Feynman Rules of Non-Abelian Gauge Theory

Transcription

1 Feynman Rules o Non-belian Gauge Theory The Lorenz gauge In the Lorenz gauge, the constraint on the connection ields is a ( µ ) = 0 = µ a µ For every group index a, there is one such equation, so there are three constraints overall. For a gauge variantion θ, the transormation is The connection ield components are given by U( θ) = + i θ(x) τ + O(θ ) a µ θ (x) = a µ(x) + ɛ abc θ b c µ g µθ a (x) The transormed gauge constraints are thereore a ( θ µ) = a ( µ ) + {ɛ µ abc θ b c µ } g µθ a (x) which we will write as a ( t µheta) = a ( µ ) + d 4 y ( M (x,y) ab θ b (y) ) δ (4) (x y) where a,b are the group indices. In the Lorenz gauge, the gauge ixing and Faddeev-Popov-ghost terms in the action are S g = d 4 x ( µ µ (x) ) ξ S FPGh = d 4 x c g a(x) { µ δ ab µ gɛ abc µ} c cb (x) a,b The term in the Fadeev-Popov-ghost line which is in parentheses basically be viewed as a covariant derivative. In the Lagrangian, this produces, among others, the monomial c µ c, a ghost-ghost-connection ield vertex. Similar as in QED, there is, once again, a spin one gauge boson. Feynman Rules or QCD The indices a,b,... denote the group indices. Let us call them color indices. Moreover, call the gauge boson gluon.

2 The gluon propagator depends on Lorentz indices as well as color indices. For the massless gluon, the Feynman rule or the propagator is { a k b i( δ µ ν ab ) g µν k } µk ν k k + iɛ This is the term which comes rom the quadratic terms in the ields. The quadratic terms in the c ields gives the ghost propagator. The ghost is a very strange particle, because in the Lagrangian, it looks like a ermion, but its propagator is similar to that or bosons. Moreover, there are no external ghosts. It only exists or virtual lines. i( δ ab ) k + iɛ Even though there is no mass term in the propagator, one cannot or sure call the ghost massless, because it is neither boson nor ermion. We have already stated that the three-gluon vertex must depend on the momenta k, the Lorentz indices µ and the color indices a. k µ a { } igɛ a a a 3 gµ µ (k k ) µ3 + g µ µ 3 (k k 3 ) µ + g µ3 µ (k 3 k ) µ 3 Next comes the Feynman rule or the our-gluon vertex. In the Lagrangian term F µν F µν, there appears a term with our connection ields, because o the non-vanishing commutator [ µ, ν ] in a non-abelian gauge theory. 4 3 g ɛ a a c ɛ a 3a 4 c { g µ µ 3 g µ µ 4 g µ µ 4 g µ µ 3 } +ɛ a a 3 c ɛ a a 4 c { g µ µ g µ3 µ 4 g µ µ 4 g µ µ 3 } +ɛ a a 4 c ɛ a 3a c { g µ µ 3 g µ µ 4 g µ µ g µ3 µ 4 } The Feynman rule or the gluon-ghost vertex is a bit strange because it is not symmetric in the ghost momenta. (k +k ) a k a 3 µ igɛ a a a 3 k µ k a It is important to notice that only the monentum o the incoming ghost particle, that is the line directed towards the vertex, contributes, with the Lorentz index o the vertex. However, since the ghost only apears internally, the sum over all graphs still works out, even though this Feynman rule look counterintuitive. The Feynman rule or the ermion o the theory looks as usual, besides the act that there is a contraction o color indices. a b iδ ab /k m + iɛ

3 Finally, the Feynman rule or the gluon-ermion-vertex is a c µ ig(t c ) ab γ µ b where T c can be any representation we choose, it just needs to be speciied with which representation one is working. O course, we have already thought about the ghost particle, what it is good or, why it is actually there at all, and whether or not the whole ghost ormalism makes sense at all. To gain some more inormation, let us look into analyticity o the S -matrix and Cutkosky rules again.. Cutkosky in QCD The S matrix is unitary: S S = I. I we write the S matrix as S = I + it, we get (in a rather symbolic notation) I(T i ) = T in T n δ (4) (p in p out ) n Now, let us assume that a ermion antiermion pair interacts somehow, where a virtual connection ield pair is produces, which again decays in some way into a ermion antiermion pair. ccording t Cutkosky, I.. Imaginary parts o propagators = nalyze the imaginary part o a gauge propagator with Lorentz indices µν and color indices ab: ab k µν = δ ab ( g µν ) k ( + iɛ ( ) ) = δ ab ( g µν ) P k + iπδ(k ) k I( ab µν) = πδ ab (g µν )δ(k )θ(ω k ) In the third line, P denotes the Cauchy principal value. Consequently, the imaginary part o the ghost propagator is.. Forward scattering I( ) = πδ ab δ(k )θ(ω k ). In the absolute square o the matrix element rom above, k k 3

