Advanced Quantum Gauge Field Theory. P. van Nieuwenhuizen

Transcription

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

2 Contents 3 Renormalization of unbroken gauge theories 3 1 Ward identities for divergences in proper graphs Multiplicative renormalizability of QCD Multiplicative renormalizability of quarks and gluons On-shell renormalization in QED Nonlinear gauges Noncovariant algebraic gauges Asymptotic freedom in the Coulomb gauge One-loop Z-factors in QCD One-loop beta function and running masses The two-loop β function A Proof that Γ = Γ ren even with external sources B Functional methods for external sources C Details of the renormalization of the Dirac-Yang-Mills system

3 Chapter 3 Renormalization of unbroken gauge theories Renormalization remains, with unitarity, one of the central issues of quantum field theory. In recent times the center of study has shifted to effective actions with composite operators which are not renormalizable, but to understand the latter, one still needs to understand renormalization theory. We study in this chapter multiplicative perturbative renormalization of unbroken nonabelian gauge theories such as QCD, i.e., gauge theories without or with matter but without spontaneous symmetry breaking. By perturbative renormalization we mean that we consider unrenormalized proper Green functions (one-particle irreducible Feynman graphs without counter terms) and construct corresponding finite renormalized proper Green functions loop-by-loop (one-particle irreducible graphs, some of which have counter terms as vertices). We shall follow induction; assuming that all (n 1)-loop proper graphs have been made finite, we shall first determine all divergences in proper graphs with n loops. These n- loop divergences are spacetime integrals of local polynomials in fields and derivatives of fields, as we prove in the chapter on unitarity. By multiplicative renormalization we mean that these n-loop divergences can be absorbed by rescaling the (n 1)-loop renormalized fields and parameters (masses, coupling constants and the gauge param- 3

4 4 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES eter ξ) such that the proper graphs computed in terms of these rescaled (n 1)-loop renormalized quantities become also finite at the n-loop level. The rescaled (n 1)- loop renormalized quantities are then the n-loop renormalized quantities, see (3.2.29) for an example. A more general approach than multiplicative renormalization is additive renormalization, usually called algebraic renormalization [1]. For theories with chiral fermions or γ 5 -matrices, such as the electroweak sector of the Standard Model or supersymmetric models, one cannot use multiplicative renormalization, but one must instead use algebraic renormalization. In this chapter we discuss multiplicative renormalization; this is sufficient for QCD and QED. We shall not specify how one regulates loop corrections. We only require that the regularized Green functions satisfy the BRST Ward identities. As we shall derive, the divergent terms in the effective action satisfy Ward identities of the form SΓ div = 0 where S is the Slavnov-Taylor operator, which satisfies S 2 = 0, 1 and Γ is the effective action. If there are terms by which the Ward identities are broken, SΓ = An, these An are finite (i.e., nondivergent) terms, which one might call candidate anomalies and which must satisfy the consistency condition that they are annihilated by the Slavnov-Taylor operator S. One must then solve the equation SAn = 0, and renormalizability requires that any such An is BRST exact: An = SX. The An are spacetime integrals over polynomials in the fields and derivatives, with the same dimension and ghost number as SΓ, namely the An have dimension 5 and ghost number +1. If there are polynomials An which are BRST closed (SAn = 0) but which are not BRST exact (meaning that An cannot be written as SX; such An are called nontrivial cohomology in mathematics) there are genuine anomalies in the theory and these prevent renormalization of the theory. If, on the other hand, the candidate anomalies are BRST exact, one can 1 As we shall show, this operator is nilpotent as a consequence of the BRST symmetry of the quantum action, and is itself sometimes called the BRST charge, although strictly speaking it is not the BRST charge, but rather a consequence of BRST symmetry. We shall follow this usage and call expressions X which are annihilated by S BRST-closed, instead of Slavnov-Taylor closed.

