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Gauge Theories. Lagrangians in relativistic field theory Local gauge invariance Yang-Mills theory Chromodynamics The mass term Spontaneous symmetry-breaking The Higgs mechanism
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Gauge Theories Lagrangians in relativistic field theory Local gauge invariance Yang-Mills theory Chromodynamics The mass term Spontaneous symmetry-breaking The Higgs mechanism These notes are not original, and rely heavily on information and examples in the texts “Introduction to Particle Physics” by David Griffiths,
1.Lagrangians in relativistic fields • Particles in classical mechanics and relativistic field (let, ħ=c=1) • Euler Lagrange equations
Klein-Gordon Lagrangian for scalar field • Suppose • and • Klein-Gordon equation, a particle of spin 0 and mass m
Dirac Lagrangian for Spinor field • Consider • and • Dirac equation, a particle of spin 1/2 and mass m
Proca Lagrangian for a Vector field • Suppose • and • Proca equation, a particle of spin 1 and mass m
Proca Lagrangian for a Vector field • Introduce • so • field equation of a particle of spin 1 and mass m becomes
Maxwell Lagrangian for Massless vector field with Source Jμ • Suppose • The Euler-Lagrange equations yield • It follows (continuity equation)
2.Local Gauge Invariance • Dirac Lagrangian is invariant under the transformation (global gauge trans.) But, if the phase factor is different at different space-time, (local gauge trans.) • Is Dirac Lagrangian invariant under local gauge trans.? ( No )
let so, under • We add something in order to make LDirac be invariant under local gauge trans. Suppose with • Now, Lagrangian is invariant under local gauge trans. But full Lagrangian must include a “free term” for the gauge field. Consider Proca Lagrangian
Where is invar. is not. Evidently guage field must be massless(mA=0) therefore, with • The difference between global and local gauge trans. Arises where, is called “covariant derivative” and
3.Yang-Mills theory • Suppose two spin ½ fields, ψ1 and ψ2 • by matrix representation
If the two masses is equal, ,where ψ is now two element column vector • General global inv. (where U is 2×2 unitary matrix) We can write (where H is Hermitian) [global SU(2) trans.] • Let [local SU(2) trans.]
£ is not invar. under local SU(2) trans. Resulting Lagrangian • Introduce vector fields,
Aμ Require their own free Lagrangian (Proca mass term is excluded by local guage invar.) • The complete Yang-Mills Lagrangian (describes two equal-mass Dirac fields in interaction with three massless vector gauge fields.) Dirac fields generate three currents
4.Chromodynamics • The free Lagrangian for a particular flavor • by matrix representation
General global invar. we can write (where H is Hermitian) thus • Let [local SU(3) trans.]
£ is not invar. under local SU(3) trans. Resulting Lagrangian • Introduce vector fields,
now we add the free gluon Lagrangian • The complete Lagrangian for Chromodynamics • Dirac fields constitute eight color currents
5.The mass term • The principle of local gauge invar. works beautifully for the strong and E.M. interactions. • The application to weak interactions was stymied because gauge fields have to be massless. • Can we make gauge theory to accommodate massive gauge fields? Yes, by using spontaneous symmetry-breaking and the Higgs mechanism. • Suppose
If we expand the exponential the second term looks like the mass term in the K.G. Lagrangian with The higher-order terms represent couplings, of the form This is not supposed to be a realistic theory
To identify how mass term in a Lagrangian may be disguised, we pick out the term propotional to Φ2 the second term looks like mass, and the third term like an interaction. If that is mass term, m is imaginary(nonsense) • Feynman calculus about a perturbation start from the ground state(vacuum) and treat the fields as fluctuations about that state: Φ=0 But for above Lagrangian, Φ=0 is not the ground state. To determine the true ground state, consider
so, And the minimum occurs at • Introduce a new field variable In terms of η Now second term is a mass term, with the correct sign.
[ graph of U(Φ)] • The third and fourth terms represend couplings of the form
6.Spontaneous symmetry-breaking • From the mass term, the original Lagrangian is even in Φ • The reformulated Lagrangian is not even in η • (the symmetry has been broken) • It happened because the vacuum does not share the symmetry of the Lagrangian
For example, the Lagrangian with spontaneously broken continuous symmetry (it is invar. under rotations in Φ1Φ2 space ) where, The minimum condition We may as well pick,
[ spontaneous symmetry breaking in a plastic strip ] • [ the potential function ]
Introduce new fields • Rewriting the Lagrangian in terms of new variables, • The first term is a free K.G. Lagrangian for the field η the second term is a free Lagrangian for the field ξ
The third term defines five couplings • In this form, the Lagrangian doesn’t look symmetrical at all (the symmetry has been broken by the selection of a particular vacuum state) • One of the fields(ξ) is automatically massless
7.The Higgs mechanism • If we combine the two real fields into a single complex field • The rotational(SO(2)) symmetry that was spontaneously broken becomes invar. under U(1) phase trans. • We can make the system invar. under local gauge trans.
Replace equations with covariant derivatives • Thus Define the new fields Lagrangian becomes
The first line describes a scalar particle and a massless Goldstone boson (ξ) • The second line describes the free gauge field Aμ, it has acquired a mass • The term in curly brackets specifies various coupling of ξ,η, Aμ • We still have unwanted Goldstone boson (ξ) as interaction, it leads to a vertex of the form
Writing equation in terms of its real and imaginary parts • Pick will render Φ’ real, Φ2’=0 In this particular gauge, (ξis zero)
We have eliminated the Goldstone boson and the offending term in £; we are left with a single massive scalar η(the Higgs particle) and massive gauge field Aμ • A massless vector field carries two degree of freedom (tranverse polarizations). When Aμacquires mass, it picks up a third degree of freedom(longitudinal polarization) Q: where did this extra degree of freedom come from? A: it came from the Goldstone boson, which meanwhile disappeared from the theroy. The gauge field ate the Goldstone boson, thereby acquiring both a mass and a third polarization state (Higgs mechanism)