Gauge theories and the standard model of elementary particle physics

Gauge theories and the standard model of elementary particle physics Mark Hamilton 21st July 2014 1 / 35

Table of contents 1 The standard model 2 3 2 / 35

The standard model The standard model is the most successful theory of elementary particle physics. Its predictions have been verified in numerous experiments at particle accelerators. The standard model describes all known interactions among elementary particles, except gravity. The mathematical foundation of the standard model is a gauge theory, based on the symmetry group U(1) SU(2) SU(3). The particles of the standard model consist of three generations of fermions (electron, neutrino, quarks, etc.), the gauge bosons (photon, gluon, etc.) that mediate the interactions between the fermions and the Higgs boson which is a part of the Higgs field that conveys mass to the fermions and some of the gauge bosons. 3 / 35

The particles of the standard model Figure: The elementary particles of the standard model (en.wikipedia.org) 4 / 35

History 1940s: Development of quantum electrodynamics. 1954: Yang and Mills develop non-abelian gauge theories. 1956: Lee, Wu and Yang discover chirality of the weak interaction. 1961: Glashow unites electromagnetic and weak interaction. 1964: Gell-Mann and Zweig postulate existence of quarks. 1964: Higgs and others develop the Higgs mechanism. 1967: Salam and Weinberg combine Glashow s model with the Higgs mechanism. 1972: t Hooft and Veltman prove renormalizability of electroweak interaction. 1973-74: Development of quantum chromodynamics. Several Nobel prizes for development of the standard model (in particular 1979 for Glashow, Salam and Weinberg, 1999 für t Hooft and Veltman as well as 2013 for Higgs and Englert). 5 / 35

Gauge theories as field theories Gauge theories are field theories that have two kinds of symmetries: Lorentz invariance Gauge invariance We will describe below two models for gauge theories: Geometrical model, in which the symmetries are implicit Physical model, in which the symmetries are explicit. We will also describe the transition from the geometrical to the physical model. We will only discuss the classical field theory, in particular the Lagrangian. In physics the field theory is quantized (quantum field theory) and the Feynman rules are deduced from the Lagrangian. Elementary particles correspond to quanta, i.e. minimal excitations of the fields. 6 / 35

Bundles Gauge theories are formulated geometrically in the language of bundles over manifolds, in particular principal bundles and vector bundles (spinor bundle, associated bundles). We have the following fundamental fact: Remark Every bundle over M = R 4 is trivial, i.e. a product M Σ consisting of base M and fiber Σ, since R 4 is contractible. However, such a bundle is not canonically trivial. In other words: There are trivializations, but in general none of them is preferred. In this regard bundles are similar to other mathematical objects. For example, every vector space has a basis, but none of them is preferred. In physics one often chooses trivializations and the independence of such a choice reflects itself in invariances and symmetries. 7 / 35

The geometrical model The geometrical model of a gauge theory consists of the following data: 1 Spacetime (M, η), a differentiable manifold with a metric of signature (+,,..., ), and the spinor bundle S over M. 2 The gauge group G, a compact, semisimple Lie group. 3 The gauge bundle P M, a G-principal bundle. 4 A unitary representation G V V on a complex vector space V. 5 The associated bundle E = P G V. 6 The fermion bundle or multiplet bundle F = S E. 7 The gauge boson field, a connection A on the principal bundle P with curvature F A. 8 / 35

Spinor bundle The spinor bundle is a certain complex vector bundle over the manifold M (it exists if M is spin). On the spinor bundle there exists a Clifford multiplication so that TM S S, (v, ψ) v ψ, v (v ψ) = 2η(v, v)ψ. If (M, η) = (R 4, η Mink ), then S = M C 4 after a choice of inertial frame. In this case, Clifford multiplication with a basis vector e µ in an inertial frame is given by multiplication of a spinor in C 4 with iγ µ, where γ µ are certain 4 4 Dirac matrices that satisfy {γ µ, γ ν } = γ µ γ ν + γ ν γ µ = 2η µν Id. 9 / 35

Principal bundle A G-principal bundle is a manifold P with a projection π : P M and an action P G P so that: 1 Every fibre of P is diffeomorphic to G and the action of G preserves the fibres and is simply transitive on them. 2 P is over small open sets U in M of the form U G. In other words, P is locally trivial. If (M, η) = (R 4, η Mink ), then P is globally trivial. In this case we have P = M G, together with the standard action of G. It is important that these trivializations are not canonical: A trivialization is given by a global section s : M P. We then get every element of P as s(x) g, for x M. 10 / 35

