PHY646 - and the

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 . Weinberg-Witten Theorem. The Faddeev-Popov Lagrangian.

At this we know how to construct Lagrangians with non-Abelian gauge . Our goal is to relate the ides of non-Abelian gauge invariance to the real of particle . 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 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 + ψ(iD/ − m)ψ, (1) 4 µν where the index a is summed over the generators of the gauge G, and the ψ 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 of G. The covariant 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 ψ ψ, and Aµ. We can imagine expanding the functional integral in , starting PHY646 - and the Standard Model Even Term 2020 with the free Lagrangian, at g = 0. The free theory contains a number of free equal to the d(r) of the representation r, and a number of free vector equal to the number d(G) of generators G. Using the methods we learned in functional , it is straightforward to derive the fermion

Z 4   d k i −ik·(x−y) ψiα(x)ψjβ(y) = 4 δije , (4) (2π) k/ − m αβ where α, β are Dirac indices and i, j are indices of the : 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 + gAλψγ t ψ − gf (∂κAλ)A A − g (f AκAλ)(f A A ). (6)

The first of the three nonlinear terms gives the fermion-gauge vertex

igγµta; (7) this is a that acts on the Dirac and gauge indices of the fermions. The second nonlinear term leads to a three 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 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 , 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

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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) Lagrangian

N 1 X a L = − (F a )2 + ψ (δ i∂/ + gA/ T a − mδ )ψ . (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 + ψ (iδ γµ∂ + gγµAa T a − mδ )ψ . (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 = −gψiγνTijψj, (12) for gauge fields and a a (i∂/ − m)ψi = −gA/ Tijψj (13) for . Because the Lagrangian has a gauge symmetry, it has a global symmetry, under which

a a ψi → ψi + iα Tijψj (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 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 fields φn = ψi and gauge fields φn = Aµ gives

a µ a abc b c Jµ = −ψiγ Tijψj + 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 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 in it instead of . 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 = −ψiγ Tijψj, (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 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 momentum can never have a massless particle of spin 2. (Note: and AdS/CFT correspondence get around this by having emerge in a different .)

The Faddeev-Popov Lagrangian

Let us derive the 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 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 †(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

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where Dµ is the covariant derivative acting on a field in the adjoint representation. Note that, as long as the gauge-fixing function G(A) is linear, the functional derivative δG(Aα)/δα is independent of α. Since the Lagrangian is gauge invariant, we can replace A by Aα in the exponential of Eq. (21). Then, as in the Abelian case, we can interchange the order of the functional integrals over A and α, and then change variables in the inner integral from A to A0 = Aα. The transformation (23) looks more complicated than in the Abelian case, but it is nothing more than a linear shift of the a a Aµ, followed by a unitary rotation of the various components of the symmetry multiplet Aµ at each point. Both of these operations preserve the measure

Y Y a DA = dAµ. (25) x a,µ

Thus DA = DA0, under the integral over α. Just as in the Abelian case, the integral over gauge motions α can be factored out of the functional integral into an overall normalization, leaving us with Z Z  Z δG(Aα) DAeiS[A] = Dα DAeiS[A]δ(G(A)) det . (26) δα This normalization factor cancels in the computation of correlation functions of gauge-invariant operators. From this point, the derivation of the gauge boson propagator proceeds as for the photon prop- agator. We choose the generalized

µ a a G(A) = ∂ Aµ(x) − ω (x), (27) with a Gaussian weight for ωa as for the case of electrodynamics. Following the steps similar to the ones we took for the case of electrodynamics then lead to the class of gauge field

