Gauge Theories

1. 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,

2. 1.Lagrangians in relativistic fields • Particles in classical mechanics and relativistic field (let, ħ=c=1) • Euler Lagrange equations

3. Klein-Gordon Lagrangian for scalar field • Suppose • and • Klein-Gordon equation, a particle of spin 0 and mass m

4. Dirac Lagrangian for Spinor field • Consider • and • Dirac equation, a particle of spin 1/2 and mass m

5. Proca Lagrangian for a Vector field • Suppose • and • Proca equation, a particle of spin 1 and mass m

6. Proca Lagrangian for a Vector field • Introduce • so • field equation of a particle of spin 1 and mass m becomes

7. Maxwell Lagrangian for Massless vector field with Source Jμ • Suppose • The Euler-Lagrange equations yield • It follows (continuity equation)

8. 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 )

9. 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

10. 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

11. 3.Yang-Mills theory • Suppose two spin ½ fields, ψ1 and ψ2 • by matrix representation

12. 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.]

13. £ is not invar. under local SU(2) trans. Resulting Lagrangian • Introduce vector fields,

14. 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

15. 4.Chromodynamics • The free Lagrangian for a particular flavor • by matrix representation

16. General global invar. we can write (where H is Hermitian) thus • Let [local SU(3) trans.]

17. £ is not invar. under local SU(3) trans. Resulting Lagrangian • Introduce vector fields,

18. now we add the free gluon Lagrangian • The complete Lagrangian for Chromodynamics • Dirac fields constitute eight color currents

19. 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

20. 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

21. 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

22. 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.

23. [ graph of U(Φ)] • The third and fourth terms represend couplings of the form

24. 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

25. 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,

26. [ spontaneous symmetry breaking in a plastic strip ] • [ the potential function ]

27. 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

28. 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

29. 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.

30. Replace equations with covariant derivatives • Thus Define the new fields Lagrangian becomes

31. 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

32. Writing equation in terms of its real and imaginary parts • Pick will render Φ’ real, Φ2’=0 In this particular gauge, (ξis zero)

33. 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)