5 5 remove them by adding X to the action as a counter term, and in this case the candidate anomalies are not genuine anomalies. Of course, it is desirable to know beforehand which theories have genuine BRST anomalies, and in which theories one can always remove the candidate BRST anomalies by counter terms. It can be shown that there are BRST anomalies if and only if there are chiral anomalies. So from now on we consider only gauge theories without chiral anomalies. 2 The proof that nonabelian gauge theories are renormalizable is due to t Hooft and Veltman [3] who received the Nobel prize for their work in Their work is based on a careful study of properties of Feynman graphs, in particular relations (Ward identities) between different Feynman graphs. These diagrammatical methods [4] are completely equivalent to functional methods developed somewhat later by J. Zinn-Justin [5, 6] and B.W. Lee [6] and others. We shall follow here the approach which uses functional methods and base our entire discussion on BRST symmetry. The physical content is the same, but functional methods allow one to summarize properties of sets of Feynman graphs in a very simple and general manner, and BRST methods [5] have the advantage that they eliminate the nonlocal expressions in the Ward identities [6]. We shall not be totally one-sided, though, because we shall use Feynman graphs when they clarify formal issues. Functional methods for generating functionals of Feynman graphs with external sources were proposed by Schwinger in [7]. Further references to original articles on functional methods can be found in chapter 6 of [8]. Renormalization of gauge theories differs from renormalization of generic spin 0 or spin 1/2 field theories in the following way. Certain Green functions must have Z factors which are related to the Z factors of other Green functions if multiplicative renormalization is to hold. For example, one can introduce separate Z factors for the proper 2, 3 and 4 point functions for gauge fields, but then these Z factors are not 2 If there are no chiral anomalies at the one-loop level, there are also no chiral anomalies at the higher-loop level [2]. We discuss chiral anomalies in a separate chapter.

6 6 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES independent, but satisfy one relation. These relations follow from the BRST symmetry of the quantum action. We shall use this symmetry to derive Ward identities for the effective action Γ. (The effective action is the sum of all proper Green functions.) Because the transformation laws of the fields under BRST symmetry are nonlinear in fields (for example, δ B A µ = + ga µ c) these Ward identities are quadratic in Γ. We shall refer to these relations as the ΓΓ equations. For field theories with linear transformation rules, the Ward identities are linear in Γ. As we already discussed, we shall assume that all formal manipulations with path integrals remain valid when the theory is properly regularized. In particular, we assume that the ΓΓ equations are satisfied, so we assume that there are no anomalies. We shall not specify in this chapter which regularization scheme we use; all we need to know is that the divergences at the n-loop level are spacetime integrals of local polynomials in the fields and derivatives thereof, which satisfy the Ward identities. A proof that these divergences (and hence also the counter terms which remove them) are local is given in the chapter on unitarity. The divergences of Green s functions with generic off-shell momenta consist only of ultraviolet divergences. In dimensional regularization, they show up as poles at n 4, and we must remove them by renormalization. If one uses a gauge where the renormalized parameter ξ ren is not equal to unity, there are k µ k ν /k 2 terms in the propagator of massless vector bosons, but also these do not lead to infrared divergences in Green s functions with generic off-shell momenta. For the computation of cross sections the situation is different. In QED, there are infrared divergences from soft (i.e., with small momenta) emitted photons (called Bremsstrahlung) and also infrared divergences from soft virtual photons in loops. Both kinds of infrared divergences show up as poles at n 4 in dimensional regularization. In the cross sections of QED these infrared poles cancel by themselves, without having to invoke renormalization. In QCD the situation is more complicated

7 7 [9]. Having made the proper graphs finite order-by-order in the number of loops, the connected graphs can be made finite as well in the following way. Consider a general connected graph, and draw blobs around all proper 2-point, 3-point or 4- point subgraphs. These are the only potentially-divergent proper graphs as we shall prove using power counting. Make these blobs as large as possible. We shall call the resulting subgraphs maximal potentially-divergent proper subgraphs. (By potentially divergent we mean divergent as far as power counting is concerned.) For example, in a 2-loop selfenergy graph with overlapping divergences, there are two ways to isolate a proper 3-point graph, but both are part of a larger proper graph (the selfenergy itself). Figure III.1. An example of extending potentially-divergent proper graphs to maximal potentiallydivergent proper graphs. The result of this procedure of identifying a set of maximal potentially-divergent proper subgraphs contained in a given connected graph is actually unique. For example in the following graph, one can draw blobs around the vertex and propagator corrections in various ways, but there is a unique way such that the proper potentiallydivergent subgraphs are maximal