D E Z d4k −i  k k  Aa (x)Ab (y) = g − (1 − ξ) µ ν δabe−ik·(x−y), (28) µ ν (2π)4 k2 + i µν k2 with a freely adjustable gauge parameter ξ. What we wrote down in Eq. (5) corresponds to the choice ξ = 1, called the Feynman-t’Hooft gauge. However, there is a subtlety. The functional determinant, evaluated using the infinitesimal form (24) of the gauge transformation, δG(Aα) 1 = ∂µD , (29) δα g µ depends on A. The functional determinant (29) thus contributes new terms to the Lagrangian. Faddeev and Popov chose to represent this determinant as a functional integral over a new set of anticommuting fields belonging to the adjoint representation

1  Z  Z  det ∂µD = DcDc¯exp i d4xc¯(−∂µD )c . (30) g µ µ

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This formal identity can be derived using our rules for fermionic functional integrals. The factor 1/g is absorbed into the normalization of the fields c and c¯. But to give the correct identity, c and c¯ must be anticommuting fields that are scalars under Lorentz transformations. The quantum excitations of these fields have the wrong relation between spin and to be physical particles. However, we can nevertheless treat these excitations as additional particles in the computation of Feynman diagrams. These new fields and their particle excitations are called Faddeev-Popov ghosts. Here, c and c¯ are ghosts and anti-ghosts, respectively. We write the Lagrangian more explicitly as

a 2 ac µ abc b c Lghost =c ¯ (−∂ δ − g∂ f Aµ)c . (31)

The first term gives the ghost propagator

Z d4k i hca(x)¯cb(y)i = δabe−ik·(x−y). (32) (2π)4 k2

In a diagram, this propagator carries an arrow that shows the flow of ghost number. In the interaction term, the derivative stands to the left of the gauge field; this implies that this derivative is evaluated with the momentum coming out of the vertex along the ghost line. The explicit Feynman rules are shown in Fig. 2.

Figure 2: Feynman rules for Faddeev-Popov ghosts.

There are no further subtleties in the construction of the perturbation theory for non-Abelian gauge theories. The final Lagrangian, including all of the effects of Faddeev-Popov gauge fixing, is

1 1 L = − (F a )2 − (∂µAa )2 + ψ¯(iD/ − m)ψ +c ¯a(−∂µDac)cc. (33) 4 µν 2ξ µ µ

The Faddeev-Popov gauge fixing, like for the case of electrodynamics, gives the correct gauge- invariant expressions for S-matrix elements in the non-Abelian case as well.

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Remarks on One-Loop Divergences and

Let us make a few remarks about the one-loop divergences and beta function in non-Abelian gauge theories. A more detailed study of these topics is reserved for QFT III (PHY658). As always, in quantum field theory, some of the loop diagrams of non-Abelian gauge theories will diverge. We must take care to teat the divergent integrals correctly. Non-Abelian gauge symmetries will severely restrict the number of divergent diagrams.

Once we complete the one-loop analysis, we get three counterterms: δ1, δ2 and δ3; and they represent the gauge boson self-energy, fermion self-energy, and the fermion-gauge boson vertex, respectively. All three counterterms contribute in non-Abelian gauge theory (unlike QED, where the first two terms cancel by the Ward identity). We know that the beta function gives the rate at which the renormalized changes as the scale µ is increased. To the lowest order we have

∂ 1 β(g) = gµ (−δ + δ + δ ). (34) ∂µ 1 2 2 3

Once we plug in the expressions for these three counterterms, the beta function takes the form

g3 11 4  β(g) = − C (G) − n C(r) , (35) (4π)2 3 2 3 f with nf denoting the number of fermion species in the representation r.

At least for small values of nf , the beta function is negative, and so non-Abelian gauge theories are asymptotically free. In an SU(N) gauge theory with fermions in the fundamental representation, we have

g3 11 2  β(g) = − N − n . (36) (4π)2 3 3 f

Asymptotic freedom is a special property of non-Abelian gauge theories. Among renormaliz- able quantum field theories in four spacetime , only the non-Abelian gauge theories are asymptotically free. (S. Coleman and D. J. Gross, Phys. Rev. Lett. 31, 851 (1973)).

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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