4 there is a phase space integrad in it which indicates orward scattering. [ ] dρ T µνt ab µ ab ν g µµ g νν S ab S ab Tµν ab is the T matrix element or scattering. S ab represents c c scattering. The entire integral has an unphysical part, whose imaginary part is o interest. I we compare the right-hand side and let-hand side o I = then this expression should be identical to dρ T ab µνtµ ab ν P µµ (k )P νν (k ) which is entirely physical. The two integrals should be the same. These are all the contributions to Cutkosky cuts o the process: k k, The curly line denotes the gluon ields, the solid lide denotes the ermion ields, and the dashed line denotes the ghost ield. Moreover, the dashed lines rom top to bottom suggest the Cutkosky cuts. Since there is no external ghost particle in existence, the contribution to orward scattering consists o + + We are able to obtain a consistent result i we can show that k µ T ab µν = is ab k ν T ab µνk ν = is ab k µ I we contract the entire expression with k µ k ν, we should get zero, because k = 0 on the mass shell (gluons are massless). Thereore, k µ Tkν = 0. s it turns out, once again, this whole integrals simply makes no sense without the ghost loop. This is the reason or its introduction. There is a more systematic way to derive the introduction o the ghost loop, but it is too long or this class. 4

5 Optical Theorem and Renormalizability The optical theorem and renormalizability to not trivially or automatically both hold. Still, there is quite a systematic approach to it. Let us work in φ 3 theory here. The two-loop PI graphs or the propagator are 6 + To get the imaginary part, we look at Cutkosky cuts: X + X + X + X + + O course, counterterms can be cut, too. For each o the cuts, we get branch cut ambiguities. For the graphs with two cut lines, like, it starts at m, or three cut lines, line, it starts at 3m, and so on. The orward scattering is given by, or example, There are also short-distance singularities: + B X + B : one-loop renormalized C + D X C + D : one-loop renormalized Only i we allow counterterms, there can be renormalization and Cutkosky at the same time. The optical theorem is meaningul even or short-distance singularities. Showing that the optical theorem also holds or unrenormalized amplitudes is not so easy. One would need to regulate, and or that, the gauge bosons have to be massless, which is not generally the case. I the gauge bosons were massive, then there would be no gauge-invariant regulator. 5

6 3 Slavnov Taylor Identities Back to gauge theories: Let us derive Slavnov Taylor identities. Write down the Lagrangian or a ermion and boson ield: where L = 4 F µν a F µνa + ψi /Dψ m ψψ /D = γ µ D µ D µ ψ = ( µ ig a µt a) ψ F µν a = µ ν a ν µ a + gɛ abc µ b ν c I we transorm the ermionic ield, it depends on the representation we use: The transormation o the connection ield is The Lagrangian or the gauge ixing is and or the Faddeev Popov ghost δψ = it a θ a ψ δ a µ = ɛ abc θ b c µ g µθ a L g = ξ ( µ a µ) L FPGh = ic a µ ( δ ab µ gɛ abc c µ) cb Now, we write the ghost ield in terms o complex ermion ields. c a = (ρ a + iσ a ) c a = (ρ a iσ a ) That allows us to decompose the ghost ields c and c in real and imaginary ermionic ields ρ and σ. The Faddeev Popov ghost Lagrangian becomes L FPGh = i µ ρ a D µ σ a with D µ σ a = µ σ a gɛ abc σ b c µ This looks nice! Our gauge condition is the real part times the covariant derivative o the imaginary part. 6

7 We introduce a variable ω as ininitesimal Grassmann variable. It anticommutes with any other Grassmann variable and commutes with any scalar ield. Ininitesimal transormations are δµ a = ωd µ σ a δψ = igω ( T a σ a) ψ δρ a = i ξ ω µ a µ δσ a = gωɛ abc σb σ c U( θ) = exp i θ a (x) τa a θ a (x) = gωσ a (x) The last two lines are the ormal transormation laws o the ixed Lagrangian L. Let us consider the ull Lagrangian L + L FPGh + L g. Is it invariant? L is certainly invariant, L FPGh and L g are certainly not invariant. We claim, without proo (it should be in any textbook), that the sum L FPGh + L g is invariant. 4 Ward Identities I we take a look at the couplings, g g g g 7

8 there has got to be a Ward identity! Imagine the couplings were dierent or each vertex. g g g 3 g 4 Then there would be dierent g s in dierent monomials in the Lagrangian, like ψ /ψ and 4 F µν F µν. This would result in a total loss o gauge invariance, which would destroy all ormal proos. The Ward identity, which holds not only or the actors, but also or the complete Green s unctions, is = = = The last two terms can be interpreted graphically: = = 8