Advanced Quantum Gauge Field Theory. P. van Nieuwenhuizen

Transcription

1 Advanced Quantum Gauge Field Theory P. van Nieuwenhuizen January 13, 2016

2 Contents 2 BRST symmetry 3 1 Invariance of the quantum action for gauge fields Nilpotency and auxiliary field The BRST Jacobian Anti-BRST symmetry Nonrenormalizability of massive gauge theory BRST, Faddeev-Popov and string-like quantization Classical and quantum Yang-Mills theory from the Noether method Gauge invariance from tree unitarity Historical and other comments A Heat kernel regularization of the BRST Jacobian

3 Chapter 2 BRST symmetry In the previous chapter we discussed some aspects of the historical development of quantum gauge field theory. In this chapter we shall follow a more recent approach to determine the quantum action and its properties, namely by requiring that it have a certain symmetry. (By quantum action we mean the action which appears in the path integral, so usually the sum of the classical action, gauge fixing term and ghost action.) This symmetry is BRST symmetry, where the letters B,R,S and T stand for its inventors Becchi, Rouet, Stora and Tyutin [1]. It is a residual rigid symmetry with an anticommuting Lorentz-scalar parameter which remains after the classical gauge symmetry has been broken by adding a gauge fixing term and a corresponding ghost action. (By a rigid symmetry we mean a symmetry whose parameter is constant, i.e., spacetime independent.) Also supersymmetric theories have a constant anticommuting parameter, but in this case the parameter is a Lorentz-spinor. Nevertheless, there is a connection between BRST symmetry and supersymmetry as we show in section 9. In classical physics the gauge action has local symmetry, and as a consequence no propagator can be constructed. Following Fermi, Heisenberg and Pauli, one can add a gauge fixing term to the classical action in order to be able to construct a propagator, but then the local invariance is lost. It would be wonderful if one could maintain 3

4 4 2. BRST SYMMETRY the essence of classical gauge symmetry and at the same time construct propagators. That is precisely what BRST symmetry does. For gauge fields the BRST transformation rule is an infinitesimal gauge transformation, with the gauge parameter λ a replaced by c a Λ where c a (x) are the ghost fields and Λ is the constant anticommuting BRST parameter. Thus for the classical gauge fields, BRST transformations are a subclass of classical gauge transformations, and since they are a symmetry of the quantum action (if one introduces suitable BRST transformation rules for the ghosts and antighosts), one might call it residual quantum gauge symmetry. It is, however, a rigid symmetry. Many properties and concepts of classical gauge invariance can be lifted to the quantum level by using BRST symmetry. The importance of BRST symmetry is, of course, not that it gives one more derivation of the quantum action, but it allows to derive Ward identities for proper graphs (and connected graphs) which simplify the proofs of renormalizability (and unitarity) enormously. In the work of t Hooft and Veltman [2] these identities can already be found, but they were initially derived by a diagrammatic method. Later path integral derivations of these Ward identities were given, but some of these Ward identities contained nonlocal terms [3]. All Ward identities derived from BRST symmetry are local, and the BRST method applied to path integrals allows a far simpler derivation. The BRST method is a general method which can be applied to any gauge theory. BRST symmetry of a given model does not always imply that such a model is unitary or renormalizable. For example, models with higher covariant derivatives in the classical gauge action, or ordinary gauge actions but with an opposite overall sign are not unitary, but are BRST invariant. 1 However, such actions are pathological and one would not use them. One can also find models without such pathologies where BRST symmetry and unitarity do not hold simultaneously. We give an example of a 1 As we already mentioned and discuss in more detail in the next section, for classical fields the BRST transformations are equal to gauge transformations with parameter λ a = c a Λ where c a are the ghost fields and Λ is an anticommuting constant parameter. Hence classical gauge invariant actions are automatically BRST invariant.

5 5 model which is BRST invariant but not unitary [4] in section 5, and an example of a model which is unitary but not BRST invariant appears in [5]. These models are, however, peculiar exceptions, and in general unitarity and BRST symmetry imply each other. The relation between renormalizability and BRST symmetry is of a similar nature: in general they imply each other although there are exceptions. For example, adding a so-called Pauli coupling 1 g ψγ µ γ ν ψ( m µ A ν ν A µ ) to QED preserves BRST symmetry because it is gauge invariant, but destroys renormalizability. Also the gauge-invariant higher-derivative gauge theories we mentioned before are BRST invariant but not renormalizable. In both examples coupling constants with a negative dimension appear (g/m), which in general violates renormalizability (the prime example being gravitation). One can also construct examples of theories with dimensionless coupling constants which are BRST invariant but not renormalizable. For example, dropping the λϕ 4 coupling of a scalar field theory coupled to gauge fields, the quantum action remains BRST invariant but renormalizability is lost. 2 (The reason is that proper box diagrams with four external ϕ fields and two internal gauge fields lead to divergences proportional to ϕ 4.) In a path integral approach it is not sufficient that the quantum action be invariant under a symmetry transformation, also the measure should be invariant. If the measure is not BRST invariant, there could be BRST anomalies, and these lead in general to a violation of renormalizability and unitarity. In section 3 we shall therefore analyze whether the Jacobian for BRST transformations is unity. This is usually checked without regularization, but such an approach is ill-defined 3, and, following Fujikawa [6], we shall regularize the BRST Jacobian. We use heat kernel regular- 2 More precisely, multiplicative renormalizability is lost but the model without classical λϕ 4 term is still additively renormalizable. 3 For chiral symmetry of models with fermions the need for regularization is well appreciated. The Jacobian is in this case proportional to the trace of γ 5, and would vanish without regularization. With regularization one obtains a nonvanishing result proportional to ɛ µνρσ F µν F ρσ.

