gauge theories and the standard model of elementary ... · the standard model gauge theories –...

The standard modelGauge theories – geometrical and physical

Examples

Gauge theories and the standard model ofelementary particle physics

Mark Hamilton

21st July 2014

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Examples

Table of contents

1 The standard model

2 Gauge theories – geometrical and physical

3 Examples

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Examples

The standard model

The standard model is the most successful theory ofelementary particle physics. Its predictions have been verifiedin numerous experiments at particle accelerators. Thestandard model describes all known interactions amongelementary particles, except gravity.The mathematical foundation of the standard model is agauge theory, based on the symmetry groupU(1)× SU(2)× SU(3).The particles of the standard model consist of threegenerations of fermions (electron, neutrino, quarks, etc.),the gauge bosons (photon, gluon, etc.) that mediate theinteractions between the fermions and the Higgs bosonwhich is a part of the Higgs field that conveys mass to thefermions and some of the gauge bosons.

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Examples

The particles of the standard model

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

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History

1940s: Development of quantum electrodynamics.1954: Yang and Mills develop non-abelian gauge theories.1956: Lee, Wu and Yang discover chirality of the weakinteraction.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 withthe Higgs mechanism.1972: t’Hooft and Veltman prove renormalizability ofelectroweak interaction.1973-74: Development of quantum chromodynamics.

Several Nobel prizes for development of the standard model (in particular1979 for Glashow, Salam and Weinberg, 1999 fur t’Hooft and Veltman aswell as 2013 for Higgs and Englert).

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Examples

Gauge theories as field theories

Gauge theories are field theories that have two kinds ofsymmetries:

Lorentz invarianceGauge invariance

We will describe below two models for gauge theories:Geometrical model, in which the symmetries are implicitPhysical model, in which the symmetries are explicit.

We will also describe the transition from the geometrical to thephysical model. We will only discuss the classical field theory, inparticular the Lagrangian. In physics the field theory is quantized(quantum field theory) and the Feynman rules are deduced fromthe Lagrangian. Elementary particles correspond to quanta,i.e. minimal excitations of the fields.

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Examples

BundlesGauge theories are formulated geometrically in the language ofbundles over manifolds, in particular principal bundles and vectorbundles (spinor bundle, associated bundles). We have the followingfundamental fact:RemarkEvery bundle over M = R4 is trivial, i.e. a product M × Σconsisting of base M and fiber Σ, since R4 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 ispreferred. In physics one often chooses trivializations and theindependence of such a choice reflects itself in invariances andsymmetries.

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Examples

The geometrical model

The geometrical model of a gauge theory consists of the followingdata:

1 Spacetime (M, η), a differentiable manifold with a metric ofsignature (+,−, . . . ,−), 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 FA.

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Examples

Spinor bundle

The spinor bundle is a certain complex vector bundle overthe manifold M (it exists if M is spin).On the spinor bundle there exists a Clifford multiplication

TM × S → S, (v , ψ) 7→ v · ψ,

so thatv · (v · ψ) = −2η(v , v)ψ.

If (M, η) = (R4, ηMink), then S ∼= M × C4 after a choice ofinertial frame.In this case, Clifford multiplication with a basis vector eµ in aninertial frame is given by multiplication of a spinor in C4 withiγµ, where γµ are certain 4× 4 Dirac matrices that satisfy

{γµ, γν} = γµγν + γνγµ = 2ηµνId.

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Examples

Principal bundle

A G-principal bundle is a manifold P with a projectionπ : P → M and an action P × G → P so that:

1 Every fibre of P is diffeomorphic to G and the action of Gpreserves the fibres and is simply transitive on them.

2 P is over small open sets U in M of the form U × G . In otherwords, P is locally trivial.

If (M, η) = (R4, ηMink), then P is globally trivial. In this casewe have P ∼= M × G , together with the standard action of G .It is important that these trivializations are not canonical: Atrivialization is given by a global section s : M → P. Wethen get every element of P as s(x) · g , for x ∈ M.

