The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten
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1 Lecture 4 QCD as a Gauge Theory Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora
2 The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Tensor The QCD Lagrangian Feynman Rules in Covariant Gauges Running Coupling in QED Running Coupling in QCD Asymptotic Freedom
3 The Gauge Principal
4 The Gauge Principal
5 The Gauge Principal Maxwell s formulation of electrodynamics was perhaps the first field theory and a gauge theory in physics. In 1929, Weyl showed that electrodynamics was invariant under the gauge transformation of the gauge field and the wave-function of the charged field. In 1954, Yang and Mills studied the gauge principle in non-abelian field theories.
6 The Gauge Principal Gauge principle provides us with a recipe to construct field theories describing the forces of nature. We shall first study this for QED to see how it works and how interactions between electrons and photons emerge naturally through employing the gauge principle. We shall then employ it to QCD, a non abelian theory describing interactions between quarks and gluons (gauge bosons) and how they bind together to make hadrons. Note that Salam-Weinberg Weinberg-Glashow theory of weak interactions is also a gauge theory and the interactions can be derived through gauging the force free Lagrangian.
7 Quantum Electrodynamics The free Dirac Lagrangian: This is invariant under global U(1) gauge transformations: where α is a real constant. This is an abelian symmetry because the two successive transformations commute. This is a continuous symmetry. Electric current is the conserved Noether current and the fermion electric charge is conserved.
8 Quantum Electrodynamics The wave-function of an electron carries a phase with it. In global gauge transformation, this phase is changed by the same amount at every space-time point.
9 Quantum Electrodynamics We now want to generalize this symmetry into a local gauge symmetry by demanding that the Lagrangian should be invariant under this symmetry even if α(x) is an arbitrary differentiable function of x. This is a strong demand. It says that we can change the phase of a Dirac field at any space-time point in an arbitrary way and the Lagrangian, i.e., the dynamics of the Dirac field should remain the same. QED, QCD and the Standard Model are constructed this way.
10 Quantum Electrodynamics The mass term is still invariant. However the kinetic energy term is no longer invariant. In order to fulfill our demand, we introduce a gauge field and define the so-called gauge covariant derivative. It contains the coupling constant e of the fermion field to the gauge field.
11 The starting point is: Quantum Electrodynamics The local gauge transformation now also involves the transformation property of the new gauge field: The above Lagrangian is invariant under this local U(1) gauge transformation.
12 Quantum Electrodynamics Gauge field with compensating transformation is introduced.
13 Quantum Electrodynamics The crucial point is that the covariant derivative now transforms as: Thus the kinetic energy term is now invariant, ensuring that the Lagrangian as a whole is also invariant. The A-field is still not dynamical so far.
14 Quantum Electrodynamics A candidate for the dynamics of the gauge field A is the vector potential of the Maxwell s equations: F μν is is invariant under the gauge transformations:
15 Assume the ψ-field to be an n-tuple tuple: SU(N) Gauge Theory The free field is described by the Dirac Lagrangian: Assume m to be a diagonal n x n matrix with identical masses as its entries. Dirac matrices are also embedded in an n x n unit matrix in a diagonal manner. We consider transformation properties of the Lagrangian under the SU(N) transformations:
16 Global Gauge Transformations For fermion fields, we can write: Here θ a are the real constants, independent of x. That is why these are called the global SU(N) transformations.
17 Global Gauge Transformations As τ a do not commute with each other, SU(N) Lie group is non-abelian abelian. This means that 2 successive transformations give a different result when one interchanges the order. SU(N) Dirac Lagrangian is invariant under these global transformations. There are conserved currents and correspondingly conserved charges:
18 Global Gauge Transformations There are different color phases associated with quarks In global gauge transformations, they are transformed independently of x.
19 Local Gauge Transformations We now want to generalize this symmetry into a local gauge symmetry by demanding that the Lagrangian should be invariant under this symmetry even if θ(x) is an arbitrary differentiable function of x. This is a strong demand. It says that we can change the quark phases a of a field at any space-time point in an arbitrary way and the Lagrangian, i.e., the dynamics of the quark field should remain the same. QCD and the Standard Model are constructed this way.
20 Local Gauge Transformations In order to fulfill our demand, introduce 8 gauge fields and define the covariant derivative. Note that again it contains the coupling constant g of the quark field to the gauge field. We have as many gauge fields as the number of generators. However, we have only one coupling. The same coupling exists between very quark field and every gauge field. We demand that the gauge fields transform under the local transformations such that:
21 Local Gauge Transformations This is achieved if: Thus the kinetic energy term is now invariant, ensuring that the Lagrangian as a whole is also invariant. Let us look at it pictorially.
22 Initially: Local Gauge Transformations Compensating Fields: Local Gauge Transformation:
23 Dynamics of Field Tensor We can easily check the following in QED: Combining the two identities: As ψ is an arbitrary spinor and exp(-iα(x)) (x)) & F μν commute, the later being a function, we obtain the gauge invariance of the field tensor.
24 Dynamics of Field Tensor How do we define F μν for SU(N)? The following definition does not work: We can check that with this definition, is not gauge invariant. Thus we follow the procedure just outlined. Define F μν follows and deduce what it is: μν as
25 Is F a μν gauge independent? But its Lorentz invariant construction is. Thus: SU(N) Lagrangian: Dynamics of Field Tensor Gauge field tensor: These terms correspond to self coupling of the gauge field.
26 The QCD Lagrangian QCD is obtained by demanding local SU(3) gauge invariance of the following free quarks Lagrangian: Quarks have 3 internal degrees of freedom, called color. It is these color degrees of freedom which are gauged with an SU(3) color gauge transformation.
27 The QCD Lagrangian Thus the QCD Lagrangian can be written as: Gluon Fields: Massless, spin 1 bosons, color octet because the group contains eight generators, flavor singlet (i.e., gluons are flavor blind, not distinguishing between flavors of whatever they interact with.
28 Feynman Rules in Covariant Gauges The Propagators:
29 Feynman Rules in Covariant Gauges The Gluon Self Interactions:
30 Feynman Rules in Covariant Gauges The Gluon Vertices with Ghosts and Quarks:
31 Running Coupling in QED
32 Running Coupling in QED
33 Landau Pole: Running Coupling in QED
34 Experimental Measurement: Running Coupling in QED
35 Running Coupling in QCD QCD:
36 Running Coupling in QCD Competition between color and flavor:
37 Running Coupling in QCD
38 Asymptotic Freedom: Running Coupling in QCD
39 What Next? How can we describe hadrons, starting from QCD dynamics? How can we invoke symmetry (and its breaking) arguments to generate pions? How can the mass of the nucleon be related to chiral symmetry breaking? Can we derive Gell-Mann Mann-Okubo mass formulae and other low energy relations? What role does the vacuum have to play in the study of chiral dynamics through the fundamental degrees of freedom of quarks and gluons?
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