6 6 2. BRST SYMMETRY ization because it is particularly well suited for the regularization of Jacobians, but other schemes could equally well be used. The conclusion will be that pure Yang-Mills theory is free from BRST anomalies. For matter-coupled gauge theories, the BRST Jacobian is unity if and only if the quantum gauge field theory does not contain triangle anomalies in the chiral gauge symmetries. Having shown that there are no anomalies in one regularization scheme is sufficient to conclude that there are also no genuine anomalies in any other regularization scheme. It can be shown using the action principle that the results for the effective action obtained from different regularization schemes differ only by local finite counter terms ΔS [7]. Thus, if in one regularization scheme there are no anomalies, one can also in any other scheme make the theory anomaly free by adding a suitable local finite counter term to the effective action. Adding a finite local counter term to the quantum action does not violate additive renormalizability. However, it would be more satisfactory if there were a direct method to study anomalies in Ward identities without relying on even one particular regularization scheme. Such a method exists, and is called the cohomology approach [1]. As we already mentioned, the conclusion is as follows: when there are no chiral anomalies there are no BRST anomalies (so then the measure is BRST invariant), and when there are no BRST anomalies, there are no chiral anomalies. In section 4 we discuss anti-brst symmetry. It interchanges the role of the ghosts and antighosts. It plays an important role in string theory, but for field theory its relevance is limited because the only known power-counting renormalizable model which is both BRST and anti-brst invariant, is the Curci-Ferrari model, and, as we shall show, this model is renormalizable but not unitary. In section 5 we consider massive Yang-Mills theory, i.e. Yang-Mills theory with a mass term m 2 A 2 µ added by hand instead of generating a mass term by spontaneous symmetry breaking. This theory has no gauge invariance and is manifestly unitary,

7 7 but the question we study is whether it is renormalizable or not. This question is of more than academic interest because it could offer an alternative to the Higgs mechanism if it were renormalizable. prove, it is nonrenormalizable. Unfortunately (or fortunately), as we shall In section 6 we discuss alternatives to BRST quantization. In particular, we compare the Faddeev-Popov method based on choosing a gauge choice such as µ A a µ = 0 with the quantization methods developed in string theory where one makes an orthogonal decomposition of a gauge field A a µ into a pure gauge part D µ ω a and its physical part. The string quantization method is formulated in terms of an operator P, defined by P A a µ = D µ ω a, and its hermitian conjugate P, and requires a study of zero modes of P P and P P, but the final results for gauge theories are the same as obtained from Faddeev-Popov quantization. In the remainder of this chapter we discuss some aspects of classical gauge field theory, which are not directly relevant for BRST symmetry but which deal with the basic structure of gauge theories. We derive in section 7 the full nonlinear structure of classical and quantum Yang-Mills theory by using the Noether method. This method uses as input the free action and its symmetries, and derives order-by-order in the coupling constant g the nonlinear terms. Some people motivate their interest in gauge theory by pointing to its aesthetic beauty, but as Boltzmann has said, beauty is only good for tailors. 4 We show in section 8 that gauge invariance follows from unitarity, because the only theories for vector fields which are unitary are gauge theories. Section 9 contains some historical comments and elaborates on results obtained in this chapter. For a first reading only sections 1 and 2 are indispensable. 4 Eleganz sei die Sache der Schuster und Schneider (Elegance is for shoemakers and tailors), quoted by Albert Einstein in the introduction to his book Relativity, the Special and the General Theory (1916), reprinted by Bonanza Books in December 1961.

8 8 2. BRST SYMMETRY 1 Invariance of the quantum action for gauge fields Consider the Yang-Mills action for nonabelian gauge fields, for example the QCD action, with the usual relativistically invariant gauge fixing term and ghost action L = 1 4 (F µν a ) 2 1 2ξ ( µ A µ a ) 2 ( µ b a )(D µ c) a F µν a = µ A ν a ν A µ a + gf a bca µ b A ν c ; (D µ c) a = µ c a + gf a bca µ b c c (2.1.1) where ξ is a real parameter, called the gauge-fixing parameter. We use the Minkowski metric η µν = { 1, +1, +1, +1}, so µ A µ a = j ja j a 0 A 0 a with j = 1, 2, 3. Although in the developments of QED untill 1950 the Coulomb gauge was dominant, work after 1950 saw the ascent of the relativistically invariant Lorentz gauge, and nowadays this gauge is almost always used. The two unphysical polarizations of gauge bosons are cancelled by the two unphysical ghost and antighost particles. More precisely, the contributions of these fields cancel in the unitarity equations, as we shall discuss at length in the chapter on unitarity. The extra minus sign needed for this cancellation is obtained by requiring that b and c anticommute. More general gauge fixing terms than the one in (2.1.1) will be discussed later. The ghost c a and antighost b a in the action are Grassmann variables: b a c b = c b b a, c a c b = c b c a and b a b b = b b b a. At the operator level, there exist corresponding Heisenberg operators, whose anticommutators need not vanish, but in the action the ghost fields always satisfy a simple Grassmann algebra. Following Kugo and Ojima [8] we take b a imaginary and c a real, in order that L (ghost) is real (hermitian). Hermiticity of the quantum action allows a straightforward proof of unitarity as we shall see. 5 We could, of course, have taken both b a and c a real, but then we would have needed 5 Actually, hermiticity of the gauge fixing term and consequently of the corresponding quantum action allows a relatively simple proof of unitarity, but it is not strictly necessary as we show in the chapter on unitarity. Because the S-matrix is independent of the choice of gauge fixing term, one can choose complex gauge fixing terms and corresponding nonhermitian ghost actions, and still satisfy unitarity. For a suitable choice of complex gauge fixing terms one can greatly simplify the interactions in the quantum action.

