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The Abelian Higgs Model: Unitarity in the Unitary Gauge

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The Abelian Higgs Model: Unitarity in the Unitary Gauge Radboud University Nijmegen The Abelian Higgs Model: Unitarity in the unitary Gauge Author: Supervisor: Oscar Boher Prof. Dr. Ronald Kleiss Second corrector: Prof. Dr. Nicolo de Groot March 16, 2017 Abstract The Standard Model of particles is one of the most successful theories in history explaining a grand variety of experimental measures. It uses Lagrangian densities to explain the dynamics of the quantum fields, the fundamental objects in the theory that correspond to particles in the physical reality. The Standard Model is also a gauge theory, which allows performing gauge transformations to the Lagrangian. These mathematical operations do not change the physics that the Lagrangian describe, but do change its form. This property is exploited in the theory to prove results using mathematical abstract concepts that do not correspond to any element in the physical reality. In concrete, proving the correct energy behavior of the matrix element using the equivalence theorem [1] uses non-physical fields in the Feynman-t'Hooft gauge - massless bosons, ghost particles, Faddeev-Popov ghosts - that allow the mathematical proof to be achieved. In this thesis we will focus on proving the correct energy behavior of a subgroup of the Standard Model - the Abelian Higgs model - working in the unitary gauge, in which every element has its counterpart in reality. 2 Contents 1 Introduction 2 1.1 Conventions . .2 1.2 Lagrangian formalism and gauge theories . .3 1.3 Standard Model . .5 2 Abelian Higgs Model 5 2.1 Interactions . .7 2.2 Feynman rules . .8 2.3 Energy dependence limit of the matrix element . .9 2.4 Energy dependece . 10 2.5 Hypothesis check . 11 3 Massless approximation, E2 terms 11 3.1 Zn;k;Hn;k ............................... 14 3.1.1 Zn;0;Hn;0 ........................... 14 3.1.2 Finding Zn;k;Hn;k ...................... 16 4 E0 terms 25 2 4.1 mh terms . 25 2 4.2 mz terms . 27 5 On-shell recursion relations 28 6 Summary 33 A Appendix 36 A.1 4Z, 1H . 36 A.2 2Z, 3H . 39 A.3 Three and Four particles processes . 43 1 1 Introduction This project tries to study the high energy behavior of a theory included in the Standard Model (SM) of Particles, the Abelian Higgs Model. This Model contains only two particles. These are two bosons, the Z and the Higgs Bosons. These two particles are massive and electrically neutral, they follow Bose- Einstein statistics - integer spin -, their interactions are limited and well-known, the Z boson carries a polarization while the Higgs does not, and at tree level there are no other particles involved. In this context, the focus of the project is in proving that this theory is well- behaved at high-energies (Escale MZ ;MH ) in the physical gauge. The SM being a renormalizable theory, on which the divergence is solved by it, the energy behavior of the expressions for processes that can be experimentally measured - with a finite value - must have an upper limit if unitarity holds. All terms that have a higher energy dependence in the Matrix element have therefore to vanish. For the Standard Model this have been proven in the Feynman- t'Hooft gauge, using the equivalence theorem [1] to observe the highest energy dependence, and checking that it is under the limit energy behavior. In this gauge the particle spectrum allows Higgs ghosts and Faddeev-Popov ghosts to appear. The apparition of these particles is critical to see that the SM is well- behaved , but those are the mathematical residues of working on this gauge, since they do not exist in the physical world. Using the physical unitary gauge, in which the experimentally found particles interact, makes the expressions more complicated to simplify. This is why in this study, only the Abelian Higgs Model is taken into account. 1.1 Conventions The conventions here used are the following: • Natural units. ~ = c = 1. • Einstein summation notation of a repeated index. X µ µ p qµ ≡ p qµ µ @ • Partial derivatives are defined as: @µ = @xµ • Incoming momentum for all particles in an interaction: n X µ pi = 0 i • External Z bosons always in the longitudinal polarization if not stated otherwise. Later on the discussion some new conventions will be introduced to simplify the equations worked on. 2 1.2 Lagrangian formalism and gauge theories In the history of physics, mathematics has been the basic tool that allow physi- cists to describe things in reality and perform operations on a system. This mathematical description also allows for predictions to be made. Developing new rigorous mathematical methods improves, simplifies and deepens the knowl- edge of a subject, at the cost of abstraction and an obscurer connection between the real objects and their mathematical representations. In the 17th century, Leibniz and Newton developed