8 8 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES Figure III.2. Maximal potentially-divergent proper subgraphs do not intersect. Note that the blob on top in the second figure is not proper, but the extension in the third graph is proper. In this example one sees that the maximal potentially-divergent proper subgraphs do not intersect. The claim is that for any graph the blobs around maximal potentiallydivergent proper subgraphs are unique and do not intersect. This means that one can make proper subgraphs finite without having to worry about overlapping divergences or about the order in which one makes them finite. To prove in general that blobs around the maximal potentially-divergent proper subgraphs do not intersect, assume the contrary. 3 Then there are at least two blobs which are overlapping but neither one is entirely contained in the other. Each is either a 2-point, or a 3-point, or a 4-point function. Furthermore each is maximal: one cannot add a further part of the original graph to a blob such that the result is again proper and has again 2, 3 or 4 external legs. Draw a blob around the vertices in the intersection, and two other blobs around the remaining parts of the original blobs. Let the blob around the intersection have p external lines (p = 0, 1, 2, ). Then the intersection-blob must be connected by at least two lines to each of the remaining blobs (since each of the original blobs was proper) and each of the two remaining blobs can have at most 2 p external lines (because the intersection blob and one of the remaining blobs form together one of the original blobs which have at most four external lines). 3 I thank G. t Hooft for this proof.

9 9 = n k l m p k 2 and k + p + m 4 imply p + m 2 l 2 and n + p + l 4 imply p + n 2 } m + n + 2p 4 hence m + n + p 4 (3.1) Figure III.3. Maximal potentially-divergent proper subgraphs are unique. In this graph, m+n+p 4 and, as explained in the text, this implies that two maximal potentially-divergent proper graphs do not intersect. The dots??? vertices, but we have not drawn propagators between the vertices to keep the notation simple. But then the union of the two blobs would be again a blob (since it has at most m + n + p 4 external lines). 4 This contradicts the assumption that the original blobs were maximal. If also five-point functions would be divergent by power counting, one would run into trouble. Suppose five-point functions would be divergent but six-point functions not. One would like also in this case to draw blobs around proper subgraphs with five external lines, but now the identification of proper subgraphs inside a connected 4 We also use that the union of two proper overlapping blobs is again proper, and leave proof to the reader.

10 10 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES graph is not unique as the following example shows Figure III.4. Example of a proper graph which contains two overlapping 5-point graphs, but the graph itself has more than 5 external legs (namely 6). Drawing a circle around the two blobs on the left or the two blobs on the right identifies two proper subgraphs with five external lines, but now the whole graph is not a blob by itself, having six external lines. Hence, admitting proper graphs with five external lines, the procedure of identifying maximal potentially-divergent proper subgraphs becomes ambiguous. Fortunately, five-point proper graphs are not divergent in 4 dimensions, as we shall prove by power counting, so we do not need to identify proper subgraphs with five external lines. The topology of proper graphs fits beautifully with the program of renormalization in 4 dimensions. Having drawn blobs around the maximal proper subgraphs which are potentially divergent, these proper graphs are made finite by the renormalization procedure which is discussed in this chapter. Consider then the set of graphs obtained from the original graph by replacing the subgraphs inside blobs by the sum of subgraphs (including counter terms) which make the blobs finite. All other proper subgraphs are finite by power counting. We can then apply a theorem by Weinberg which states that if all subgraphs of a proper graph are finite according to power counting (or by renormalization), then the graph itself has only local overall divergences, and these occur only in 2, 3 and 4-point Green functions [10]. Removing these overall divergences by the process of renormalization, one arrives at finite Green s functions for arbitrary connected n-point functions in gauge theories with or without matter. We now turn to the task of making the divergent proper graphs finite.

