used to having time-independent operators and time-dependent wave functions. This doesn t mean that ĤS can We choose the non-relativistic regime.

17 6 CHAPTER. INTRODUCTION never be time-dependent. Now in the Heisenberg picture we remove the time-dependence from the wave function and basically put it into the operators. The interaction picture is something in between the Heisenberg and Schrødinger pictures. Here the time-dependence is smeared over the wave functions and operators. One can get from one picture to another through a unitary transformation time-dependent) Ut). Sofar it s rather basic Quantum Mechanics we have been taking about relativistic and non-relativistic). To get to QFT we first remember how we calculate transition amplitudes the usual way φ ψ = d 3 r φ r, t) ψ r, t),.) the integral goes over all space. In classical field theory we search for the physical path of the particle, but since we are in the quantum world the word path must be handled with care. We imagine that we have a state x, t and we want to calculate transition amplitude from that state to x, t and we denote the amplitude by Fx, t; x, t ). We let the state be an eigenstate to the position operator ˆX. So the eigenvalue equations in the Heisenberg and Schrødinger picture are respectively ˆX H t) x, t = x x, t ˆX S x = x x,.3) the x, t is a Heisenberg state vector at time t, x is a Schrødinger state vector. If the Hamiltonian is time-independent we can write Fx, t; x, t ) as Fx, t; x, t ) = x, t x, t = x exp ī ) hĥt t ) x. In the above equation there is no sign of a path integral yet, but to generate the path integral we use the completeness relation = dx j x j, t x j, t, now we divide the time interval t t ) into n + ) equal parts with the length ε given by ε = t t n +, We look at a system with one degree of freedom.

18 .. PATH INTEGRALS IN QUANTUM MECHANICS 7 so the time step t j is given by t j = t + jε. If we let n and set Ĥ = Ĥ ˆP, ˆX) = f ˆP)+g ˆX) we get the following result for the transition amplitude Fx, t; x, t ) = Dxt)Dpt) π h in the above equation we have defined Dxt)Dpt) π h = ī t ) exp dt [pt)ẋt) Hpt), xt))], h t i= j= dxt i )dpt j ). π h The above integral for the transition amplitude is a path integral like the one we know for classical mechanics, but in the classical world we are used to having a integral over one path, the physical path, where the Action is extremal. The expression we have derived goes over all paths both physical and unphysical. At a first glance this seems rather strange that unphysical paths contribute to the transition amplitude. But we have to remember that in the Quantum world we have no geometrical picture that enables us to find a specific path, we simply can t tell the difference between the physical or unphysical paths between two states. Thereby we have to take all paths into account. With these thoughts, the integral above does make sense. In the exponent we have an expression that is very similar to the Lagrangian from classical physics, but it doesn t depend on xt) and ẋt). Above we have derived the path integral in phase space, we need it in configuration space. To get to configuration space we look at a more explicit Hamiltonian, we choose Ĥ = ˆP then we do the momentum integration and we get m + V ˆX),.4) Fx, t; x, t ) = ) Dxt) ī N π h exp h S [x],.5) N = Dpt) exp ī ) t dt p, h t m S [x] = t t Lx, ẋ), Lx, ẋ) = mẋ V x).

19 8 CHAPTER. INTRODUCTION Above we have derived Feynman s path integral in Quantum Mechanics. Now transition amplitudes are interesting, but so are matrix elements of operators, like the position operator. If we look at a time-ordered product of position operators and go through the basic steps we took to derive the path integral above for the transition amplitude we find not surprisingly x, t T { ˆXt )... ˆXt N )} x, t = Dxt)Dpt) xt )...xt N ) π h ī t ) exp dt [pt)ẋt) Hpt), xt))], h t if we now apply the same change we did before in order to get from phase space to configuration space we get the N-point function G N t,...,t N ) = x, t T { ˆXt )... ˆXt N )} x, t = ) Dxt) ī N π h xt )...xt N ) exp h S [x]..6) The reader will notice that we can write the matrix elements for the timeordered position operators with the help of the transition amplitude, if we add an external source driving force) to the Hamiltonian, i.e. H H hjx so Fx, t; x, t ) becomes Fx, t; x, t ) J = Dxt)Dpt) π h ī t ) exp dt [pt)ẋt) Hpt), xt)) + hjt)xt)], h t the matrix element can now be written as a functional derivative of the new amplitude Fx, t; x, t ) J, i.e. G N t,...,t N ) = ) N δ N i δjt )...δjt N ) Fx, t; x, t ) J..7) J= In the next section we extend this path integral description to n degrees of freedom and we substitute the position operators with field operators, thereby jumping directly into Quantum Field Theory..3 Quantum Field Theory When looking at infinite degrees of freedom or n, where n is a large number, we must replace the x coordinate with a n-component vector. The integral now stands for a sum over all trajectories in the n-dimensional space, with