Gauge The standard model Definition (Gauge) We call a global section s of the principal bundle P a gauge. Every gauge defines a trivialization of P. The notion of gauge is of central importance for gauge theory (standard notion?). It plays a similar role to the notion of inertial frame in relativity. In both cases there is a manifold which is trivial in a certain sense, but that does not have a preferred trivialization. The change between two trivializations is described by a Lorentz and gauge transformation, respectively. The choice of gauge is the choice of a coordinate system. Gauge invariance will later mean: Invariance under the choice of gauge. 11 / 35

Associated bundle Let G V V be a representation. Then G acts on P V from the right via (p, v) g = (p g, g 1 v). The associated bundle is the quotient E = (P V )/G = P G V. It is a vector bundle over M with fibre isomorphic to V. If (M, η) = (R 4, η Mink ), then E is trivial, E = M V. A trivialization is not canonical, but given by a choice of gauge: If s : M P is a gauge, then sections Φ: M E correspond precisely to mappings φ: M V via Φ = [s, φ]. 12 / 35

Fermion bundle The fermion bundle F is given by F = S E. Let (M, η) = (R 4, η Mink ). We choose an inertial frame and a gauge s : M P. Let r denote the complex dimension of the representation space V. Then a section Ψ in F is given by ψ 1 Ψ =., ψ r where each component ψ i : M C 4 is a spinor. A fermion, i.e. a section of F, is thus described by a vector that has r components, each one of which is a spinor (multiplet). The representation of G mixes these components. 13 / 35

Connection and curvature I A connection A on the principal bundle P is a certain invariant 1-form on P with values in the Lie algebra g. The curvature of A is defined as F A = da + 1 [A, A]. 2 Here the commutator is to be taken in g. The curvature F A is a 2-form on P with values in g. One can think of the curvature as a 2-form on the basis M with values in the associated bundle Ad(P) = P G g, defined by the adjoint action. The difference between two connections is a 1-form on M with values in this bundle. One therefore says that gauge bosons transform under the adjoint action of the gauge group G. 14 / 35

Connection and curvature II Let (M, η) = (R 4, η Mink ) and choose a gauge s : M P. Then the differential of s defines the following forms on M with values in the Lie algebra g: With A = A ds F A = F(ds( ), ds( )). A µ = A(e µ ), F A µν = F A (e µ, e ν ) we have the fundamental equation Curvature F A µν = µ A ν ν A µ + [A µ, A ν ]. 15 / 35

Covariant derivative Every connection A on P defines a covariant derivative A on the associated bundle E: Let (M, η) = (R 4, η Mink ) and s : M P be a gauge. Then every section Φ in E is described by a map φ: M V, so that Φ = [s, φ]. In an inertial frame the corresponding covariant derivative is given by Covariant derivative A µφ = µ φ + A µ φ. On the right hand side the g-valued function A µ acts on the V -valued function φ via the representation of the group G. 16 / 35

Gauge fields Let n be the dimension of G. Choosing a basis T a of g, with a = 1,..., n, we can write A µ = A a µt a (Einstein summation convention). The connection A µ corresponds via the Lorentz metric to n vector fields A 1µ, A 2µ,..., A nµ on M. These vector fields describe the gauge bosons. There are thus precisely dim(g)-many gauge bosons in the gauge theory. The corresponding covariant derivative on F = S E describes the coupling of the gauge bosons to the fermions (the fermions interact with the gauge field and thus indirectly with each other, emission/absorption of gauge bosons). The term [A µ, A ν ] describes the interaction of the gauge bosons with each other in non-abelian gauge theories. 17 / 35

Dirac operator The covariant derivative A on E defines together with the spin connection S on the spinor bundle S a covariant derivative F on the fermion bundle F = S E. This defines with Clifford multiplication a twisted Dirac operator D A : C (S E) C (S E). We only need a formula for (M, η) = (R 4, η Mink ): Let s : M P be a gauge. Then we have in an inertial frame Dirac operator D A Ψ = iγ µ A µψ = iγ µ ( µ + A µ )Ψ, for Ψ = ψ 1. ψ r. The Dirac matrices γ µ here act on each 4-spinor component ψ i, while the gauge fields A µ act on the r components of Ψ. 18 / 35