9 1. INVARIANCE OF THE QUANTUM ACTION FOR GAUGE FIELDS 9 a factor i in front of the ghost action and we prefer not to have to deal with such factors of i. In some of the literature the notation c is used for the antighost, and sometimes c is viewed as the complex conjugate of c. Then the ghost action is not hermitian, and we prefer to use a different symbol (b a ) for the antighost to stress that the antighost is not the complex conjugate of the ghost. In the BRST approach it becomes particularly clear that the ghost c a has a definite reality (purely real or purely imaginary) because as we shall see c a Λ replaces the real gauge parameter λ a, where Λ is an anticommuting constant. Clearly both c a and Λ must have definite reality properties in order that c a Λ be real. We choose c a to be real and Λ to be purely imaginary; then c a Λ is real. Another property we shall need is the dimension of fields. Since the action is dimensionless in units with = c = 1, and coordinates have dimension 1, the gauge fields have dimension +1 in four dimensions and the sum of the dimensions of the ghost and antighost field is +2. Without loss of generality (because all terms in the quantum action conserve ghost number) we may take the dimension of ghosts and antighosts to be equal, hence +1. In the literature one sometimes chooses c a to have dimension zero and b a dimension 2, but we prefer to treat c a and b a on the same footing as ordinary scalar fields, and so assign dimension 1 to both. This is of course also the choice which is made in most textbooks. The classical action is, of course, invariant under infinitesimal gauge transformations 6 6 We recall that the gauge action can also be written in terms of Lie-algebra valued fields A µ A a µt a as S = 1 2 T rfµν F µν with the antihermitian generators for the fundamental representation T a normalized to T rt a T b = 1 2 δ ab. The gauge action is clearly invariant under the infinitesimal gauge transformation δ gauge F µν = [F µν, λ] where λ = gλ a T a. Since F µν = µ A ν ν A µ + g[a µ, A ν ], we can introduce the one-form A A µ dx µ and the two-form F = 1 2 F µνdx µ dx ν and find then F = da + gaa and δ gauge F = [F, λ]. Finite gauge transformations are given by F = U 1 F U where U = exp λ. It is then relatively easy to find the finite gauge transformation of A which leads to this expression for F, namely the covariant derivative transforms as a vector in the adjoint representation: (d + ga) = d + ga = U 1 (d + ga)u. Indeed, the curvature is the commutator of two covariant derivatives, F = 1 g (d + ga)(d + ga), and thus F = U 1 F U. For A a µ expansion

10 10 2. BRST SYMMETRY δ gauge A µ a = (D µ λ) a µ λ a + gf a bca µ b λ c (2.1.2) The basic idea of BRST symmetry is to replace the classical gauge parameter λ a by c a Λ where c a is the ghost field and Λ is a constant, anticommuting and imaginary parameter with ghost number 1 (ghosts having by definition ghost number +1) and dimension 1 (since λ a is dimensionless while ghosts have dimensions +1). Taking the ghost field to be real, it is clear that c a Λ is again real, just as λ a is real. Hence the combination c a Λ can be viewed as a particular choice of λ a, and thus the classical action is invariant under the following BRST transformations of the gauge fields δ B A µ a = (D µ c) a Λ (2.1.3) If there are scalars ϕ i transforming as δϕ i = g(t a ) i jϕ j λ a under classical gauge transformations (with [T a, T b ] = f ab c T c ), then the corresponding BRST transformations read δ B ϕ i = g(t a ) i jϕ j c a Λ. The classical gauge invariant action 1 2 (D µϕ i ) 2 with D µ ϕ i = µ ϕ i +g(t a ) i jϕ j A µ a is then also BRST invariant. Similarly, for fermions, BRST transformations are gauge transformations with λ a replaced by c a Λ. We shall now derive the remaining BRST transformation rules from the requirement that the quantum action be BRST invariant. This is thus a dynamical approach. Afterwards we shall check the nilpotency of these transformation rules. We could instead have started with a kinematical approach, namely by requiring that the BRST transformation rules be nilpotent, and then afterwards construct a quantum action that is invariant under these rules. The results of both approaches are the same. To make the sum of the gauge fixing term and the ghost action in (2.1.1) invariant, it is clearly sufficient that 1. the BRST variation of µ (D µ c) vanishes separately and to first order in λ yields the result (2.1.2). The finite gauge transformation of A a µ reads explicitly g(a a µ) T a = e λ ( µ e λ ) + e λ ga µ e λ.

11 1. INVARIANCE OF THE QUANTUM ACTION FOR GAUGE FIELDS the variation of the gauge fixing term be canceled by that variation of the ghost action which is due to a suitable variation of the antighost. L(gauge) 1 }{{} 2ξ ( µ A a µ) 2 + b a µ D µ c a }{{} invariant }{{} invariant invariant (2.1.4) In fact, this is the only way the quantum action can be BRST invariant once one has imposed (2.1.3), because variation of the term µ D µ c still leaves the field b in the variations, and µ A µ does not vary into b. So, the variation of b in L(ghost) must necessarily cancel against the variation of the gauge fixing term. Then one is left with the variation of the fields in µ D µ c. The BRST variation of the gauge fixing term is according to (2.1.3) given by δ B L(fix) = 1 ξ ( A a) µ (D µ c) a Λ (2.1.5) Variation of the antighost field by a transformation law δ B b a which we do not yet know, yields after partial integration (we discuss boundary terms later) δ B L(ghost) = (δ B b a ) µ (D µ c) a (2.1.6) Clearly, the sum of the variation of the gauge fixing term and the variation due to δ B b a in L (ghost) cancels for the following transformation law of the antighost (Moving Λ past µ (D µ c) a yields an extra minus sign.) δ B b a = 1 ξ ( µ A µ ) a Λ (2.1.7) All that is left, is to make sure that δ B µ (D µ c) a = 0. Let us first make the at first sight stronger requirement that δ B D µ c = 0. Since we already have fixed δ B A a µ, we must now see whether we can fix δ B c a such that δ B D µ c a = 0. Evaluating the BRST variation of D µ c is straightforward δ B (D µ c) a = δ B ( µ c a ) + δ B (gf a bca µ b c c ) = µ (δ B c a ) + gf a bc(d µ c) b Λc c + gf a bca µ b (δ B c c ) (2.1.8)