20 .3. QUANTUM FIELD THEORY 9 appropriate boundary conditions. We substitute xt) with Φ x, t) Φx), x is now a point in Minkowski space. To keep everything as simple as possible we only look at scalar fields. The scalar field operator is reinterpretation of the scalar wave function, which is a solution to the Klein-Gordon equation. The solution is written out as a Fourier integral over plane-wave solutions. One then performs canonical quantization of Φx) this gives a field operator. Our transition amplitudes and matrix element have sofar been between two arbitrary states, usually in QFT we look at transitions going from one vacuum to another, so the transition amplitude is ) ī = DΦ exp d h 4 xl,.8) in the above equation we have defined the Lagrangian density in order to get a more space-time like representation of the amplitude. The matrix elements are now called the Green s functions or n-point functions, i.e. ) ī G n) x,..., x n ) = N DΦ Φx )...Φx n ) exp d h 4 xl,.9) is equal to.8), applying the same trick as before, we can define a N generating functional W[J] by W[J] = N = consequently we get n= ī DΦ exp h i n n! G n) x,...,x n ) = ) d 4 x [L + hjx)φx)] dx...dx n G n) x,...,x n )Jx )...Jx n ),.) i ) n δ n δjx )...δjx n ) W[J] J=,.) one might wonder if we actually can solve these equations. The answer to that is, No. There are few cases were we actually can write down an explicit solution down, so most of the time we solve these things approximately, i.e. perturbation theory..3. Perturbation Theory in QFT Let us first derive the Euler-Lagrange equations for QFT, with the following Action S = d 4 xlφ, µ Φ),.)

21 CHAPTER. INTRODUCTION using δs = and the chain rule we get L δs = d 4 x Φ δφx) + L ) µ Φ) δ µφx)) { }) L L = d 4 x Φ µ δφx),.3) µ Φ) above we ve used δ µ Φx)) = µ δφx)) and we did an integration by parts. Now we can conclude { } L L Φ µ =,.4) µ Φ).4) is the Euler-Lagrange equation of motion. The Lagrangian density for the KG-equation one real scalar field) is then for the Dirac equation it is L scalar = µφ µ φ µ φ, L fermion = Ψ iγ µ µ m) Ψ, a little change in notation has happened, we have set h = c = G = and we write from now on γ µ µ =. To introduce perturbation theory we look at the simplest case, one scalar field real) with an interaction term, so the Lagrangian is L = L scalar + L int = µφ µ φ µ φ λ 4! φ4,.5) λ is the coupling constant. The reason why we have chosen a cubic interaction term, is simply because it s a renormalizable theory in d = 4 dimensions. From integration theory we know that we can integrate terms that are linear or quadratic in the exponent think of the Gaussian integral). So we can solve the Green s functions in the free field theory no interaction term). As in Quantum Mechanics we let the strength of the interaction be small, i.e. λ small. This allows us to Taylor expand a part of W[J], we have then ) ī W[J] = N DΦ exp d h 4 x [L scalar + L int + hjx)φx)] = N exp ī d 4 x λ [ ] 4 δ W [J] h 4! i δjx) = N n= ī n! h d 4 x λ 4! [ δ i δjx) ] 4 n W [J],.6)