Lagrangian I We can now write down the Lagrangian L of the gauge theory. The Lagrangian is a real valued function on M. It is given by where L = L fermion + L YM L fermion = Ψ, (D A m)ψ L YM = c 4g 2 FA F A. Here, is a hermitian scalar product on F = S E, m is the mass of the fermion, c a constant depending on the group G, g the coupling constant and a scalar product on the Ad(P)-valued 2-forms on M. 19 / 35

Lagrangian II We can also formulate the Lagrangian for (M, η) = (R 4, η Mink ). We choose a gauge s : M P and an inertial frame. Then we have L fermion = Ψ(iγ µ A µ m)ψ L YM = 1 4g 2 F Aaµν F Aa µν = 1 2g 2 tr(f Aµν F A µν). Here Ψ = Ψ γ 0, so that terms like ΨΨ and Ψγ µ A µψ transform as a scalar. In addition we choose a basis T a of the matrix algebra g, so that tr(t a T b ) = 1 2 δab. Then we write F A µν = F Aa µν T a. 20 / 35

The physical model The physical model is given precisely by this second Lagrangian. It is supposed to have the following symmetries: Definition (Lorentz invariance) The Lagrangian is independent of the choice of inertial frame. Definition (Gauge invariance) The Lagrangian is independent of the choice of gauge. It is clear that the Lagrangian is Lorentz invariant. We only have to check gauge invariance. 21 / 35

Gauge invariance I Let s, s : M P be two gauges. Then there is a gauge transformation U : M G so that s = s U. If Φ is a section of E, then Φ is described by φ, φ : M V with Φ = [s, φ] = [s, φ ]. We therefore have φ = U φ. The connection A is described in the gauges by 1-forms A, A on M with A = A ds, A = A ds. One can check (for a matrix group G): A µ = U A µ U 1 + U µ (U 1 ). 22 / 35

Gauge invariance II It follows that A µ φ = U A µ(u 1 φ ) F A µν = U F A µν U 1. These equations imply the gauge invariance of L fermion und L YM (for L fermion we use that the representation of G on V is unitary). Theorem The Lagrangian L = L fermion + L YM is gauge invariant. This was implicitly clear from the geometric formulation. 23 / 35

Normalized gauge fields In physics one often uses normalized gauge fields Then we have W µ = 1 ig A µ F W µν = 1 ig F A µν. F W µν = µ W ν ν W µ + ig[w µ, W ν ] W µ φ = µ φ + igw µ φ W µ = UW µ U 1 i g U µ(u 1 ) L YM = 1 2 tr(f W µν F W µν ). 24 / 35

In this section we describe some examples of gauge theories as well as the standard model of elementary particle physics. In every example we will indicate in particular the Lie group G and the vector space V. The representations of G on V depend on certain (rational) numbers that are called charges. Example (Charges) Quantum electrodynamics has gauge group U(1). The charge Q is called electric charge. The electroweak interaction has gauge group U(1) Y SU(2) L. The charges are called weak hypercharge Y and weak isospin T 3. We have Q = T 3 + Y 2. 25 / 35

QED The standard model The simplest example is quantum electrodynamics (QED). We have G = U(1) (abelian). The 1-form W µ has values in u(1) = R, hence W µ is after a choice of basis for R a standard 1-form. The gauge field W µ is called photon. The curvature F µν is thought of as the field strength. V = C. Therefore we have F = S E = S, i.e. fermions are described by 4-component spinors Ψ. We have g = e (elementary charge). One often writes A µ instead of W µ. We have L = L fermion + L YM = Ψ(iγ µ µ m)ψ 1 4 F µν F µν where µ = µ + iqa µ (charge q) and F µν = µ A ν ν A µ. 26 / 35

QCD I The standard model The next example is quantum chromodynamics (QCD). It describes the strong interaction between quarks. We have G = SU(3). The gauge field W µ has dim(su(3)) = 8 components (gluons). Since the group is non-abelian, there is an interaction between the gluons. V = C 3 with the standard representation. A quark (section in F = S E) is of the form q f = q r f q g f q b f, where f = u, d, c, s, t, b is one of six flavours, r, g, b one of three colours (red, green, blue) and q i f a 4-component spinor. The group SU(3) mixes the colours. 27 / 35