12 12 2. BRST SYMMETRY One can solve this equation in one step by rewriting it as D µ δ B c a D µ(gf a bcc b Λc c ) = 0 (2.1.9) We pulled the covariant derivative D µ in front of f a bc which is allowed because f a bc is an invariant tensor of the gauge group. 7 The solution is then (2.1.11). However, for readers who are not familiar with the concept of invariant tensors, we now give an explicit elementary derivation. Assuming no knowledge about δ B c a at all, it could in principle contain terms with A a µ and terms without A a µ. Let us first assume that δ B c a does not contain terms depending on A a µ. The terms in (2.1.8) without A b µ yield the condition µ (δ B c a ) gf a bc( µ c b )c c Λ = 0 (2.1.10) Since f a bc( µ c b )c c = µ ( 1 2 f a bcc b c c ) due to the antisymmetry of the structure constants, one finds a solution for δ B c a δ B c a = 1 2 gf a bcc b c c Λ (2.1.11) It remains to prove that the remaining terms in (2.1.8) (the terms depending on A µ a ) cancel also. As we already mentioned, it could have happened that one needs A µ c dependent terms in δ B c a (just like in δ B b a ), but the result for δ B c a in (2.1.11) is already the complete answer, as inspection of the A µ dependent variations in (2.1.8) 7 To prove that (2.1.8) and (2.1.9) are equivalent note that both ghosts contribute equally in (2.1.9). In (2.1.9) one finds a term 1 2 gf a pqa p µ(gf q bcc b Λc c ), while in (2.1.8) one finds gf a bc(gf b pqa p µc q )Λc c. The difference is proportional to ( 1 2 f a pqf q bc f a qcf q pb)c b Λc c. Writing the second term as two terms, each with a factor 1/2, by antisymmetrization in bc, one obtains 3 terms whose sum yields the Jacobi identity, and thus vanishes. The Jacobi identity reads f a pqf q bc + f a bqf q cp + f a cqf q pb = 0. It can be written as M a qf q bc f a bqm q c f a qcm q b = 0 with M a q = f a pq, and states that if one transforms all indices of f a bc with a matrix M in the adjoint representation, it is invariant: the structure constants are invariant tensors of the gauge group. For x-independent invariant tensors, the covariant derivative vanishes, D µ f a bc = 0, and this we used to obtain (2.1.9).

13 1. INVARIANCE OF THE QUANTUM ACTION FOR GAUGE FIELDS 13 shows ( ) gf a bc(gf b pqa p µ c q Λ)c c + gf a b 1 bca µ 2 gf c pqc p c q Λ = g (f 2 a bsf b pq 1 ) 2 f a pbf b qs A p µ c q c s Λ (2.1.12) We claim that these terms vanish as a consequence of the Jacobi identities for the structure constants. After using the antisymmetry in c q and c s to write the first term as two terms, each with a factor 1/2, one obtains in (2.1.12) a factor f a bsf b pq f a bqf b ps f a pbf b qs = f a bsf b pq + f a bqf b sp + f a bpf b qs = 0 (2.1.13) The last line is cyclic in the indices s, p, q and is the usual form of the Jacobi identities for the structure constants. The most general solution of (2.1.9) for δ B c a is a sum of a particular solution of the inhomogeneous solution (which we found) and the most general solution of the homogeneous equation D µ δ B c a = 0. Since there is no homogeneous solution which is a polynomial in fields and their derivatives, the solution for δ B c a is unique. Coming back to the requirement that only δ B µ (D µ c) a need vanish, instead of δ B (D µ c) a, we find from the A µ a -independent variations the condition µ µ δ B c a + µ (gf a bc µ c b Λc c ) = 0. Since the solution of the homogeneous equation µ µ δ B c a = 0 for general off-shell fields c a is only δ B c a = 0, 8 the solution is still only (2.1.11). Hence, the vanishing of D µ c a is equivalent to the vanishing of µ D µ c a. We conclude that the quantum action has a rigid symmetry (with constant parameter Λ) given by δa µ a = (D µ c) a Λ, δc a = 1 2 gf a bcc b c c Λ, δb a = 1 ξ µ A µa Λ (2.1.14) These rules preserve the reality properties of the fields provided one declares that Λ is purely imaginary. 8 The solution for δ B c a should be a polynomial in the fields, and after a Wick rotation the operator µ µ in Euclidean space has no eigenfunctions with vanishing eigenvalue. (The integral (δb c a ) µ µ (δ B c a ) = ( µ δ B c a ) 2 is negative definite, and only vanishes for δ B c a = 0.)

14 14 2. BRST SYMMETRY It is now clear that BRST symmetry is also present if one uses other gauge fixing terms. In general, δ B A µ a and δ B c c are the same, but if (using a notation in the next equation which will be explained in the next paragraph) L(fix) = 1 2 γ abf b F a, L(ghost) = b a δ B F a /Λ (2.1.15) where F a is any gauge fixing term and γ ab any field-independent matrix, then δ B b a = γ ab F b Λ (2.1.16) Indeed L(fix) + L(ghost) is still BRST invariant δ B ( 1 2 γ abf b F a )+(δ B b a )δ B F a /Λ = γ ab F b δ B F a +( γ ab F b Λ)δ B F a /Λ = 0 (2.1.17) The relativistic gauge fixing term corresponds to F a = µ A µ a and γ ab = 1 ξ δ ab. For field-dependent γ ab, see section 9, eq. (2.9.4). We shall, however, mostly use the most used gauge fixing term in (2.1.1). A few words about notation. We have explicitly written the parameter Λ in all variations. To discard Λ in δ B F a we use the notation δ B F a /Λ which indicates that one should move Λ to the far right and then discard it. One sometimes introduces a symbol s by δ B A = (sa)λ (another convention is δ B A = ΛsA) for any object A, so that in sa one omits the Λ (and calls the resulting rules antiderivations ), but then one should specify the rules how to take variations of products of fields. With Λ present, no such additional information is needed. For paedagogical reasons we shall use the perhaps more cumbersome notation with Λ present. Readers who are more used to the notation with s should have no problem converting our notation to theirs because they only must delete Λ at various places. In terms of s the BRST rules read sa µ a = (D µ c) a, sc a = 1 2 gf a bcc b c c, sb a = 1 ξ µ A a µ. (2.1.18) Similarly, (2.1.15) and (2.1.16) can be written in terms of s as follows L(fix) = 1 2 γ abf b F a, L(ghost) = b a sf a, sb a = γ ab F b (2.1.19)