22 .4. NOTATION W [J] which is given in.) where L = L free. It is possible to show that W [J] can be written as i h ) W [J] = exp d 4 xd 4 yjx)gx y)jy),.7) the Gx y) is the Green s function for the KG-equation and it s given by d 4 k exp ikx y)) Gx y) =..8) π) 4 k µ + iε The last few steps need a bit of explanation, upon deriving.6) we substituted φx) with δ, then we could move it outside the integral and i δjx) Taylor expand it. We can now write the Green s function down and in terms of functional derivatives with respect to J. The explicit expression is ) G n) n δ n x,...,x n ) = N k! i δjx )...δjx n ) k= ī h d 4 x λ 4! [ δ i δjx) ] 4 k W [J] J=,.9) the reader can then convince himself that the n-point function will be a product of Gx i x j ) s Wicks theorem). We then do are graphical interpretation of these functions, this gives us the Feynman diagrams. Perturbation calculations can be long and hard depending on what order the result has to be in, but there is so much systematic about the procedure, that one can set up rules for the theory he or she is calculating, these are called the Feynman rules. These rules help us to write down any n-point function we need and to any order one desires. I m going end the general introduction to QFT here, because I could keep going on for many more pages but not all of it would be relevant to this thesis. I m ending this chapter with a section about the notation I will be using in the upcoming chapters. This will diverge a bit from the notation sofar..4 Notation Quantum Field Theory is calculated usually) in Minkowski space, i.e. flat space-time. We will denote the metric tensor of the space-time with η µν and we choose the following signs η µν =..3)

23 CHAPTER. INTRODUCTION In my thesis I will only work with scalar theories, we will denote a real scalar field with φ i, where i designates what field we are talking about, i.e. Higgs, Goldstone etc. A complex scalar or general will be denoted with Φ = φ +iφ, and the non-zero Vacuum Expectation Value VEV, see next chapter) will be denoted with υ. The momenta running the different diagrams, which the reader will encounter during this thesis will be labelled the following way. External momenta will be denoted by p i, i =,,... and the internal momenta, we denote them by k i, i =,,.... The photon field is denoted as always by A µ. The coupling constant for the photon will denoted by e.

24 Chapter The Effective Potential Our universe consists of four fundamental forces, these are; gravity and the electromagnetic, strong and weak forces. With QFT we can describe three of these forces in one model. This model is called the Standard Model SM). Here we are able to describe electromagnetic, strong and weak forces up energies about TeV. If we look at two electrons interacting with each other and suppose the only force acting is the electromagnetic one. The way these electrons communicate with each other is through photons. One electron sends one photon out, the other electron absorbs it. The strong and the weak force have similar field quanta, the ones acting in the strong force are named gluons and in the case of the weak force, they are called Z and W ±. One rather peculiar thing is that the Z and W ± particles have mass, but the photon is massless gluons are also massless). What mechanism is responsible for given mass to Z and W ±, one might ask. The answer is, the Higgs mechanism. This mechanism occurs when spontaneous symmetry breaking is present. What does all this have to do with the effective potential and the Higgs particle? These questions will be answered later in this chapter, but first we start of with some formal definitions of the potential.. Defining the Effective Potential In the last chapter we defined the generating functional W[J] for the Green s functions G n) x,..., x n ) it held all the information about the theory, i.e. all the Feynman diagrams to all orders. This means there are diagrams present where the incoming particles don t interact, they go in and then out. Such diagrams don t contribute to the scattering amplitude of a process. So it would be nice if we could define a generating functional which holds 3

