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The Chirality-Flow Method for Massive and Electroweak Amplitudes

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The Chirality-Flow Method for Massive and Electroweak Amplitudes LU-TP 20-41 July 2020 The Chirality-Flow Method for Massive and Electroweak Amplitudes Joakim Alnefjord Department of Astronomy and Theoretical Physics, Lund University Master thesis supervised by Malin Sj¨odahl,Christian Reuschle, and Andrew Lifson Abstract In this thesis I extend the chirality-flow method to the full standard model at tree-level by including massive particles and electroweak interactions. The chirality-flow method is a new diagrammatic method for calculating Feynman diagrams that is based on the spinor- helicity formalism, that before this thesis was written had been worked out for massless QED and QCD at tree-level. I summarize what is known about massive spinor-helicity calculations and use this as the basis for the extended diagrammatic interpretation for massive particles. I use the Lorentz structure of the electroweak vertices to rewrite them in the chirality-flow picture. Popul¨arvetenskaplig beskrivning Det finns m˚angav¨agaratt g˚ai jakten efter en b¨attre f¨orst˚aelsef¨orv˚armateriella v¨arlds minsta best˚andsdelar,elementarpartiklarna. Vi kan bygga kraftigare och mer exakta par- tikelacceleratorer och d¨armedf¨orb¨attraden experimentella delen av partikelfysiken. Vi kan ocks˚af¨ors¨oka hitta nya teorier om hur v¨arldenfungerar, n˚agotsom i praktiken ¨ar v¨aldigt komplicerat. Vi kan ist¨alletfokusera p˚asmartare s¨attatt g¨orautr¨akningarinom dagens teoretiska ramar, vilket ¨arvad den h¨ar uppsatsen handlar om. Inom partikelfysiken ¨arvi ofta intresserade av att r¨aknaut s˚akallade ¨overg˚angsamplituder. Dessa kan exempelvis liknas med en sannolikhet att ett visst tillst˚andav partiklar ska ¨overg˚atill ett annat tillst˚andav andra partiklar vid en kollision i partikelacceleratorer. F¨ordet mesta har vi inget exakt svar f¨or¨overg˚angsamplituderna,s˚aist¨alletapproximeras dem. M˚alet¨aratt f¨ors¨oka g¨ora s˚abra approximationer som m¨ojligt. Ofta anv¨andsen typ av diagram, som kallas f¨orFeynman diagram efter nobelpristagande fysikern Richard Feynman, f¨oratt hj¨alpatill i ber¨akningarna. Diagram ¨arett smidigt s¨att att l¨attarevisualisera utr¨akningarna och det finns flera typer av diagram som anv¨ands.Nu har vi en ny metod som kallas f¨or"chirality-flow method" med en ny typ av diagram som f¨orhoppningsvisska kunna g¨orautr¨akningarl¨attare.Den h¨arnya metoden har utvecklats f¨oratt kunna appliceras p˚avissa av elementarpartiklarna, d¨arpartiklarnas massa ocks˚a har ignorerats. M˚aletmed den h¨aruppsatsen ¨aratt inkludera alla typer av k¨andaele- mentarpartiklar samt deras massa, s˚aatt den nya metoden kan appliceras i alla m¨ojliga fall. Acknowledgments I would like to thank my supervisors for their feedback, patience, and helpful contributions to our many discussions; I have learned a lot during this thesis work. I would also like to thank my closest friends for their constant support. Contents 1 Introduction 1 2 Background 1 2.1 The Lorentz Group . .2 2.2 The Spinor-Helicity Formalism . .2 3 The Massless Chirality-Flow Method 5 3.1 Notation and Diagrammatic Interpretation . .5 3.2 Diagram Examples . 11 4 Massive Particles 12 4.1 Bispinor Decomposition . 13 4.1.1 Eigenvalue Decomposition . 13 4.1.2 General Light-like Decomposition . 15 4.2 Dirac Equation Solutions . 17 4.2.1 In The Eigenvalue Decomposition . 17 4.2.2 In The General Light-like Decomposition . 18 4.3 Polarization Vectors . 20 4.3.1 In The Eigenvalue Decomposition . 20 4.3.2 In The General Light-like Decomposition . 21 5 Massive and Electroweak Chirality-Flow Rules 21 5.1 Massive Fermion Propagator . 22 5.2 Massive Fermion Legs . 23 5.3 Electroweak Vertices . 24 6 Massive and Electroweak Diagram Examples 30 7 Conclusion and Outlook 35 1 Introduction In particle physics we are often interested in scattering amplitudes which are needed to calculate cross sections. The complexity of the calculations increases not only from loops in higher order terms of the perturbation series, but also when we increase the number of external particles involved in a specific process. To counter the complexity of calculations, we would like to have better and more efficient methods of calculating the scattering amplitudes, which will help us increase the accuracy of our theoretical predictions. The chirality-flow method [1] is a new diagrammatic method for calculating scattering amplitudes, which has been developed for amplitude calculations for massless QED and QCD at tree-level. It is based on the spinor-helicity formalism [2, 3, 4, 5, 6] where objects used in amplitude calculations such as Dirac spinors, Feynman-slashed momenta, and po- larization vectors are all decomposed and written in terms of two-component