calculus. This became an extremely powerful method by which many physical systems could be solved. Infinitesimals introduced continuous variables and their use in equations of mo- tion allowed description and prediction of moving bodies. Classical mechanics was born. Later in 1788, Joseph-Louis Lagrange reformulated classical mechanics. He introduced Lagrangian mechanics. The physics described were exactly the same as in classical mechanics, but the Lagrangian was a much more powerful and easy description to extract information from. A system could be described by kinetic and potential energies, and constrains, all this information contained in the Lagrangian (in non-relativistic mechanics L(qj; q_j; t) = T − V , with qj dqj the coordinates of particle j,q _j = dt the time derivative of the coordinate of particle j, and T and V the kinetic and potential energy respectively). The integral (over time) of the Lagrangian is the action (S = R t2 Ldt). Applying t1 then the principle of least action (δS = 0), which looks for the path between the points t1 and t2 with a stationary action, led to the Euler-Lagrange equations of motions, equivalent to Newton's ones (full derivation can be found in [2]). In non-relativistic mechanics those are: d @L @L ( ) = (1) dt @q_j @qj This equations that describe point-like objects also can be extended to con- tinuum fields. The coordinates become now fields ( j) and their derivatives are now on a space-time point s on a Manifold M. The action is now defined as the four dimensional integral over the Lagrangian density (L ) - usually just called Lagrangian. With the equivalent of the least action principle for continuum fields ( δS = 0), we find the new equations of motion: δ j Z @ j α n S[ j] = L ( j; ; s )d s (2) @sα @L @L @µ( ) = (3) @(@µ j) @ j In the first quarter of the 20th century physicist were revolutionized with the new theories of general relativity and quantum mechanics. The latter showed that elementary particles could be described by point-like objects as well as by fields. Even though the Hamiltonian formulation was used first to describe 3 quantum mechanics, Richard Feynman worked on a Lagrangian-based formu- lation, manifestly symmetric in time and space. The principle of least action was generalized to the path integral formulation. The idea behind it was that since particles are also waves, all possible paths between two points must be considered, each with a phase corresponding to the action, eiS. This sum (func- tional integral) over all the paths is the calculation of the quantum amplitude, which squared is a probability density that allows to compute transition prob- abilities between states. Those paths further from the classic trajectory are suppressed, and in the classical limit the classic trajectory is recovered (more on the derivation at [3]). Feynman also developed the Feynman diagrams, an easy and intuitive way to represent mathematical expressions for elementary-particle interactions in the form of pictures, or diagrams. Each element in the diagram (vertices, internal lines, external lines) has a mathematical equivalent that comes directly from the Lagrangian terms. The sum of all the possible Feynman diagrams for a pro- cess is the Matrix Element M, that when squared yields the quantum amplitude. Once seen how the Lagrangian formalism arises, it is possible to look at its properties. Specifically at the continuous symmetries a Lagrangian has. A La- grangian is symmetric under a transformation when it remains invariant under it, and the equations of motion prevail unchanged. Each symmetry is related to a conserved quantity by Noether's theorem. In the classical formulation, a symmetry under time transformation (t ! t + δt) implies that the total en- ergy is conserved. Space symmetry (qk ! qj + δqj) implies conservation of the momentumq _j. When a theory has a Lagrangian invariant under a continu- ous group of transformations is called a gauge theory. For example, Maxwell's formulation of classical electrodynamics is a gauge theory, where the vector po- tential Aµ = (V; A~) can be transformed as Aµ ! Aµ − @µΛ leaving the electric and magnetic fields unaffected .This group of gauge transformations form a Lie group, which has a number of generators. Each generator will have an associ- ated field that ensures Lagrangian invariance. When the theory is quantized, the quanta of the fields are the gauge bosons. A simple complex scalar theory can illustrate this. Start with the Lagrangian density for a complex scalar field Φ: 1 ∗ µ 1 2 ∗ L = (@µΦ )(@ Φ) − m Φ Φ (4) 2 2 This is globally gauge invariant since a transformation Φ ! Φ = eiΛΦ0 , where Λ is real and does not depend on space nor time, leaves the Lagrangian invariant. But Local symmetry is also required. When the field Λ depends on space-time coordinates, the Lagrangian is not invariant anymore due to the derivatives acting on this field.
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