25 4 CHAPTER. THE EFFECTIVE POTENTIAL all the information about the diagrams where interaction happens. These functions are called the connected Green s functions G n) connx,..., x n ) and the generating functional is called Z[J] and is defined the same way as W[J], i.e. i n Z[J] = h dx...dx n G n) n! connx,..., x n )Jx )...Jx n ),.) n= another type of Green s functions are the PI Green s functions one-particleirreducible) Γ n) x,..., x n ). They are a subset of the connected Green s functions, they remain connected after an arbitrary internal line is cut. Here the generating functional Γ[Φ] is called the effective action. Why it s called the effective action will become more clear later and it s defined Γ[Φ] = n= n! dx...dx n Γ n) x,...,x n )Φx )...Φx n )..) Φx) is now some arbitrary scalar field. The Green s functions connected and PI) are then given by G n) connx,...,x n ) = h ) n δ n i δjx )...δjx n ) Z[J] J=,.3) Γ n) δ n x,...,x n ) = δφx )...δφx n ) Γ[Φ] Φ=,.4) it s possible to show the following connection between the generating functionals see [6]) ) ī W[J] = exp h Z[J],.5) Γ[Φ cl ] = Z[J] h d 4 xjx)φ cl x),.6) Φ cl x) is called the classical field, which minimizes Γ[Φ] + h d 4 xjx)φx), or equivalently δγ[φ] = hjx).7) δφx) Φ=Φcl the equation above.7) is very similar to what we have in the exponent of W[J] S[Φ] + h d 4 xjx)φx) = d 4 x [L free + L int + hjx)φx)].

26 .. DEFINING THE EFFECTIVE POTENTIAL 5 Equation.6) is nothing more than a Legendre transform which we are familiar with from Statistical Mechanics. We are especially interested in the effective action because we define the potential from that. First we look more closely at the classical field, if we do a functional differentiation of.6) with respect to Jx) and using the chain rule we get d 4 y δγ[φ cl] δφ cl y) δφ cl y) δjx) = δz[j] δjx) hφ clx) h using.7) we get an expression for the classical field Φ cl x) = h δz[j] δjx) = G ) conn x) J d 4 yjy) δφ cly) δjx),.8) = Φx) J J,.9) the J subscript means that the source is turned on. If we now turn off the source, we then have the vacuum expectation value of the field VEV), i.e. Φx) = υ,.) we assume for now that υ =, so we get from.9) δz[j] =,.) δjx) J= since.7) and.9) are in some sense each others inverse, this implies then δγ[φ] =..) δφx) Φcl = As mentioned earlier we will have a closer look at Γ[Φ] and how it is related to S[Φ], then we look at, how we get the effective potential from that... The Effective Action Γ[Φ] To start of the discussion about the effective action we go back to λφ 4 theory, so the action is given by plus source term [ S[φ c, J] = d 4 x µφ c µ φ c µ φ c λ ] 4! φ4 c + h d 4 xjx)φ c x),

27 6 CHAPTER. THE EFFECTIVE POTENTIAL J φ ) c x) = x J + x J Figure.: Graphical version of eq..8) J applying the Euler-Lagrange equations from chapter one, we get the following equation of motion µ µ + µ ) φ c = λ 3! φ3 c + hjx)..3) Now φ c is a classical field do not confuse φ c and Φ cl ). The above equation can be solved perturbatively through the method of Green s functions, we write φ c x) = φ ) c x) + φ ) c x) +, inserting that and ordering order by order we find µ µ + µ ) φ ) c = hjx),.4) µ µ + µ ) φ ) c = λ 3! φ) c ) 3 + hjx),.5) and their solution φ ) c x) = h φ ) c x) = φ ) c x) λ 3! d 4 ygx y)jy),.6) [ d 4 ygx y) d 4 zgy z)jz)] 3,.7) a graphical interpretation of that would be see above), if we go higher up in the perturbation expansion, it is easy to see that we would get more tree diagrams and they would all be connected. If look at the generating functional W[J] in this approximation, i.e. replacing Φ with φ c in the integrand. We get and using W[J] = exp i h Z[J]) we have ) ī W[J] = exp h S[φ c, J],.8) Z[J] = S[φ c, J] = S[φ c ] + h d 4 xjx)φ c x)..9)