11 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 11 1 Ward identities for divergences in proper graphs Consider the quantum action for pure Yang-Mills theory L(qu) = 1 4 (F µν a ) 2 1 2ξ (µ A µ a ) 2 ( µ b a )(D µ c) a (3.1.1) We use the Lorentz-covariant gauge fixing term because this is the one most often used. One can also consider axial-type gauges such as 1 2ξ (nµ A a µ) 2 where n µ is a constant vector, in particular the axial gauge 1 2 (Aa 3) 2 itself, but then there are many more counter terms possible, and the whole analysis becomes much more complicated, see section 6. (For example, there are divergences proportional to Fµν 2 and separate divergences proportional to Fµ3.) 2 In principle one could even use gauge fixing terms which are not even invariant under rigid group transformations such as ( µ A 1 µ) 2 + ( µ A 2 µ) 2 + λ( µ A 3 µ) 2, but we prefer not to consider such complications. In fact, only for Lorentz-invariant and rigid-group-invariant gauge fixing terms renormalizability of nonabelian gauge theories has been proven to all orders in loops. Assuming finiteness of the effective action at the (n 1)-loop level, one could compute all proper diagrams at the n -loop level, isolate the divergences, and then multiplicative perturbative renormalizability would mean that one could remove these divergences (and hence render all n-loop 1PI Green s functions finite) by rescaling the objects in the quantum action. These rescalings one would expect to be given by 5 A µ a = Z 3 A µ a,ren ; b a = Z gh b a ren, c a = Z gh c ren a g = Z 1 (Z 3 ) 3/2 µ 1 2 (4 n) u, ξ = Z ξ ξ ren (3.1.2) Since in the action in (3.1.1) only the product of b a and c a appears, the product of their Z factors, Z b Zc, can always be equally distributed over both, hence without loss of generality we can assume that Z b = Z c Z gh. New as compared to λϕ 4 5 We shall prove renormalizability using these rescalings, so there is no need to consider rescalings of µ or the structure constants f a bc.

12 12 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES theory is, of course, the appearance of the gauge parameter ξ. One might think that one could choose a gauge with ξ = 1, and then one would not have to deal with the renormalization of ξ. This is false; even if one would choose the gauge ξ ren = 1 one still would need its Z -factor Z ξ. This can be understood as follows. A direct computation of the proper one-loop selfenergy for gauge fields (for example with dimensional regularization) yields a transverse result < A a µ A b ν >= (η µν k 2 k µ k ν )δ ab Π(k 2 ) (3.1.3) This corresponds to a renormalization of the kinetic terms 1( 4 µa a ν ν A a µ ) 2, hence at one-loop there is no renormalization of the gauge fixing term 1 2ξ (µ A a µ ) 2. Renormalization of the kinetic term requires renormalization of A a µ. However, renormalization of A a µ would lead to a counter term of the form 1 Z 2ξ 3( µ A a,ren µ ) 2, and since the explicit calculation showed that such a term is absent at the one-loop level, one must rescale ξ in the opposite way, such that the total effect of rescaling both ξ and A a µ in L(fix) cancels. Two-loop calculations confirm that the selfenergy is transverse. Hence we conclude that (i) we need a parameter ξ and (ii) we must renormalize it as Z ξ = Z 3 (3.1.4) One can actually prove that at any loop level the complete proper selfenergy of A a µ is transversal by using a Ward identity for proper graphs, see (3.1.41). In this chapter we focus on the divergences and show that the n -loop divergences in the (n 1) loop renormalized proper selfenergy of the gauge fields are transversal. The proof is given by induction, namely we show that the divergences (and therefore the counter terms) are proportional to the various terms in S(quantum) S(f ix), so the divergences are transversal (not proportional to S(f ix)). In other words, there are no counter terms proportional to S(f ix). For renormalization one needs proper graphs, but for unitarity one needs connected graphs. One can also prove that the

