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## Gauge Theories

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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)