28 .. DEFINING THE EFFECTIVE POTENTIAL 7 We see now that S[φ c ] is indeed given by all the connected tree diagrams and we can also see in this approximation that S[φ c ] = Γ[φ c ]..) Now the connection between S[Φ] and Γ[Φ] is clear. In the classical approximation S[Φ] is the generating functional for the PI Green s functions tree graphs only). Generally Γ[Φ] holds all the loop corrections too, hereby explaining the term effective action. So S[Φ] has all the tree diagrams and Γ[Φ] has all the loop corrections... Γ[Φ] and V eff Φ) Suppose we have a non-zero expectation value for the field, i.e. δz[j] = υ,.) δjx) J= δγ[φ cl ] =..) δφ cl x) Φcl =υ The classical field was defined in.9) and the effective action was defined in.6), we can Taylor expand Γ[Φ cl ] around the vacuum Φ cl = υ and we get Γ[Φ cl ] = n= n! d 4 x...d 4 x n Γ n) x,...,x n ) Φ cl x ) υ)...φ cl x n ) υ),.3) alternatively we could expand Γ[Φ cl ] in powers of momentum around a point where all external momentum vanishes, i.e. in position space) Γ[Φ cl ] = [ d 4 x V eff Φ cl ) + ] Z 3Φ cl ) µ Φ cl µ Φ cl +..4) The expression V eff Φ cl ) is what we call the effective potential and it is a function not a functional, because Φ cl = φ = constant. We can also see that the effective potential is equal to minus the non-derivative terms in the Lagrangian. If we now look at.4) and.5) together, we get V eff φ) = n= n! Γn) p i = )φ υ) n,.5) we have reached a very important conclusion here. The effective is given by an infinite sum over the PI Green s functions with zero external momenta.

29 8 CHAPTER. THE EFFECTIVE POTENTIAL A neat trick can be applied here, if take the derivative of V eff φ) with respect to υ and set φ = υ we get V eff υ) υ = Γ ) p i = ),.6) so the derivative is given by all the one-point functions tadpoles) and this a more convenient expression than.6). The above equation was derived by Lee and Sciaccaluga [7] in 975 and it s probably the easiest way of calculating the potential through Feynman diagrams tadpoles). Why do we want to use the effective potential instead the action. First of all because we will be working with a function instead of a functional. Second we have clear picture of what diagrams that need to be calculated. In the next section we give an explicit example of spontaneous symmetry breaking in a scalar model.. Spontaneous Symmetry Breaking This is how we today believe the field quanta Z and W ± acquired mass. Sometime in the early universe, spontaneous symmetry took place and then the Higgs-mechanism gave them mass. The trouble with SSB Spontaneous Symmetry Breaking) is it needs the Higgs particle to give mass to the respective field quanta. Sofar we have been unable to find this particle. Though physicists hope that it will be found around 7, because then we will be able to get the energies to make the respective processes where the Higgs particle should be involved. Now we perform an explicit calculation on a simple toy model which demonstrates SSB and the Higgs-mechanism. We start of with our λφ 4 theory with a complex KG-field, so the Lagrangian looks as always L = µφ µ Φ µ Φ λ 4! Φ Φ),.7) we want to look at what symmetries our Lagrangian has. We can see if we do the following Φ exp iθ) Φ,.8) the Lagrangian stays the same. This type of transformation is called a global U)-transformation. It s merely a phase we multiply our field with. The transformation above doesn t depend on where we are in our space-time, if we are here or at the other end at the universe. This is rather unsatisfying

30 .. SPONTANEOUS SYMMETRY BREAKING 9 Figure.: A 3D-plot of equation.3) because mostly were interested in the physics in our local surroundings and not at the other end of the universe. So we sharpen our transformation above by letting it depend on space-time, i.e. Φ exp iθx)) Φ..9) Applying this change to the Lagrangian we immediately see that it no longer is invariant under this new transformation local U)-transformation). The reason is the derivative, it transforms µ Φ exp iθx)) µ Φ i exp iθx)) Φ µ Θx), to fix this problem we define a covariant derivative by D µ Φ = µ + iea µ )Φ, we have added a gauge field A µ which transforms the following way A µ A µ + e µθx), this gauge field is of course the photon field, so we must remember adding the kinetic energy for the field to the Lagrangian. So the new Lagrangian which is invariant under local U)-transformation reads L = 4 F µνf µν + D µφ) D µ Φ) µ Φ Φ λ 4! Φ Φ),.3)