13 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 13 renormalized connected selfenergy graphs are transversal at all loop levels. We give this proof in the chapter on unitarity. We shall prove the nonrenormalizability of the gauge fixing term as part of our general proof of renormalizability. In the next chapter we consider spontaneously broken gauge theories. One finds then gauge-fixing terms with several parameters L(fix) = 1 2ξ (µ A a µ + αgvχ a ) 2 (3.1.5) where α is a gauge parameter like ξ, v is the vacuum expectation value of the Higgs field and χ a are the would-be Goldstone bosons. Again one begins by restricting the renormalization of these parameters such that L(f ix) does not renormalize: after renormalization L(f ix) has the same form as before renormalization, except that it is written in terms of renormalized quantities (thus: all Z factors in L(f ix) cancel). We shall prove that also in this case the effective action becomes finite after renormalization. There is a big difference between on the one hand the renormalization of, for example, QED or models with scalars which have a rigid symmetry (for example linear σ models), and on the other hand the renormalization of nonabelian gauge theories. In the latter case, the transformation rules of the symmetry (BRST symmetry) are nonlinear in the fields, and this means that the path integral average of the variation of a field is not equal to the product of the path integral averages of the fields in the variation. For example for δ B A a µ = µ c a Λ + gf a bca b µ c c Λ we have < gf a bca b µ c c > gf a bc < A b µ > < c c > (3.1.6) We shall derive Ward identities for generating functionals of connected and proper diagrams using path integrals, and we shall encounter terms like < δ B A a µ >. To still be able to deal with such terms, there is a general method: one adds new terms to the action which are products of external sources and the nonlinear objects. For pure Yang-Mills theory in the usual relativistic gauge, the nonlinear terms in the BRST

14 14 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES transformation rules are only present in δ B A µ a and δ B c a but not in δ B b a. Hence we add to the action the following two terms L(extra) = K a µ (D µ c) a + L a 1 2 gf a bcc b c c (3.1.7) By definition, K a µ and L a do not transform under BRST transformations. Clearly, K a µ is anticommuting and L a commuting; moreover both are arbitrary x-dependent fields which only enter into the quantum action in this way, so they have no propagators. In terms of Feynman graphs this means that we only consider proper graphs with external K and/or L lines (in addition to the usual graphs without any external K or L lines). To keep the action real, we declare that K a µ and L a are purely imaginary. Note that since BRST transformations on A a µ and c a are nilpotent, even without BRST auxiliary field, L(extra) by itself is BRST invariant. The external sources K a µ and L a were introduced by Zinn-Justin and B. Lee [5, 6]. They are called antifields in the more recent antifield formalism [11] and are then denoted by (A a µ) and (c a ). They can be considered as a kind of covariant momenta conjugate to A µ a and c a (but with opposite statistics from the usual momenta). In more complicated theories with open gauge algebras, or reducible gauge algebras, this antifield formalism provides a systematic derivation of the correct quantum action. We are now ready to derive the Ward identities. As in λϕ 4 theory we add the usual external sources which couple to the fields in the quantum action L(source) = J a µ A µ a + β a c a + b a γ a (3.1.8) The external source β a is imaginary and γ a real to make L(source) real, and both are anticommuting. (Recall that we take c a to be real and b a imaginary.) We shall first consider linear gauges, i.e., gauges in theories like QCD in which the gauge fixing function F a is linear in fields. For these theories the proof of renormalizability is simplest if one does not introduce the auxiliary fields d a. Later we shall consider nonlinear gauges where it is simpler to keep d a as an independent field.