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Examples

Gauge

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 inertialframe in relativity. In both cases there is a manifold which is trivialin a certain sense, but that does not have a preferred trivialization.The change between two trivializations is described by a Lorentzand gauge transformation, respectively. The choice of gauge is thechoice of a coordinate system. Gauge invariance will later mean:Invariance under the choice of gauge.

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Examples

Associated bundleLet G × V → V be a representation. Then G acts on P × V fromthe 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) = (R4, ηMink), then E is trivial, E ∼= M × V .A trivialization is not canonical, but given by a choice ofgauge: If s : M → P is a gauge, then sections Φ: M → Ecorrespond precisely to mappings φ : M → V via

Φ = [s, φ].

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Examples

Fermion bundle

The fermion bundle F is given by F = S ⊗ E . Let(M, η) = (R4, ηMink). We choose an inertial frame and a gauges : M → P. Let r denote the complex dimension of therepresentation space V . Then a section Ψ in F is given by

Ψ =

ψ1...ψr

,where each component ψi : M → C4 is a spinor. A fermion, i.e. asection 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.

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Examples

Connection and curvature I

A connection A on the principal bundle P is a certain invariant1-form on P with values in the Lie algebra g. The curvature of Ais defined as

FA = dA +12 [A,A].

Here the commutator is to be taken in g. The curvature FA is a2-form on P with values in g.

One can think of the curvature as a 2-form on the basis M with values inthe associated bundle

Ad(P) = P ×G g,

defined by the adjoint action. The difference between two connections isa 1-form on M with values in this bundle. One therefore says that gaugebosons transform under the adjoint action of the gauge group G .

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Examples

Connection and curvature II

Let (M, η) = (R4, ηMink) and choose a gauge s : M → P. Then thedifferential of s defines the following forms on M with values inthe Lie algebra g:

A = A ◦ dsF A = F(ds(·), ds(·)).

WithAµ = A(eµ), F A

µν = F A(eµ, eν)

we have the fundamental equation

CurvatureF A

µν = ∂µAν − ∂νAµ + [Aµ,Aν ].

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Examples

Covariant derivative

Every connection A on P defines a covariant derivative ∇A onthe associated bundle E : Let (M, η) = (R4, ηMink) and s : M → Pbe a gauge. Then every section Φ in E is described by a mapφ : M → V , so that

Φ = [s, φ].

In an inertial frame the corresponding covariant derivative is givenby

Covariant derivative∇A

µφ = ∂µφ+ Aµφ.

On the right hand side the g-valued function Aµ acts on theV -valued function φ via the representation of the group G .

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Examples

Gauge fieldsLet n be the dimension of G . Choosing a basis T a of g, witha = 1, . . . , n, we can write

Aµ = AaµT a (Einstein summation convention).

The connection Aµ corresponds via the Lorentz metric to n vectorfields

A1µ,A2µ, . . . ,Anµ

on M.These vector fields describe the gauge bosons. There arethus precisely dim(G)-many gauge bosons in the gauge theory.The corresponding covariant derivative on F = S ⊗ Edescribes the coupling of the gauge bosons to the fermions(the fermions interact with the gauge field and thus indirectlywith each other, emission/absorption of gauge bosons).The term [Aµ,Aν ] describes the interaction of the gaugebosons with each other in non-abelian gauge theories. 17 / 35


Examples

Dirac operatorThe covariant derivative ∇A on E defines together with the spinconnection ∇S on the spinor bundle S a covariant derivative ∇F

on the fermion bundle F = S ⊗ E . This defines with Cliffordmultiplication a twisted Dirac operator

DA : C∞(S ⊗ E ) −→ C∞(S ⊗ E ).

We only need a formula for (M, η) = (R4, ηMink): Let s : M → Pbe a gauge. Then we have in an inertial frameDirac operator

DAΨ = iγµ∇AµΨ = iγµ(∂µ + Aµ)Ψ, for Ψ =

ψ1...ψr

.The Dirac matrices γµ here act on each 4-spinor component ψi ,while the gauge fields Aµ act on the r components of Ψ.

