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Grand Unified Theory

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Grand Unified Theory Grand Unified Theory Manuel Br¨andli [email protected] ETH Z¨urich May 29, 2018 Contents 1 Introduction 2 2 Gauge symmetries 2 2.1 Abelian gauge group U(1) . .2 2.2 Non-Abelian gauge group SU(N) . .3 3 The Standard Model of particle physics 4 3.1 Electroweak interaction . .6 3.2 Strong interaction and color symmetry . .7 3.3 General transformation of matter fields . .8 4 Unification of Standard Model Forces: GUT 8 4.1 Motivation of Unification . .8 4.2 Finding the Subgroups . .9 4.2.1 Example: SU(2) U(1) SU(3) . .9 4.3 Branching rules . .× . .⊂ . 10 4.3.1 Example: 3 representation of SU(3) . 11 4.4 Georgi-Glashow model with gauge group SU(5) . 14 4.5 SO(10) . 15 5 Implications of unification 16 5.1 Advantages of Grand Unified Theories . 16 5.2 Proton decay . 16 6 Conclusion 17 Abstract In this article we show how matter fields transform under gauge group symmetries and motivate the idea of Grand Unified Theories. We discuss the implications of Grand Unified Theories in particular for proton decay, which gives us a tool to test Grand Unified Theories. 1 1 Introduction In the 50s and 60s a lot of new particles were discovered. The quark model was a first successful attempt to categorize this particle zoo. It was based on the so-called flavor symmetry, which is an approximate symmetry. This motivated the use of group theory in particle physics [10]. In this article we will make use of gauge symmetries, which are local symmetries, to explain the existence of interaction particles and show how the matter fields transform under these symmetries, section 2. This will lead us to a very success- ful model, the Standard Model of particle physics, section 3. This model has some issues, the most important one is that we have to put a lot of parameters into the model. We would like to find a theory, which already includes these pa- rameters. In an attempt to find such a theory, we use a symmetry group, which includes the gauge groups found in the Standard Model. Theories obtained in such manner are called Grand Unified Theories (GUT), section 4. We will dis- cuss the implications of these models in section 5. In particular Grand Unified Theories predict new interaction particles, which lead to new interactions i.e. proton decay, section 5.2. 2 Gauge symmetries This section is based on Ref. [1]. A symmetry transformation is a transformation of a field φ φ0 which leaves the Lagrangian invariant δ = 0. The idea of a gauge symmetry7! is that this transformation can be a functionL of spacetime. This means that we have a local symmetry. To see this in detail let's have a look at the Abelian gauge group U(1). 2.1 Abelian gauge group U(1) We have a Lagrangian which has the following form: L y µ y = @µφ @ φ V (φ φ); (2.1) L − where the dagger stands for hermitian conjugate. Our global symmetry is the transformation of our field by a phase: φ φ0 = eiαφ, which results in a µ 7! conserved current @µj = 0. This phase factor is a complex number, which means eiα U(1). If we now consider that our phase factor α can be a function of the spacetime2 coordinate, i.e. α(x), we have φ eiα(x)φ. This modification results in an additional term from the derivative7! of α(x). We would like to obtain the same Lagrangian, so to compensate for this term we introduce an electromagnetic potential Aµ which transforms in the following way: Aµ(x) 1 7! Aµ(x) e @µα(x) where e is the coupling constant and stands for the electric charge.− We combine the derivative with the potential to define a new covariant derivative 2 Dµ = (@µ + ieAµ)φ, which transforms as: 0 0 iα(x) Dµφ (Dµφ) = (@µ + ieAµ)(e φ) = 7! (2.2) iα(x) 1 iα(x) iα(x) = @µ(e φ) + ie(Aµ(x) @µα(x))e φ = e Dµφ. − e If we therefore replace the derivatives in the Lagrangian (2.1) with the new covariant ones, we find a Lagrangian which is invariant under the gauge trans- formation: y µ y = (Dµφ) (D φ) V (φ φ): (2.3) L − There is a kinetic term missing for our field Aµ. To find this kinetic term let's have a look at the field strength tensor Fµν , which has the following form: 1 Fµν = [Dµ;Dν ] = @µAν @ν Aµ: (2.4) − e − The change of the field strength tensor under gauge transformation is zero, which follows from: 1 δFµν = @µδAν @ν δAµ = (@µ@ν @ν @µ)α(x) = 0; (2.5) − − e − where we used that derivatives commute. Our kinetic term is, therefore, sim- 1 µν ply A;kin = 4 Fµν F , because it has to be Lorentz invariant. The full LagrangianL is then− given by: y µ 1 µν y = (Dµφ) (D φ) Fµν F V (φ φ): (2.6) L − 4 − 2.2 Non-Abelian gauge group SU(N) We can generalize this idea to non-Abelian gauge groups. We will focus on SU(N) gauge groups, because they will play an important role later on. We have a set of fields: Φ1 Φ = . which transforms as: Φ UΦ with U SU(N): 0 . 