15 1. INVARIANCE OF THE QUANTUM ACTION FOR GAUGE FIELDS 15 Before we proceed to study the BRST transformation laws further, we should point out that they are infinitesimal transformation rules. Ordinary symmetry transformation rules can be exponentiated to yield finite transformation rules, but this does not yield a more general result for the transformation rules in (2.1.14) because if Λ is in the exponent, expansion of the exponent would yield terms with Λ 2, Λ 3, etc. which all vanish. Note that the position of indices on b a, c a and F a is such that one does not need a metric to raise or lower group indices in δ B A µ a or δ B c a. Hence one can extend these BRST rules to the case of non-semisimple groups (for which the group metric g ab T rt a T b has no inverse by definition). However, in the gauge fixing terms in the action in (2.1.15) (and thus also in the ghost action) a symbol γ ab appears which plays the role of a metric. In practical applications, this γ ab is often the Killing metric, and can be taken to be equal to δ ab (see the next paragraph). In the next section we shall see that one can remove this γ ab also from δ B b a by introducing an auxiliary field. Then γ ab is absent from all transformation rules (i.e., γ ab is absent at the kinematical level). Thus γ ab is then entirely a dynamical object (an object appearing only in the action). In the classical action one should contract the group indices of the two Yang- Mills curvatures with a metric γ ab which is proportional to the Killing metric g ab = f ap q f bq p. Then the classical action is gauge invariant. 9 For SU(n), γ ab = δ ab if one normalizes the generators T a in the defining representation of SU(n) by Tr T a T b = 1 2 δ ab. This follows from group theory. 10 (For SU(2), this normalization corresponds 9 The gauge invariance of γ ab F a µνf b,µν follows from the fact that γ ab is an invariant tensor. Namely, transforming each index of γ ab with a matrix in the adjoint representation, the result vanishes as a consequence of the total antisymmetry of the structure constants: δγ ab = f ac a γ a b + f bc b γ ab = f acb + f bca = Note that for any group g ab = Tr T a (adj) T (adj) b because (T a (adj) ) c b = f c ba are the matrix elements = γ (R) of the generators in the adjoint representation. If for a representation R one has Tr T a (R) T (R) b ab, all these γ (R) ab are proportional to each other because there is only one symmetric invariant 2-index tensor with indices a and b, namely g ab.

16 16 2. BRST SYMMETRY to f a bc = ɛ abc and T a = i τ 2 a with τ a the Pauli matrices.) We have implicitly assumed this normalization in (2.1.1). Thus group indices are raised and lowered by the Kronecker delta δ ab and δ ab, respectively. Because the quantum action has a rigid BRST symmetry, there is a Noether current for BRST symmetry and a Noether charge Q. This BRST charge is nilpotent, Q 2 = 0, and plays a crucial role in string theory, and is also used in gauge theories to define physical states. In the next chapter we shall construct a differential operator which is also nilpotent and follows from the BRST symmetry of the effective action. It is sometimes called the Slavnov-Taylor operator and it plays a crucial role in the proof of renormalization of non-abelian gauge theories. We denote it by S. By using forms, the structure of the BRST transformations comes out more clearly. Define A = T a A a µdx µ and c = T a c a, then sa = Dc = dc + {A, c} and sc = cc, where d is the exterior derivative, d = dx µ / x µ, and we assumed that ghosts anticommute with dx µ. (This is natural if one views ghosts as one-forms, c a = c a b dϕb where ϕ b are the group coordinates. We discuss this further in the chapter on anomalies.) The antighost and the auxiliary field which we introduce in the next section form a contractible pair, by which in general one means a pair A and B such that sa = B and sb = 0. The geometric sector with A and c is completely decoupled from the sector with a contractible pair. Having derived the BRST symmetry of the quantum action for gauge theories, the reader may have noticed a tacit assumption which we made right at the beginning. Namely, we began by deriving the BRST symmetry of the quantum action by requiring that the classical action L(class) and the sum L(fix) + L(ghost) are each separately invariant. Is it possible to find other BRST-like transformations which leave the whole quantum action invariant, but not L(class) by itself? Indeed, this is possible, but only in very particular models, an example being ordinary (i.e., non-supersymmetric) 3-

17 2. NILPOTENCY AND AUXILIARY FIELD 17 dimensional Chern-Simons theory in the Landau gauge (but not in any other gauge). It has a symmetry which has been called vector supersymmetry, 11 a somewhat misleading term as it has nothing to do with supersymmetry (although also supersymmetric Chern-Simons theory in the Landau gauge has this symmetry). Another question one might already raise at this moment is the following: can one interchange the role of ghost and antighost in the BRST formalism, and begin with δa a = D µ (b a Λ) instead of δa a µ = D µ (c a Λ)? This is indeed possible as we shall discuss in section 4, and leads to another symmetry which has been called anti-brst symmetry. 2 Nilpotency and auxiliary field The BRST transformation laws of A a µ and c a are nilpotent. For A a µ one BRST variation yields (D µ c) a and we already showed that the BRST variation of (D µ c) a vanishes. Hence, the BRST transformations are nilpotent on A a µ. Note that we consider here the product of two transformations, not a commutator. However, since Λ 1 Λ 2 = Λ 2 Λ 1, the product δ(λ 1 )δ(λ 2 ) is equal to the commutator 1[δ(Λ 2 1), δ(λ 2 )]. On c a we find for the product of two BRST transformations δ(λ 1 )δ(λ 2 )c a = δ(λ 1 ) 1 2 gf a bcc b c c Λ 2 = gf a bc(δ(λ 1 )c b )c c Λ 2 ( ) 1 = gf a bc 2 gf b pqc p c q Λ 1 c c Λ 2 = 1 2 g2 (f a bcf b pq)c p c q c c Λ 2 Λ 1 (2.2.1) 11 The action and vector supersymmetry transformation rules are [9] L CS = 1 (F 4 ɛµνρ µνa a a ρ 1 ) 3 gf abca a µa b νa c ρ, δa a µ = ɛ µνρ ν b a ɛ ρ, δd a = ( ν b a )ɛ ν L fix + L ghost = d a µ A a µ ( µ b a )(D µ c a ), δc a = A a µɛ µ, δb a = 0 where ɛ µ is anticommuting and ɛ µνρ ɛ µστ = 2δ[σ ν δρ τ] in Minkowski space. One can also interchange the role of b a and c a to obtain anti-vector supersymmetry. The classical Chern-Simons action can be made supersymmetric by adding only a mass term λλ for Majorana fermions λ. (This is the only possibility on dimensional grounds.) The classical action has then both ordinary and vector supersymmetry. (As an aside we mention that at the quantum level it is better to use superfields because then the gauge-fixing term and the ghost action are also supersymmetric. However, this introduces more ordinary fields into the action.)