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Examples

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

L = Lfermion + LYM

where

Lfermion = 〈Ψ, (DA −m)Ψ〉

LYM =c

4g2FA · FA.

Here 〈· , ·〉 is a hermitian scalar product on F = S ⊗ E , m is themass of the fermion, c a constant depending on the group G , gthe coupling constant and · a scalar product on theAd(P)-valued 2-forms on M.

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Examples

Lagrangian IIWe can also formulate the Lagrangian for (M, η) = (R4, ηMink).We choose a gauge s : M → P and an inertial frame. Then we have

Lfermion = Ψ(iγµ∇Aµ −m)Ψ

LYM =1

4g2 F AaµνF Aaµν =

12g2 tr(F AµνF A

µν).

Here Ψ = Ψ†γ0, so that terms like

ΨΨ and Ψγµ∇AµΨ

transform as a scalar. In addition we choose a basis T a of thematrix algebra g, so that tr(T aT b) = 1

2δab. Then we write

F Aµν = F Aa

µν T a.

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Examples

The physical model

The physical model is given precisely by this second Lagrangian. Itis 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 haveto check gauge invariance.

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Examples

Gauge invariance I

Let s, s ′ : M → P be two gauges. Then there is a gaugetransformation U : M → G so that s = s ′ · U. If Φ is a section ofE , 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 inthe 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 · Aµ · U−1 + U · ∂µ(U−1).

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Examples

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 Lfermion und LYM(for Lfermion we use that the representation of G on V is unitary).

TheoremThe Lagrangian L = Lfermion + LYM is gauge invariant.

This was implicitly clear from the geometric formulation.

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Examples

Normalized gauge fieldsIn physics one often uses normalized gauge fields

Wµ =1ig Aµ

F Wµν =

1ig F A

µν .

Then we have

F Wµν = ∂µWν − ∂νWµ + ig [Wµ,Wν ]

∇Wµ φ = ∂µφ+ igWµφ

W ′µ = UWµU−1 − i

g U∂µ(U−1)

LYM = −12tr(F W µνF W

µν ).

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Examples

Examples

In this section we describe some examples of gauge theories as wellas the standard model of elementary particle physics. In everyexample we will indicate in particular

the Lie group G andthe 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 iscalled electric charge. The electroweak interaction has gauge groupU(1)Y × SU(2)L. The charges are called weak hypercharge Y andweak isospin T3. We have Q = T3 + Y

2 .

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Examples

QEDThe simplest example is quantum electrodynamics (QED). Wehave

G = U(1) (abelian). The 1-form Wµ has values in u(1) ∼= R,hence Wµ is after a choice of basis for R a standard 1-form.The gauge field Wµ is called photon. The curvature Fµν isthought of as the field strength.V = C. Therefore we have F = S ⊗ E = S, i.e. fermions aredescribed by 4-component spinors Ψ.We have g = e (elementary charge).

One often writes Aµ instead of Wµ. We have

L = Lfermion + LYM

= Ψ(iγµ∇µ −m)Ψ− 14F µνFµν

where ∇µ = ∂µ + iqAµ (charge q) and Fµν = ∂µAν − ∂νAµ.26 / 35


Examples

QCD IThe next example is quantum chromodynamics (QCD). Itdescribes the strong interaction between quarks. We have

G = SU(3). The gauge field Wµ has dim(SU(3)) = 8components (gluons). Since the group is non-abelian, there isan interaction between the gluons.V = C3 with the standard representation. A quark (section inF = S ⊗ E ) is of the form

qf =

qrf

qgf

qbf

,where f = u, d , c, s, t, b is one of six flavours, r , g , b one ofthree colours (red, green, blue) and qi

f a 4-component spinor.The group SU(3) mixes the colours.