31 CHAPTER. THE EFFECTIVE POTENTIAL now we find the potential, it was given by minus the non-derivative terms in the Lagrangian, i.e. V eff Φ) = µ Φ + λ 4! Φ Φ), = ) µ φ + φ λ ) + φ 4! + φ.3) a 3-dimensional plot of this function is given above. Condition.3) is transformed into, when we use.5) dv eff φ) =..3) dφ φ=υ Now we determine the minimums of this function, i.e. V eff φ = µ φ + λ 3! φ + φ )φ,.33) V eff φ = µ φ + λ 3! φ + φ )φ,.34) each of these two equations have to be zero, this implies φ = φ = and φ + φ = 6µ λ υ,.35) we see that we have a non-zero ground-state for the vacuum, if µ <. This ground-state is a degenerate circle. The vacuum can only be in one of these infinite many minimums, so we have to choose the vacuum has to). So we choose φ = υ and φ =. We have φ = υ,.36) φ =,.37) we have a non-zero VEV, when µ < expectation values are real), so SPONTANEOUS SYMMETRY BREAKING has occurred. Why? The Lagrangian density is invariant under local U)-transformations by construction) but the vacuum is not. If we would apply the transformation, we would go from the old vacuum to a new vacuum. The new vacuum is just as good as the old one degenerate circle), but it s not the same. Now the Higgs-mechanism insures that we get a massive field quanta A µ. Of course we know that the photon is massless in real life, but one has to remember that this is a toy model. We describe only a very small part of the physics in the real world.

32 .. SPONTANEOUS SYMMETRY BREAKING.. The Higgs-mechanism We redefine our fields now, we choose Φx) = hx) υ+iχx), i.e. φ = h υ and φ = χ. This merely an expansion around the chosen vacuum, where h and χ are KG-fields when we are close to the vacuum. We insert that into our Lagrangian L = 4 F µνf µν + µφ µ Φ + e A µ A µ Φ Φ + ie Aµ Φ µ Φ Φ µ Φ) µ Φ Φ λ 4! Φ Φ) = 4 F µνf µν + µφ µ φ + µφ µ φ + e A µ A µ φ + φ +ea µ φ µ φ φ µ φ ) ) µ φ + φ λ φ 4! + φ) = 4 F µνf µν + µh µ h + µχ µ χ + e A µ A µ h + υ hυ + χ ) + ea µ h υ) µ χ χ µ h) ) + λ υ h + υ hυ + χ ) λ 4 h + υ hυ + χ ) = 4 F µνf µν + µh µ h + µχ µ χ λ 6 υ h + e υ A µ A µ +interaction terms. We can now see what the Higgs-mechanism has done, our photon has acquired mass m A = e φ and also one of the scalar particles has mass m h =. The λυ 3 other scalar field remains massless because all quadratic terms in the field have the wrong sign. The important thing here is that we gained a field quanta with mass without breaking the U)-symmetry. Because we could easily have added a mass term but it would break the U)-symmetry. Which we imply that the Lagrangian has to have. Though it seems there is a new problem, the number of degrees of freedom have changed because the photon has mass now. SSB can t change the number of degrees of freedom. But we can choose a gauge where we remove the χ-field, and this leaves the new Lagrangian with same degrees of freedom as the old one. The h-field is what we in general call the Higgs particle, the χ-field in known as the Goldstone particle, it s a nonphysical particle because it depends on gauge and physical quantities cannot and may not depend on gauge. It s clear now why the Higgs particle is so important at the moment. If we are unable to find it, a lot of QFT will breakdown and we are forced to find new ways to give mass to the field quanta in the weak force.

33 CHAPTER. THE EFFECTIVE POTENTIAL The above calculation shows that even at tree no loop corrections) level do we have SSB. In the next chapter, we will show that if there is no SSB at tree level, then higher order corrections may induce it, i.e. give a non-zero VEV.