QCD II The notions of colour and hence quantum chromodynamics come from the triality of basic colours and because one only observes white combinations in nature (color confinement). One often writes G µ instead of W µ. We have L = L fermion + L YM = f q f (iγ µ µ m f )q f 1 2 tr(f µν F µν ) where µ = µ + igg µ F µν = µ G ν ν G µ + ig[g µ, G ν ]. Emission of a gluon can change the colour of a quark, different from the case of photons and electric charge. 28 / 35

Chirality Every 4-component spinor Ψ (Dirac spinor) over a 4-manifold M with a Lorentz metric decomposes into the direct sum of two 2-component spinors (Weyl spinors) Ψ R and Ψ L, Ψ = ( ΨR which are eigenvectors of the chirality operator γ 5 = iγ 0 γ 1 γ 2 γ 3 (orientation) with eigenvalues ±1 (right- and left-handed spinors). In the examples so far, right- and left-handed spinors transform in the same representation of the gauge group, which is why we can combine them into a 4-component spinor. The electroweak interaction on the other hand is a chiral gauge theory right- and left-handed spinors transform in different representations of the gauge group. Ψ L ), 29 / 35

Electroweak interaction I G = U(1) Y SU(2) L. One writes B µ for the gauge boson that belongs to U(1) Y (with coupling constant g ) and W µ for the gauge bosons that belong to SU(2) L (coupling constant g). The representations of G distinguish between right- and left-handed spinors. For left-handed spinors we have V = C 2 with the standard representation of SU(2). The fermions are of the form ( ) ( ) νel ul el, d L. Here ν e is the electron-neutrino, e the electron, u the up-quark and d the down-quark. Similar doublets exist for the other generations. Every component is a left-handed 2-component spinor. The isospin is T 3 = ± 1 2. The hypercharge is Y = 1 (leptons) and Y = 1 3 (quarks), respectively. 30 / 35

Electroweak interaction II For the left-handed quarks we have d L s L = V b L with the so-called CKM-matrix V. A quark of type u L can thus turn via weak interaction into different quarks of type d L, and vice versa (flavour change, β-decay d u + e + ν e ). For right-handed spinors we have V = C with the trivial representation of SU(2). The fermions are right-handed 2-component spinors of the form d L s L b L e R, u R, d R (with T 3 = 0 and Y = 2, 4 3, 2 3 ). One does not observe right-handed neutrinos (sterile). Similar singlets exist for the other generations. 31 / 35

The Higgs field I There are two problems: In chiral theories only mass terms with m = 0 in the Lagrangian are gauge invariant. But all fermions except the neutrinos have a mass different from zero. In gauge theories the gauge bosons have zero mass. However, one observes that the gauge bosons W ±, Z 0 of weak interaction have a non-zero mass. Solution: The fermions and gauge bosons have by themselves mass zero and acquire a mass only through interaction with a scalar field (Higgs field). This field is the only one which has a non-zero vacuum expectation value (the vacuum with Higgs field equal to zero is not stable). Since the field is non-zero in the vacuum, the vacuum is only invariant under a subgroup U(1) em of U(1) Y SU(2) L (spontaneous symmetry breaking). In addition a new particle arises, the Higgs boson, with non-zero mass. 32 / 35

The Higgs field II The Lagrangian of the Higgs field φ = ( φ1 φ 2 ) is: L = 1 2 ( µ φ) ( µ φ) V (φ), with V (φ) = 1 2 µφ φ + 1 2 λ(φ φ) 2. The minimum of the potential (vacuum) is at v = φ = µ 2λ. The mass of the fermions and weak gauge bosons is proportional to v. The mass of the Higgs boson is µ. Figure: Potential V (φ) of the Higgs field (en.wikipedia.org) 33 / 35

Further topics Field quantization, perturbation theory: free fields interacting fields (path integrals, Feynman diagrams, renormalization). Grand Unified Theories (SU(5) U(1) SU(2) SU(3), proton decay p e + + 2γ). Supersymmetry, Minimal Supersymmetric Standard Model (MSSM, superpartners: squark, slepton, gluino, etc.), candidates for dark matter (WIMPs, Weakly Interacting Massive Particles) in addition to sterile neutrinos. Quantum theory of gravity, superstrings. 34 / 35

References Helga Baum, Eichfeldtheorie, Springer-Verlag 2014 (in German). Ulrich Mosel, Fields, Symmetries, and Quarks, Springer-Verlag 1999. Thank you! 35 / 35