1 7! 2 Φ B nC @ A We can also write U as an infinitesimal transformation using its generators T a: a iαaT a U = e 1 + iαaT : (2.7) ≈ These generators have the following commutation relation: [T a;T b] = if abcT c: (2.8) We see, that we again have a global symmetry, if our Lagrangian has the form: y µ y = (@µΦ) (@ Φ) V (Φ Φ): (2.9) L − 3 a Our gauge transformation is then given by: Φ U(x)Φ = eiαa(x)T Φ, where we will get an additional term as before, because7! the derivative acts also on U(x). We again introduce a vector field, which this time needs to be matrix- a a a valued: Aµ = AµT , where T are the generators of the group. Therefore, we get for each generator of SU(N) a vector field. The quantization of these vector fields result in interaction particles. For example the quantization of the Aµ results in the photon. Further, the gluons, the interaction particles for the strong interaction, result from quantization of the non-Abelian gauge symmetry group SU(3)C , section 3.2. Consider the transformation of DµΦ under the gauge transformation: 0 0 0 [DµΦ] = [(@µ + igAµ)Φ] = (@µ + igA )UΦ = µ (2.10) −1 −1 0 = U(@µ + U (@µU) + igU AµU)Φ We would like to be able to pull U(x) out, i.e. −1 −1 0 U(@µ + U (@µU) + igU AµU)Φ = UDµΦ: For this to be true we need: 0 −1 i −1 A = UAµU U@µU ; (2.11) µ − g where g is the so-called coupling constant. The field strength tensor is: i a a Fµν = [Dµ;Dν ] = @µAν @ν Aµ + ig[Aµ;Aν ] = F T : (2.12) −g − µν The components of the field strength tensor are given by: a a a abc b c F = @µA @ν A gf A A : (2.13) µν ν − µ − µ ν The additional term in the field strength tensor comes from the fact that the gauge vector interacts with itself. An example of this is the gluons which have themselves color and therefore interact with each other. To find the kinetic part of the interaction field we can use the trace, because tr(A) tr(A0) = tr(UAU −1) = tr(A). Therefore, this quantity is invariant under gauge7! trans- formation. We finally find our Lagrangian: y µ 1 µν y = (Dµφ) (D φ) tr(Fµν F ) V (Φ Φ): (2.14) L − 2 − 3 The Standard Model of particle physics The Standard Model of particle physics is a very successful model. In figure 1 we see the general structure of the Standard Model. There are six quarks and six leptons which are both fermions and there are 12 interaction particles called vector-bosons, because they have spin 1. These include 8 gluons, the interaction particles for the strong interaction, and the 4 interaction particles of 4 Figure 1: particles in the standard model [20] the electroweak interaction, namely the photon, the W ± bosons and the Z boson. There is also the Higgs particle which is a scalar boson (spin=0), which we will not cover in this article. We can arrange the fermions into three generations, where the fermions of different generations differ only by mass. The Standard Model is a chiral theory, which means that it violates parity. This means if we do a parity inversion of our system, there are processes that have different probability occurring in the two systems [2]. On the way to a full formulation of the Standard Model, Sheldon Glashow, Steven Weinberg and Abdus Salam unified the weak interaction and the elec- tromagnetic interaction into the electroweak interaction the so-called Glashow- Weinberg-Salam model. It predicted the existence of a Z-boson, because this model has the underlying gauge group SU(2)L U(1)Y , which means that we have 4 generators and therefore 4 interaction particles.× In 1974 neutral cur- rent, the interaction mediated by the Z-bosons, was observed in the Gargamelle neutrino experiment at CERN [3]. The GIM mechanism, which is part of the Standard Model, predicted a fourth quark, the charm quark [4] and laid the foundation for the prediction of the top and bottom quark by Makoto Kobayashi and Toshihide Maskawa [5].
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