18 18 2. BRST SYMMETRY (We used here two different parameters Λ 1 and Λ 2 instead of only one Λ because Λ 2 would vanish. Using antiderivations (BRST variations sa µ = D µ c without any Λ) would avoid this complication.) In the first line we used that variation of c c gives the same result as variation of c b. Since c p c q c c is totally antisymmetric in p, q, c, the Jacobi identities can be used to show that this expression vanishes. Hence, BRST transformations are also nilpotent on the ghost fields. On the antighost one finds no nilpotency but, rather, as we shall show, the product of two BRST transformations of the antighost is proportional to its field equation. This situation is the same as in supersymmetry, where the commutator of two supersymmetry transformations of the fermion fields contains a term proportional to their field equation. The procedure to remove the field equation in the BRST algebra is also the same as in supersymmetry: one adds an auxiliary field to restore nilpotency of the BRST transformations (which corresponds to closure of the supersymmetry algebra). First, let us demonstrate that the product of two BRST transformations of the antighost does not vanish. Using (2.1.7) one finds δ(λ 1 )δ(λ 2 )b a = δ(λ 1 ) [ 1ξ ] ( µ A a µ )Λ 2 = 1 ξ µ (D µ c) a Λ 1 Λ 2 (2.2.2) This expression is not zero, but it is proportional to the antighost field equation δ δb a S(ghost) = µ (D µ c) a. (The letter S denotes actions and by δ δb a we mean the functional derivative with respect to the field b a (x).) We now observe that if one replaces the gauge fixing term in (2.1.1) by L(fix, aux) = 1 2 ξ(d a) 2 + d a ( A) a (2.2.3) where d a is a new, commuting, and auxiliary field, then the quantum action is again BRST invariant under δ B b a = d a Λ, δ B d a = 0 (2.2.4)

19 2. NILPOTENCY AND AUXILIARY FIELD 19 (By auxiliary field we mean that it has no kinetic term with a Dalembertian, and does not correspond to a physical degree of freedom.) Indeed, the term 1 2 ξ(da ) 2 is invariant by itself, while the variations d a δ B ( A) a + (δ B b a ) µ (D µ c) a (2.2.5) cancel each other as before. Eliminating d a by its algebraic field equation yields d a = 1 ξ Aa, and substituting the result back into δ B b a = d a Λ, one regains the previous result δ B b a = 1 ξ µ A a µ Λ. Similarly, substituting the d a field equation into the action, one recovers (2.1.1). The substitution of this field equation into an action does not mean that one is on-shell; rather, it amounts to a Gaussian integration over the nonpropagating field d a in the path integral and does not imply that any of the remaining fields are on-shell. In supersymmetry, auxiliary fields play an important role, because they close the supersymmetry algebra. Here the auxiliary field makes BRST transformations nilpotent, which one can write algebraically as Q 2 = 0. Thus also here the algebra closes. The analogy goes further because just as supersymmetry can be reformulated in superspace with superfields, one can also reformulate BRST symmetry in a similar superspace [10]. Using the new field d a to linearize the action in terms of the gauge fixing term F a is an example of first order formalism. Another example is writing 1 2 q2 as p q 1 2 p2 in classical mechanics. The formulation with d a present has the advantage that the BRST laws are also nilpotent on b a (and also on d a ). Nilpotency of the BRST laws is a useful property in the proofs of renormalizability and unitarity. In fact, we have arrived at transformation rules which are purely kinematical, in the sense that they no longer depend on the particular form of the gauge fixing term. (In the action there is still a dependence on a metric γ ab in the term 1 2 γba d a d b where γ ab is the inverse of γ ab. We recall that in (2.2.3) γ ab is equal to ξδ ab.)

20 20 2. BRST SYMMETRY Another advantage of the BRST formalism with auxiliary field is that if one partially integrates to write the gauge fixing term as L(fix, aux) = 1 2 ξ(d a) 2 ( µ d a )A µ a (2.2.6) then the Lagrangian density, not only the action, is BRST invariant. Without auxiliary field the same is true if one writes the ghost action as b a µ D µ c a, but this is less desirable for canonical quantization since the ghost field then carries a double time derivative. Thus, in the formulation with auxiliary field and using the gauge fixing term in (2.2.6) there are no boundary terms in the BRST variation of the quantum action. 12 We have derived the BRST laws from requiring that the action be invariant, and subsequently discovered that these laws are nilpotent. One can also start from the requirement of BRST nilpotency, and then try to construct invariant actions afterwards. We summarize the BRST transformation rules with auxiliary field δ B A µ a = (D µ c) a Λ, δ B c a = 1 2 gf a bcc b c c Λ, δ B b a = d a Λ, δ B d a = 0 (2.2.7) We can write the complete quantum action as follows L(qu) = L(cl) + δ B ( b a ( F a ξda )) /Λ = L(cl) + s(b a (F a ξda )) (2.2.8) where F a = A a. This is also the quantum action for more general gauge-fixing terms, see section 9. In this form the BRST invariance is manifest: the classical action is BRST invariant because it is gauge invariant whereas the gauge artefacts are BRST invariant because the BRST transformations are nilpotent. In some modern 12 More precisely, there are no boundary terms due to varying the gauge fixing term and the ghost action. There can be boundary terms from varying the classical action. For example in general relativity there are such boundary terms, but not in Yang-Mills theory.