15 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 15 Consider the following path integral for connected and disconnected graphs Z(J µ a, β a, γ a ; K µ a, L a ) = N da a µ db a dc a exp i [L(qu) + L(extra) + L(source)]d 4 x (3.1.9) The constant N is chosen such that Z = 1 when all its arguments vanish. We now make a change of integration variables, from (A a µ, b a, c a ) to (A a µ ) = A a µ + ɛδ B A a µ, b a = b a + ɛδ B b a, (c a ) = (c a + ɛδ B c a ) where ɛ is an infinitesimal commuting constant. 6 We assume that the Jacobian for this infinitesimal BRST transformation is unity, see our discussion in chapter 2. Then da µ a db a dc a = d(a µ a ) db ad(c a ) (3.1.10) Next we use the BRST invariance of the quantum action and L(extra) in (3.1.7) to replace all fields in these actions by BRST-transformed fields L(qu) = L ((A µ a ), b a, (c a ) ) L(extra) = K a µ (D µ c a ) + L a ( 1 2 gf a bc(c b ) (c c ) ) (3.1.11) Finally, we replace in L (source) the fields A µ a by (A µ a ) ɛδ B A µ a, b a by b a ɛδ B b a and c a by (c a ) ɛδ B c a. None of these steps changes the value of Z. However, the terms with ɛ = 0 are also equal to Z, since writing Z in terms of primed variables amounts only to a change of name (the Shakespeare theorem 7 ). Hence, the expression for Z in (3.1.9) equals the expression for Z with extra ɛ-dependent terms. Thus the ɛ dependent terms should cancel by themselves. We shall work to first order in ɛ and 6 In the literature one usually only works with Λ but one does not introduce a second constant commuting parameter ɛ. One views Λ then as an infinitesimal parameter which is anticommuting. Because the notion of an infinitesimal anticommuting parameter is unclear we prefer for paedagogical reasons to introduce another infinitesimal commuting parameter ɛ and work to first order in ɛ. After having derived the Ward identity, we will no longer need ɛ ô be some other name. Whats in a name? that which we call a rose, By any other word would smell as sweete... [12].

16 16 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES find then the following identity for Z da a µ db a dc a (J µ a (y)δ B A a µ (y) + β a (y)δ B c(y) a + δ B b a (y)γ a (y)) d 4 y exp i [L(qu) + L(extra) + L(sources)] d 4 x = 0. (3.1.12) This identity holds if there are no BRST anomalies (in which case the BRST Jacobian equals unity). Note that this Ward identity would also hold if one had not included the sources K µ a and L a, but they will soon become crucial. To simplify the notation we write this expression as < J a µ δ B A µ a + β a δ B c a + δ B b a γ a > d 4 y = 0 (3.1.13) where < > denotes the path integral average. We can bring J µ a, β a and γ a outside the brackets and we encounter then the before-mentioned terms < δ B A µ a > and < δ B c a >. Now we see the use of L(extra), since we can write ( ) i < δ BA a µ (y) >= K aµ (y) Λ Z; i ( ) < δ Bc a (y) >= L a (y) Λ Z (3.1.14) In linear gauges (gauges which are linear in quantum fields), < δ B b a > can be written as a differential operator acting on Z. For example, for δ B b a = 1 ξ (µ A a µ )Λ we obtain ( i < δ Bb a >= 1 ) ξ µ J Λ Z (3.1.15) µ a since i < A µ a >= simplifies to Z. Putting all these results together, the Ward identity for Z J a µ ( J a µ K + β µ a + 1 a L a ξ µ J a µ γa ) d 4 y Z = 0 (3.1.16) (Pulling Λ past γ a yields an extra minus sign.) This Ward identity is a linear firstorder partial differential equation with infinitely many variables, a notoriously complicated mathematical object, but we shall be able to extract all information on renormalizability we need from it without actually solving it in general. We pause at this moment to answer a question the reader may have had from the beginning of this chapter. Namely, why does one not construct the composite

17 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 17 operator δ B A a µ(x) by differentiating Z simultaneously w.r.t. the sources J µ a (x) and β a (x), instead of using the sources K µ a (x) and L a (x)? The answer is that we can easily apply the Legendre transformation to single derivatives of Z such as to double derivatives such as J a µ (x) β a(x) go from Z to the effective action Γ. Z, but not K a µ (x) Z. We need the Legendre transformation to First we go over to connected graphs. functional W which is the logarithm of Z They are generated by the generating Z = exp i W (3.1.17) If no loops were involved, W would simply be equal to the connected tree graphs with sources at the ends, constructed from the action L(qu) + L(extra) + L(sources). Dividing the Ward identity for Z by Z, one finds the Ward identity for W which has the same form since it is only linear in derivatives ( µ J a K + β µ a + 1 ( ) ) µ γ a d 4 y W = 0. (3.1.18) a L a ξ J µ a Note that W depends on the same variables as Z W = W (J a µ, β a, γ a ; K a µ, L a ). (3.1.19) Next we go over to the generating functional Γ for proper (one-particle irreducible) graphs. It is related to W by a Legendre transformation Γ(A a µ, c a, b a ; K µ a, L a ) = W (J µ a, β a, γ a, K µ a, L a ) (J µ a A a µ + β a c a + b a γ a ) d 4 x (3.1.20) If there were no loop corrections, Γ would be equal to S(qu)+S(extra). The p, q of this Legendre transformation are thus the pairs (A a µ, J µ a ), ( b a, γ a ) and (c a, β a ), but the (K µ a, L a ) play the role of the q which are not transformed under the Legendre transformation. If we consider W as the Lagrangian and Γ as minus the Hamiltonian, then the Legendre transform is of the form H = qp + L, and the usual

