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Feynman Rules for Non-Abelian Theories

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Feynman Rules for Non-Abelian Theories PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 41 Wednesday, April 1, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topics: Feynman Rules for Non-Abelian Theories. Weinberg-Witten Theorem. The Faddeev-Popov Lagrangian. At this point we know how to construct Lagrangians with non-Abelian gauge symmetry. Our goal is to relate the ides of non-Abelian gauge invariance to the real interactions of particle physics. We need to work out the rules for computing Feynman diagrams containing non-Abelian gauge vector particles to compute scattering amplitudes and cross sections. Feynman Rules for Non-Abelian Theories Most of the Feynman rules for non-abelian gauge theory can be read directly from the Yang-Mills Lagrangian. The Yang-Mills Lagrangian, we encountered in the previous lecture is 1 L = − (F a )2 + (iD= − m) ; (1) 4 µν where the index a is summed over the generators of the gauge group G, and the fermion multiplet belongs to an irreducible representation r of G. The field strength is a a a abc b c Fµν = @µAν − @νAµ + gf AµAν; (2) where f abc are structure constants of G. The covariant derivative is defined in terms of the repre- a sentation matrices tr by a a Dµ = @µ − igAµtr : (3) (From now on we will drop the subscript r.) The Feynman rules for this Lagrangian can be derived from a functional integral over the fields a , and Aµ. We can imagine expanding the functional integral in perturbation theory, starting PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 with the free Lagrangian, at g = 0. The free theory contains a number of free fermions equal to the dimension d(r) of the representation r, and a number of free vector bosons equal to the number d(G) of generators G. Using the methods we learned in functional quantization, it is straightforward to derive the fermion propagator Z 4 d k i −ik·(x−y) iα(x) jβ(y) = 4 δije ; (4) (2π) k= − m αβ where α, β are Dirac indices and i, j are indices of the symmetry group: i; j = 1; ··· ; d(r). In analogy with electrodynamics, we would guess that the propagator of the vector fields is D E Z d4k −ig Aa (x)Ab (y) = µν δabe−ik·(x−y); (5) µ ν (2π)4 k2 with a; b = 1; ··· ; d(G). To find the vertices, we write out the nonlinear terms in (1). If L0 is the free field Lagrangian, then a λ a abc a κb λc 2 eab a b ecd κc λd L = L0 + gAλ γ t − gf (@κAλ)A A − g (f AκAλ)(f A A ): (6) The first of the three nonlinear terms gives the fermion-gauge boson vertex igγµta; (7) this is a matrix that acts on the Dirac and gauge indices of the fermions. The second nonlinear term leads to a three gauge boson vertex. To work out this vertex, we first choose a definite convention for the external momenta and Lorentz and gauge indices. A suitable convention is shown in Fig. 1, with all momenta pointing inward. This contribution has the form −igf abc(−ikν)gµρ: (8) In all, there are 3! possible contractions, which alternate in sign according to the total antisym- metry of f abc. The last term in the interaction Lagrangian leads to a four gauge boson vertex. Following the conventions given in Fig. 1, one possible contraction gives to the contribution −ig2f eabf ecdgµρgνσ: (9) There are 4! possible contractions, of which sets of 4 are equal to one another. The sum of these contributions is shown in Fig. 1. Non-Abelian gauge theories should also satisfy Ward identities similar to those of QED. Although we do not prove this, we state that like the photon, the non-Abelian gauge boson has only two physical polarization states. We also note that the coupling constants of all three nonlinear terms in the Yang-Mills Lagrangian must be equal in order to preserve the Ward identity and avoid 2 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Figure 1: Feynman rules for fermion and gauge boson vertices of a non-Abelian gauge theory. the production of bosons with unphysical polarization states. Conversely, the non-Abelian gauge symmetry guarantees that these couplings are equal. Weinberg-Witten Theorem We can write down a locally SU(N) invariant Lagrangian N 1 X a L = − (F a )2 + (δ i@= + gA= T a − mδ ) : (10) 4 µν i ij ij ij j i;j=1 If we expand this Lagrangian we find 1 L = − (@ Aa − @ Aa + gf abcAb Ac )2 + (iδ γµ@ + gγµAa T a − mδ ) : (11) 4 µ ν ν µ µ ν i ij µ µ ij ij j The equation of motion are a abc b c a @µFµν + gf AµFµν = −g iγνTij j; (12) for gauge fields and a a (i@= − m) i = −gA= Tij j (13) for spinors. Because the Lagrangian has a gauge symmetry, it has a global symmetry, under which a a i ! i + iα Tij j (14) 3 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 and a a abc b c Aµ ! Aµ − f α Aµ (15) a a −1 a abc b c for infinitesimal α. (Note that, for local symmetry, we have Aµ ! Aµ + g @µα − f α Aµ.) Noether’s theorem tells us that a global symmetry implies a conserved current given by X @L δφn J = : (16) µ @(@ φ ) δα n µ n In the non-Abelian case, there will be N 2 − 1 currents, one for each symmetry direction αa. a Summing over both matter fields φn = i and gauge fields φn = Aµ gives a µ a abc b c Jµ = − iγ Tij j + f AνFµν: (17) a It is not hard to check that the current is conserved on the equation of motion, @µJµ = 0. In contrast to the QED current, the Noether current associated with a global non-Abelian symmetry in a theory with gauge bosons is not gauge invariant (or even gauge covariant). Thus, it is not physical and there is not a well defined charge that one can measure. Although it is true that the charges Z a 3 a Q = d xJ0 (18) a are conserved. That is, @tQ = 0. these charges depend on our choice of gauge. Thus, in a non-Abelian gauge theory such as QCD there is no such thing as a classical current, like a wire with quarks in it instead of electrons. There is no simple analog of Gauss’ law either; the gauge fields are bound up with the matter fields in an intricate and nonlinear way. One can define a matter current constructed only out of fermions as a µ a jµ = − iγ Tij j; (19) which is gauge covariant. However, this current satisfies a Dµjµ = 0; (20) a a abc b c where Dµjν = @µjν + gf Aµjν is the covariant derivative in the adjoint representation. a Thus the matter current is not conserved, @µjµ 6= 0, and there is no associated conserved charge. There is a general theorem known as the Weinberg-Witten theorem; it tells us that a theory with a global non-Abelian symmetry under which massless spin-1 particles are charged does not admit a gauge-invariant conserved current. Another way to phrase the theorem without reference to gauge invariance is that there cannot be a conserved Lorentz-invariant current in a theory with massless spin-1 particles with non-vanishing values of the charge associated with that current. A similar theorem holds for spin-2 particles as well. The Weinberg-Witten theorem for spin 2 4 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 states that a theory with a conserved and Lorentz-covariant energy momentum tensor can never have a massless particle of spin 2. (Note: String theory and AdS/CFT correspondence get around this by having gravity emerge in a different spacetime.) The Faddeev-Popov Lagrangian Let us derive the gluon propagator. The expression for the gluon propagator we encountered in Eq. (5) is correct. But it is incomplete: It must be supplemented by additional rules of a completely new type. To define the functional integral for a theory with non-Abelian gauge invariance, we will use the Faddeev-Popov method, as introduced earlier to quantize electromagnetic field. Consider the quantization of the pure gauge theory, without fermions. To derive the Feynman rules, we must define the functional integral Z Z 1 DA exp i d4x − (F a )2 : (21) 4 µν As in the Abelian case, the Lagrangian is unchanged along the infinite number of directions in the space of field configurations corresponding to local gauge transformations. To compute the functional integral we must factor out the integrations along these directions, constraining the remaining to a much smaller space. As in electrodynamics, we will constrain the gauge directions by applying a gauge-fixing condi- tion G(A) = 0 at each point x. Following Faddeev and Popov, we can introduce this constraint by inserting into the functional integral the identity Z δG(Aα) 1 = Dα(x)δ(G(Aα)) det : (22) δα Here Aα is the gauge field A transformed through a finite gauge transformation as in i Aa (x)ta ! V (x) Aa (x)ta + @ V y(x); µ µ g µ giving a a i c c (Aα)a ta = eiα t Ab tb + @ e−iα t : (23) µ µ g µ In evaluating the determinant, the infinitesimal form of this transformation will be more useful 1 (Aα)a = Aa + @ αa + f abcAb αc µ µ g µ µ 1 = Aa + D αa; (24) µ g µ 5 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where Dµ is the covariant derivative acting on a field in the adjoint representation.
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