21 3. THE BRST JACOBIAN 21 developments one views δ B as a kind of exterior derivative, satisfying δb 2 = 0, and then the classical action is closed but not exact, whereas the gauge artefacts are exact. The expression b a (F a ξda ) is called the gauge fermion. Denoting it by F, the most general form of the BRST-quantized Yang-Mills action is L(qu) = L cl + sf (2.2.9) 3 The BRST Jacobian From the path integral for gauge theories we shall obtain Ward identities by making a change of integration variables which corresponds to an infinitesimal BRST transformation. 13 We must then evaluate the Jacobian. For pure gauge theories the deviation of the Jacobian from unity is given by T r ( δ B A µ a (x)/ A ν b (y) δ B c a (x)/ c b (y) δ B b a (x)/ b b (y) ) (2.3.1) where the trace T r consists of a summation over the indices a = b and µ = ν and an integral over the spacetime points x = y. (For anticommuting fields one gets an extra minus sign because the Jacobian is in general the superdeterminant, which reduces for infinitesimal variations to unity plus a supertrace [11]. This minus sign has the same origin as the minus sign for a fermion loop in a Feynman graph.) Formally each term in the trace vanishes since the structure constants of semisimple Lie algebras are traceless. For example δ B A µ a (x)/ A ν b (y) = (D µ c a )(x)/ A ν b (y)λ = gf a bcc c (x)δ µ ν δ(x y)λ (2.3.2) and the trace over a, b yields f a ac = 0. (By δ(x y) we mean the 4-dimensional Dirac delta function δ 4 (x y).) Similarly the contribution of the ghost fields is proportional 13 The BRST transformation is infinitesimal in the sense that it is linear in Λ. However, one cannot claim that Λ is small because it is Grassmann parameter. Fortunately, the Jacobian can be defined without Λ, by just dropping Λ. No nontrivial definition of a finite BRST transformation is known.

22 22 2. BRST SYMMETRY to f a ac and vanishes. For the term with the antighosts there is even no b-dependent term in δ B b c (y), hence this contribution to the trace vanishes even more clearly. With auxiliary field, one should add a term δ B d a (x)/ d b (y) to (2.3.1), but since δ B d a = 0, also this term vanishes. Hence, the Jacobian seems unity. (For gravity one finds similar results. 14 ) However, this argument is incomplete since one should regulate the trace because it contains a sum over all spacetime points which itself is infinite and zero times infinity can be a finite but nonvanishing number. This is the origin of anomalies. For example, for chiral transformations δψ = iαγ 5 ψ, the Jacobian is proportional to T rγ 5 and formally this trace vanishes. However, after regularization with for example the regulator exp( /D /D/M 2 ) with /D = γ µ D µ, the trace no longer vanishes, as first shown by K. Fujikawa [6] and as we shall discuss in the chapter on anomalies. Likewise, regularization of the trace in (2.3.1) might yield a nontrivial result because off-diagonal terms in the Jacobian might combine with off-diagonal terms in the regulator to produce a nonvanishing trace. As with all situations involving regularization, there are two ways to proceed. One may pick a particular regularization scheme and explicitly compute the regularized Jacobian. Or one might first write down the most general expression for the Jacobian which any possible regularization scheme could ever produce, only restricted by consistency conditions which we discuss below. Then one should study whether or not this most general expression can always be canceled by adding a suitable local finite counter term to the effective action whose BRST variation equals minus the contribution from the regularized Jacobian. The latter approach is based on the 14 The vielbein field e m µ transforms classically as δe m µ = ξ ν δ ν e m µ + ( µ ξ ν )e m ν + λ m ne n µ. The parameter ξ µ for general coordinate transformations becomes ξ µ = c µ Λ and the parameter λ a b for local Lorentz transformations is replaced by λ a b = c a bλ. Nilpotency of the BRST transformations on e µ m determines the transfomation rule δc µ = c ν Λ ν c µ for the coordinate ghosts and δc m n = c m tλc t n + c ν Λ ν c m n for the local Lorentz ghosts. These transformation rules are again nilpotent. Furthermore, now the contributions to the Jacobian from the vielbein and ghosts do not cancel separately, but their sum still cancels naively (i.e., before regularization).

23 3. THE BRST JACOBIAN 23 mathematical theory of cohomology. We shall discuss this approach in the next chapter when we analyze the Zinn-Justin equation {Γ, Γ} = Δ. In this section we shall follow [12] and compute the BRST Jacobian with a particular regularization scheme, namely the heat kernel method. The calculations are straightforward if somewhat tedious, but since heat kernel methods are widely used, a complete calculation which uses them is of interest for its own sake. The impatient reader may skip all details and only read the last paragraph of this section. We shall see that the BRST Jacobian for pure Yang-Mills theory is nonvanishing, but it is the BRST variation of a local finite counter term. Hence, by starting with the effective action minus this counter term, the BRST anomalies are canceled, and BRST Ward identities (to be derived in the next chapter) hold without extra terms. Before performing the actual calculation, we should for completeness discuss what can be said about taking any other particular regularization scheme. It can be shown that the results for the Jacobian (and also for the effective action) obtained by using different regularization schemes are all equivalent modulo local finite counter terms in the action provided the anomalies satisfy certain consistency conditions. We call a regularization scheme that leads to such consistent anomalies a consistent regularization scheme. 15 An anomaly is the response of the effective action under a symmetry transformation of the quantum action. Hence, if δ λ S(qu) = 0 defines a symmetry transformation of the quantum action with parameter λ, then the oneloop anomaly An is given by An(λ) = δ λ Γ where Γ is the one-loop effective action. 16 When the symmetries of the quantum action form a closed algebra (when the commu- 15 In the chapter on anomalies we shall encounter a regulator which produces so-called covariant anomalies that do not satisfy the consistency conditions. Hence this regulator is not a consistent regulator in our terminology. 16 At higher loops, the effective action is invariant under BRST transformation laws which themselves receive quantum corrections. Namely, as we show in the next chapter, for 0, the BRST a invariance can be written as δˆγ/δa µ ˆΓ+ ˆΓ/ c a K a µ L a ˆΓ = 0 where ˆΓ is the effective action Γ minus the gauge fixing term S(fix), and ˆΓ is equal to the classical transformation law δ B A a µ = D µ c a Λ K µ a together with all one-particle irreducible diagrams with one vertex given by δ B A µ a. Similarly for δ B c a. We discuss this in the next chapter.