18 18 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES relations of classical mechanics can be written down 8 q L = p J W = A µ µ a, W = c a, a β a γ W = b a a a H/p = q Γ/A µ = J µ a, Γ/c a = β a, Γ/b a = γ a (3.1.21) We indicate right- derivatives by Γ/A µ a etc., while A a µ Γ denotes left- derivatives. (Other notations used in the literature are A Γ and Γ A, or ( L /A)Γ and ( R /A)Γ, or A Γ and Γ/ A). For b a and c a these derivations differ by a sign: b a Γ = Γ/b a and similarly for c. (The easiest way to check such relations is to take Γ = b a γ a as an example; then, since b a γ a = γ a b a one finds b a (b a γ a ) = γ a but (b a γ a )/b a = γ a.) The fields A a µ, b a and c a which appear in (3.1.20) are the path average of the fields A a µ which appear in the action. This is clear from the relation /J µ a W = A a µ etc. One calls the former fields A a µ sometimes the classical fields ; not a very fortunate choice of name because we are at the quantum level. It is customary to use the same notation A a µ for both kinds of fields, although the reader may introduce different symbols to avoid confusion. We shall follow the literature and use the same symbols for both kinds of fields. Another set of identities we shall use correspond to the relation /q L(q, q) = /q H(p, q) in classical mechanics K a µ Γ = K a µ W, L a Γ = L a W (3.1.22) Using these identities, the Ward identity for W goes over into a Ward identity 8 For anticommuting variables it matters whether one writes qp or p q, and also the left derivative Defining both for commuting and anticommuting q L differs from the right derivative L/ q. variable qp L(q, q) = H(p, q, q), variation w.r.t. q shows that H(p, q, q) is, in fact, independent of q if we define p = / ql. This shows that if one defines p by left-differentiation of L, then one needs qp and not p q in the definition of H. Variation w.r.t. q yields /ql = /qh and variation w.r.t. p yields q = H/p. (Variation of the left-hand side yields δ qp+ qδp δq q L δ q q L = qδp δq q L. Variation of the right-hand side yields δh = δq q H + δp p H and if we replace δp p H by H/pδp, the Hamiltonian equations of motion follow.)

19 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 19 for Γ [ ( Γ/A a µ (x)) K a µ (x) Γ + ( Γ/ca (x)) L a (x) Γ + 1 ( ) ] ξ x µ Aµa (x) Γ/b a (x) d 4 x = 0 (3.1.23) At tree level (for = 0) this relation reduces to the statement that the action S(qu)+ S(extra) is BRST invariant. Note that an important complication has occurred: the Ward identity for Γ is nonlinear (quadratic) in Γ, whereas the Ward identity for Z (and W ) was linear in Z (and W ). However, we will only be interested in an analysis of divergences, and for these we shall derive a Ward identity which will again be linear in Γ as we shall see. For comparison, we quote the corresponding Ward identity for linear sigma models with a rigid symmetry δϕ i = λ a (T a ) i jϕ j with constant symmetry parameters λ a (see chapter 4) (Γ/ϕ i )(T a ) i jϕ j = 0 (3.1.24) Clearly, these Ward identities are linear in Γ. As we already explained, for local nonabelian symmetries, one must use BRST transformations, and these are nonlinear in fields and lead to a Ward identity quadratic in Γ. At this point, we simplify the Ward identity for Γ by using the knowledge (or, rather, the assumption, to be justified by induction afterwards) that the gauge fixing terms do not renormalize. We subtract them from the effective action, and thus define a functional ˆΓ by Γ = ˆΓ + L(fix)d 4 x (3.1.25) where L(fix) = 1 2ξ (µ A a µ ) 2. Note that at order = 0 (in the absence of loop corrections), ˆΓ is equal to the quantum action without gauge fixing terms, while at higher order in, there is no difference between Γ and ˆΓ. Since L(fix) does not depend on K µ a, L a or b a, we find that only in the term Γ/A a µ in (3.1.23) it makes a difference whether we replace Γ by ˆΓ or not, and we claim that this difference