34 Chapter 3 One-loop Calculations in λφ 4 Theory Now we begin calculating radiative corrections to the effective potential in massless λφ 4 theory µ = ). To calculate V eff we re going to use equation.6) for that. In order to calculate the potential we need the Feynman rules for the theory. These can be found from the derivations in chapter one. We will not derive them explicitly here, that would take up to much space and time. So for the readers convenience they are given in Appendix A with µ, they can also be found in any decent QFT textbook, see [4-6]. This chapter and the next are based on the calculations by Coleman and Weinberg [8] from 973, this paper is regarded as one of the classical papers in particle physics and is among one of the most quoted ones too. Our regularization procedure of the integrals is different from [8]. They used a cut-off method, we will use what is standard in QFT today, dimensional regularization, a method developed by t Hooft and Veltman [9]. The case with µ follows directly from the massless case, though there is another way doing these calculations at one-loop level, we will do a short comment on that method later. 3. V eff Calculation, the Naive Treatment Already before we begin, we encounter a problem,.6) is an infinite sum over all Feynman diagrams all loops) this calculation clearly lies beyond our capability because we only have perturbation theory available. Though we can show that it s possible to perform the sum in an orderly fashion, i.e. loop-expansion. So first we sum all graphs with no loops tree), then one, then two and so on. The idea is to prove that the loop-expansion is equal to 3

35 4 CHAPTER 3. ONE-LOOP CALCULATIONS IN λφ 4 THEORY a power series expansion. So we introduce a parameter in the Lagrangian, through the following definition Lφ, µ φ, h) h Lφ, µ φ), 3.) next we denote the power of h h is equal to one) by P which is associated with any graph. Then one can see P = I V, 3.) where I is the number of internal lines and V is the number of vertices. The plus sign for I arises because the propagator is the inverse of the quadratic term in L and thereby carrying a factor h. The number of loops is defined L = I V ) L is the number loops, which is equal to the number of integration momenta after conservation of momenta has been reached. This explains the extra in 3.3). Every line carries one integration momentum, but the vertex carries a δ-function that removes all these integration momenta until conservation is reached. The reader will now notice that the Feynman rules in Appendix A don t have these δ-functions etc. This is because the version of the Feynman rules I have given, apply to PI Green s function, here we normal cut of all the external lines and throw the momentum conserving δ-function away. So the above integration has already been performed in the Feynman rules in Appendix A. Now we see P = L, 3.4) so the power of a graph is equal to the loop minus one. Still we have shown, that the loop expansion is equal to a power series expansion which we will now apply. 3.. Tree Level Our theory is still the λφ 4 theory with one real scalar field, but we set the bare mass µ to zero, so we have L = µφ µ φ λ 4! φ4 + δz 3 µφ µ φ δz µ φ δz λ 4! φ4, 3.5) the first two terms are we familiar with, but the last three are new. These are respectively the wave function, mass and vertex counterterms. They are

36 3.. V EFF CALCULATION, THE NAIVE TREATMENT 5 added to make our theory convergent, because one of the first problems we run into is that some of the Feynman diagrams are UV-divergent diverge for large k). This is of course a problem. Two conclusions can be drawn from this, QFT doesn t work, but there is nothing that sofar supports that statement. Another conclusion could be that the fields, coupling constants etc. in our previous Lagrangian aren t the physical ones. This means for example that the mass µ isn t the pole in the propagator, i.e. p = µ relativistic conservation of energy). So we must do some regularization of these parameters and we choose φ B = Z 3 φ, 3.6) λ B = Z λ, 3.7) Z3 µ B = Z Z 3 µ, 3.8) actually the new fields etc. aren t the physical ones either, they are just some renormalized parameters without the divergencies). Writing Z = + Z ) = + δz we get 3.5). This procedure shouldn t generate a mass counterterm µ =, but we add it because nothing prevents that radiative corrections give mass to the system. The counterterms are calculated through the loop expansion and will be presented later. We previously showed that the effective potential was equal to minus the non-derivative terms in the Lagrangian, i.e at tree level. V tree eff φ) = λ 4! φ4, 3.9) equal to the graph below). The same result would have come out of.6) Figure 3.: Tree-level contribution to the potential because Γ 4) = λ and is the only contributor. The other non-derivative terms don t contribute because they are loop correcting terms, i.e. they start contributing at L =, we look at L =.