24 24 2. BRST SYMMETRY tator of two symmetries is again a symmetry), one has [δ λ1, δ λ2 ]Γ = δ λ1 λ 2 Γ, where the notation (λ 1 λ 2 ) a denotes gf abc λ b 1λ c 2. This yields the consistency condition δ λ1 An(λ 2 ) 1 2 = An(λ 1 λ 2 ). For BRST transformations the product of two BRST transformations, one with parameter Λ 1, and the other with parameter Λ 2, vanishes (if one wishes one can rewrite this product as a commutator because Λ 1 and Λ 2 anticommute). Thus δ(λ 1 )[δ(λ 2 )Γ] = 0. Then the consistency condition for BRST anomalies states that they are themselves BRST invariant. The crucial theorem is then: when two particular regularization schemes produce each a BRST anomaly which is BRST invariant, then the difference of these anomalies is equal to the BRST variation of a local counter term in the action. We shall not prove this theorem but refer to the literature [7]. We call a regularization scheme which produces anomalies which satisfy their consistency conditions a consistent regularization scheme. An anomaly which is BRST exact will be called a trivial anomaly. We can then conclude that if one consistent regularization scheme only produces a trivial anomaly, then any other consistent regularization scheme will also only produce trivial anomalies. So if one wants to prove that there are no anomalies, it is sufficient to check that one particular consistent regularization scheme only produces a trivial anomaly. We shall now take for this particular consistent regularization scheme heat kernel regularization. For simplicity we work without auxiliary fields. We regulate the Jacobian J for an infinitesimal BRST transformation with a regulator R whose matrix elements we compute by using heat kernel methods. Denoting the result for the regulated Jacobian by 1 + An (where An stands for anomaly) we must calculate An = where T r denotes the supertrace in (2.3.1). lim T rje R/M 2 (2.3.3) M 2 As regulator we choose the operator R i j = (T 1 ) ik S kj, where φ i T ij φ j may be any nonsingular mass matrix for the fields

25 3. THE BRST JACOBIAN 25 φ i = {b a, A µ a, c a } and S kj is the kinetic operator / ϕ k S(qu) / ϕ j. (The notation indicates that one should differentiate w.r.t. ϕ k from the left and w.r.t. ϕ j from the right. Note that for anticommuting fields left- and right-derivatives of the action differ by a sign.) There is a good reason for picking this regulator, and not, for example S kj alone, but we will not go into the reasons for this choice. 17 There are, of course, other regulators one might take, but as explained before, if there is no anomaly for one consistent regulator, there is none for any other consistent regulator. So it is sufficient to pick one regulator (and check that it is consistent: that the result for the anomaly is BRST closed). We choose a nondegenerate mass term which is invariant under rigid Yang-Mills transformations and which has a vanishing ghost number. There is a unique candidate, up to rescalings, namely T r(a µ A µ + 2bc)d 4 x. Writing the matrix entries in order of decreasing ghost number, namely as (b, A, c)t (b, A, c) T, we obtain the following 6 6 matrices T = 0 η µν 0 δ(x y) ; T 1 = η µν δ(x y) (2.3.4) The operator S kj follows by differentiating the quantum action [ 1 4 (F µν a (A)) ( Aa ) 2 ( µ b a )(D µ c a )]d 4 x (2.3.5) once from the left and once from the right. One obtains then 0 ν c ρ D ρ (A) S kl = c µ R µν [ µ b] δ(x y) (2.3.6) D ρ (A) ρ [ ν b] 0 where R µν (x)δ(x y) = A ν (y) Dρ F µ ρ (x) + x µ xδ(x ν y) = ] [2F µν (A) + η µν D ρ (A)D ρ (A) D µ (A)D ν (A) + µ ν δ(x y) (2.3.7) 17 We follow here the theory of [13] for consistent regulators. The regulators are obtained by comparison with Pauli-Villars regularization, where the Pauli-Villars fields χ j have kinetic terms χ k S kj χ j and mass terms M 2 χ k T kj χ j. One can show in general that these regulators yield consistent anomalies. We shall explicitly check that the result for the anomaly is consistent.

26 26 2. BRST SYMMETRY To obtain this result for R µν, we replaced D ν (A)D µ (A) by D µ (A)D ν (A)+F µν (A). Factors of 1 2 and 2 are easily checked by noting that S = 1 2 ϕk S kl ϕ l if S is quadratic in ϕ. The term µ ν in R µν is, of course, due to the gauge fixing term. The square brackets in [ µ b] indicate that this term contains no free derivatives: the derivative µ acts on b but not beyond b. Furthermore, all entries lie in the adjoint representation of the Lie algebra, for example c = c a b gf a cbc c. For readers who have difficulty with deriving (2.3.7) we recommend to begin with (2.3.5) with explicit indices a, b, and take the derivative w.r.t. A ν b (y), and then transcribe the result into the form (2.3.7). The Jacobian for the infinitesimal BRST transformation δb a = A a Λ, δa µ a = D µ c a Λ and δc a = 1 2 gf a bcc b c c Λ is obtained by right-differentiation [11] of these expressions with respect to φ j and reads J j(x)δ(x y) = where we recall that c = c a b = gf a cbc c. 0 ν 0 0 cδ ν µ D µ (A) 0 0 c (x) δ(x y)λ (2.3.8) Having obtained explicit expressions for T 1, S and J we can calculate the anomaly in (2.3.3). This calculation is given in the appendix to this chapter, and the result reads An = 1 (4π) T r( ν c)[4a µ A ν A µ 4A µ ( µ A ν ν A µ ) 4A ν µ A µ + µ µ A ν 3 ν µ A µ ]d 4 x Λ (2.3.9) In the appendix we show that it is BRST closed, i.e. it is BRST invariant, which is the consistency condition. So, this is a consistent candidate anomaly. The claim that there is no genuine BRST anomaly in pure Yang-Mills theory boils now down to the statement that the expression in (2.3.9) for the anomaly is also BRST exact, i.e., of the form An = δ B ΔS. It indeed is of this form, with [12] ΔS = 1 [ 1 (4π) 2 12 T r ( A) A µa ν A µ A ν 1 ] 2 (A2 )(A 2 ) d 4 x (2.3.10)