Weyl spinors [7]. The chirality-flow method interprets the Weyl spinor inner products as flow lines in chirality-flow diagrams. These flow diagrams are very powerful because they drastically simplify calculations. In this thesis, I have generalised the chirality-flow method to also include masses and the electroweak sector, so that it can be used for the full Standard Model at tree-level. This has been achieved by using what is known about massive spinor-helicity calculations and including that in the diagrammatic framework of the chirality-flow formalism. I find that massive Feynman diagrams with a specific helicity configuration can be decomposed into sums of massless chirality-flow diagrams, which allows us to apply the tools that have been developed for the massless chirality-flow method. Before we begin, I will give a brief overview of the thesis. In section 2, I start with the representations of the Lorentz group and how objects involved in Feynman diagrams can be written in terms of two-component Weyl spinors using the spinor-helicity formalism. Then in section 3 I summarise how this formalism can be used to create a diagrammatic method, called the chirality-flow method, which has been recently developed for massless QED and QCD at tree-level. In section 4 I use what is known about massive spinor- helicity calculations to put it into a form which can be inserted into the chirality-flow framework. I then in section 5 describe how massive particles can be interpreted and used diagrammatically, and I also give chirality-flow rules for the electroweak vertices. Lastly in section 6 I show some examples of the diagrammatic method being used. 2 Background We will begin with brief summaries of two components that lie at the core of the chirality- flow method: the Lorentz group and the spinor-helicity formalism. These two parts will be used as the starting points for the description of the chirality-flow method. 1 2.1 The Lorentz Group The Lorentz group [8] describes Lorentz transformations in Minkowski-space and it has 6 generators: 3 rotation generators Ji and 3 boost generators Ki. The generators satisfy the following commutation relations: [Ji;Jj] = iijkJk; [Ji;Kj] = iijkKk; [Ki;Kj] = iijkJk: (2.1) − Together, the generators and commutation relations describe the Lie algebra of the Lorentz Group. It is possible to define a new set of operators as combinations of the rotation and boost generators, ± 1 N = (Ji iKi) ; (2.2) i 2 ± which gives us a new set of commutation relations: + + + − − − + − Ni ;Nj = iijkNk ; Ni ;Nj = iijkNk ; Ni ;Nj = 0: (2.3) + − We see here that Ni and Ni form two copies of su(2)C Lie algebras that commute with each other. The Lie algebra isomorphism so(3; 1)C su(2)C su(2)C is a more formal way to display this. This isomorphism between Lie algebras' can⊕ be used to characterize the 1 3 irreducible representations of the Lorentz Group as (s1; s2), where s1; s2 = 0; 2 ; 1; 2 ; ::: give the specific irreducible representations of the two copies of su(2)C. Some examples of common representations are: The scalar representation is given by (0; 0), where both copies of su(2)C are given in • the trivial representation. Left-chiral and right-chiral Weyl spinors are 2-component spinors that transform • 1 1 under the ( 2 ; 0)-representation and the (0; 2 )-representation, respectively. Dirac spinors are 4-component spinors that are formed by combining a left-chiral and • a right-chiral Weyl spinor, and they transform under the ( 1 ; 0) (0; 1 )-representation. 2 ⊕ 2 1 1 The ( 2 ; 2 )-representation is the vector representation. We use this to represent a • vector object as a product of a left-chiral and a right-chiral Weyl spinor. 2.2 The Spinor-Helicity Formalism There are two parts that lie at the core of the spinor-helicity formalism: the use of 2- component Weyl spinors and the use of the helicity basis. Dirac spinors, Feynman-slashed momenta, and polarization vectors can all be written in terms of Weyl spinors. Using these spinors, it is possible to form two types of Weyl spinor contractions that are Lorentz invariant, which is usually helpful and appreciated. The helicity basis is a good choice for 2 a spin basis, as the helicity operator commutes with the Dirac Hamiltonian, which means that their eigenstates are the same. The use of helicity is especially useful for massless particles because it is then Lorentz invariant, whereas for massive particles helicity is frame dependent. Since we are working with Weyl spinors, we use the Weyl/chiral basis for the gamma matrices, 0 σµ 0 p2τ µ γµ = = ; σµ = 1; σi ; σ¯µ = 1; σi ; (2.4) σ¯µ 0 p2¯τ µ 0 − where σi are the Pauli matrices: 0 1 0 i 1 0 σ1 = ; σ2 = ; σ3 = : (2.5) 1 0 i −0 0 1 − We write the σ-matrices as p2τ µ to avoid having to carry around unnecessary factors of 2 through the calculations.
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