37 6 CHAPTER 3. ONE-LOOP CALCULATIONS IN λφ 4 THEORY Figure 3.: The one-loop level contribution to the potential 3.. One-loop Level At this level we have the non-derivative terms from L last two) and the infinite series of the following diagrams see above). Using equation.6) we get V loop eff φ) = δz) µ φ + δz ) λ 4! φ4 + i λ φ d 4 k π) 4 n= n k + iε ) n, 3.) a little explanation is required for the numerical factors in the integral. The factor i comes from.5). Only the even PI Green s function contribute ) because the uneven are zero. Now we want to understand the factor n, n since the diagrams above represent [n )n 3)...] n )! there are crossings of these diagrams) diagrams with a symmetry factor. We have then n! [n )n 3)...]n )! = ) n, n the n! factor stems from.6), we have simply changed the indexing of the sum since it runs over, 4,.... The sum looks quite gruesome but it s actually a sum of a function we know. Before we do the sum we rotate to Euclidean time explanation follows). In the denominator we have factor +iε. If we were to set that equal to zero and do the loop integration we would integrate through two singularities. So the integral would be ill defined. So we bypass these singularities by analytical continuation to the complex plane C. This is done by deformation of the integration contour from real to imaginary axis. This is equal to adding ±iε in the denominator, or equivalently ±iεφ in the Lagrangian. One then concludes that iε can t be used and we are left with +iε. Going to Euclidean time is done through a Wick rotation. One goes from Minkowski space to Euclidean space. The poles in the denominator in 3.) are k = ± k iε. This can also be seen in the following graph and the integration contour is also shown. We have four different paths to integrate

38 3.. V EFF CALCULATION, THE NAIVE TREATMENT 7 ¹ ¹ k Figure 3.3: The poles in the k complex plane along, we are specifically interested in the one along the real axis. Now using Cauchy integral-theorem, we know that the integral is zero around the hole path sum of the four paths). There are no singularities inside the region which the paths enclose. Through parameterization of the path along the quarter of a circle, we see that the contribution goes to zero when the radius of the circle goes infinity. We conclude that integrating along the real axis is equal to integral along the complex axis. So we can set ε =, we have now i V loop eff φ) = δz) µ φ + δz ) λ d 3 k 4! φ4 + i dk i π) 4 n= n λ ) n φ, 3.) k k next we set k = ik, this gives us dropping the prime) V loop eff φ) = δz) µ φ + δz ) λ 4! φ4 + λ φ d 4 k π) 4 ) n n= n k ) n, 3.) this can be seen as ln + x) = ) n n= n xn, < x. 3.3)

39 8 CHAPTER 3. ONE-LOOP CALCULATIONS IN λφ 4 THEORY so we have V loop eff φ) = δz) µ φ + δz ) λ 4! φ4 + = δz) µ φ + δz ) λ 4! φ4 + d 4 [ k π) ln + 4 d 4 k π) 4 ln ] λ φ k [ k + λ ] φ, k the integral will be performed through dimensional regularization see Appendix A for details), one finds V loop eff φ) = δz) µ φ + δz ) + λ φ 4 56π ln λ 4! φ4 [ λ ] φ 4πM 3 + γ ǫ ). 3.4) M is and arbitrary mass-scale which define in order to keep λ dimensionless in d = 4 ǫ dimensions. is the pole the divergency of the integral). Next ǫ we invoke the renormalization constants, i.e. we fix them so that the effective potential is convergent. So we need some renormalization conditions. We define the mass µ of φ as the value of the inverse propagator with zero momentum, i.e. Γ ) p = ) = µ using the relation between Γ n) and V eff we get d V eff φ) dφ = µ, 3.5) φ= similarly we do with λ and we get d 4 V eff φ) dφ 4 = λ. 3.6) φ= The general idea of the renormalization conditions is, at tree they self explanatory, now when radiative corrections are taken into account we simply would like that µ stays µ and so forth for the coupling constant. Now the full effective potential up to one loop is V eff φ) = λ 4! φ4 + δz) µ φ + δz ) λ 4! φ4 + λ φ 4 [ λ ] ln φ 3 56π 4πM + γ ), 3.7) ǫ applying 3.5) we get δz ) = as expected) and we are left with V eff φ) = λ 4! φ4 + δz ) λ 4! φ4 + λ φ 4 [ λ ] ln φ 3 56π 4πM + γ ), 3.8) ǫ