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Examples

QCD IIThe notions of “colour” and hence “quantum chromodynamics”come from the triality of basic colours and because one onlyobserves white combinations in nature (color confinement). Oneoften writes Gµ instead of Wµ. We have

L = Lfermion + LYM

=∑

fqf (iγµ∇µ −mf )qf −

12tr(F µνFµν)

where

∇µ = ∂µ + igGµ

Fµν = ∂µGν − ∂νGµ + ig [Gµ,Gν ].

Emission of a gluon can change the colour of a quark, different from thecase of photons and electric charge.

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Examples

Chirality

Every 4-component spinor Ψ (Dirac spinor) over a 4-manifold Mwith a Lorentz metric decomposes into the direct sum of two2-component spinors (Weyl spinors) ΨR and ΨL,

Ψ =

(ΨRΨL

),

which are eigenvectors of the chirality operator γ5 = iγ0γ1γ2γ3

(orientation) with eigenvalues ±1 (right- and left-handedspinors). In the examples so far, right- and left-handed spinorstransform in the same representation of the gauge group, which iswhy we can combine them into a 4-component spinor. Theelectroweak interaction on the other hand is a chiral gaugetheory – right- and left-handed spinors transform in differentrepresentations of the gauge group.

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Examples

Electroweak interaction I

G = U(1)Y × SU(2)L. One writes Bµ for the gauge bosonthat belongs to U(1)Y (with coupling constant g ′) and Wµ

for the gauge bosons that belong to SU(2)L (couplingconstant g).The representations of G distinguish between right- andleft-handed spinors. For left-handed spinors we haveV = C2 with the standard representation of SU(2). Thefermions are of the form(

νeLe−L

),

(uLd ′L

).

Here νe is the electron-neutrino, e− the electron, u theup-quark and d the down-quark. Similar doublets exist for theother generations. Every component is a left-handed2-component spinor. The isospin is T3 = ± 1

2 . The hypercharge isY = −1 (leptons) and Y = 1

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Examples

Electroweak interaction II

For the left-handed quarks we have d ′Ls ′Lb′L

= V ·

dLsLbL

with the so-called CKM-matrix V . A quark of type uL canthus turn via weak interaction into different quarks of type dL,and vice versa (flavour change, β-decay d → u + e− + νe).For right-handed spinors we have V = C with the trivialrepresentation of SU(2). The fermions are right-handed2-component spinors of the form

e−R , uR , dR (with T3 = 0 and Y = −2, 43 ,−

23 ).

One does not observe right-handed neutrinos (sterile). Similarsinglets exist for the other generations.

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Examples

The Higgs field IThere are two problems:

In chiral theories only mass terms with m = 0 in theLagrangian are gauge invariant. But all fermions except theneutrinos 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 weakinteraction have a non-zero mass.

Solution: The fermions and gauge bosons have “by themselves”mass zero and acquire a mass only through interaction with ascalar field (Higgs field). This field is the only one which has anon-zero vacuum expectation value (the vacuum with Higgs fieldequal to zero is not stable). Since the field is non-zero in thevacuum, the vacuum is only invariant under a subgroup U(1)em ofU(1)Y × SU(2)L (spontaneous symmetry breaking). In additiona new particle arises, the Higgs boson, with non-zero mass.

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Examples

The Higgs field II

The Lagrangian of the Higgs field φ =

(φ1φ2

)is:

L =12(∇µφ)†(∇µφ)− V (φ), with V (φ) = −1

2µφ†φ+

12λ(φ†φ)2.

The minimum of the potential (vacuum) is at v = |φ| =√

µ2λ . 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)

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Examples

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 InteractingMassive Particles) in addition to sterile neutrinos.Quantum theory of gravity, superstrings.

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Examples

References

Helga Baum, Eichfeldtheorie, Springer-Verlag 2014 (inGerman).Ulrich Mosel, Fields, Symmetries, and Quarks, Springer-Verlag1999.

Thank you!

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