20 20 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES precisely cancels the last term in the Ward identity for Γ [ ( (Γ ˆΓ)/A a µ (x)) K aµ (x) ˆΓ + 1 ] ξ (µ A a µ (x)) ˆΓ/b a (x) d 4 x = 0 (3.1.26) We shall prove this relation in a moment, but accepting this claim, (3.1.23) simplifies to [ ( ) ( ) ˆΓ/A a µ (x) K aµ (x) ˆΓ To prove the claim in (3.1.26), we begin with + ( ) ( )] ˆΓ/c a (x) L a (x) ˆΓ d 4 x = 0. (3.1.27) da a µ db a dc a b a (y) e i [S(qu)+S(extra)+S(sources)] = 0 (3.1.28) This follows from the property of the Grassman integral that db a (y)b a (y) = 1 but dba (y)f = 0 if F is independent of b a (y). Dividing spacetime into cells, in each cell we have variables A µ a, b c, c a, and since b a (y) is anticommuting, there are only terms in F which are independent of b a (y) or linear in b a (y). Since b a(y) F is always independent of b a (y), the path integral in (3.1.28) vanishes. (In theories with local fermionic invariances such as supergravity and string theory, b a (y) is commuting, and then one has to argue that the integrand falls off sufficiently fast for large b a (y). For fermionic b a (y), the Berezin integral avoids this.) Recalling that b a (y) S(qu) = µ (D µ c) a (y), b a (y) S(sources) = γa (y) (3.1.29) we find the following Ward identity for connected and disconnected graphs < µ (D µ c) a (y) + γ a (y) >= 0 (3.1.30) Note that this is a local identity (not an integral over spacetime). We shall give an interpretation of this identity in terms of Feynman diagrams in a moment. i < D µc a (y) >= K µ a Z, we can rewrite this step-by-step as follows ( µ K aµ (x) + i ) γa (x) Z = 0 Since

21 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 21 x µ K aµ (x) W + γa (x) = 0 x µ K aµ (x) Γ b a (x) Γ = 0 x µ K aµ (x) ˆΓ b a (x) ˆΓ = 0 (3.1.31) Again one may check the signs by taking the = 0 limit, in which case one finds µ D µ c a µ D µ c a = 0. The result in (3.1.31) is equivalent to (3.1.26) as one may verify by using Γ ˆΓ = S(fix) and working out the first term in (3.1.26) S(fix)/A a µ (x) K aµ (x) ˆΓ 1 d 4 x = ξ Aa µ K ˆΓ d 4 x (3.1.32) µ a Using (3.1.31) we indeed find that (3.1.26) is correct. Let us now clarify the meaning of the Ward identity in (3.1.30) by checking that it holds for connected (not proper) Feynman diagrams at any loop order. First we can take (3.1.30) and set all external sources to zero; this yields µ D µ c a (x) = 0, which is obviously correct because there are no graphs with one external ghost field but no external antighost field (or K source). Next differentiate (3.1.30) with respect to γ b (x), and set afterwards again all external sources to zero. This yields i b b(x)( µ D µ c a )(y) + δ a b δ 4 (x y) = 0. (3.1.33) At tree graph level one finds the relation i b b(x) µ µ c a (y) = δ a b δ4 (x y), which is correct since the ghost propagator is given by 9 c a (y)b b (x) = δ a b i k 2 eik(y x) d4 k (2π) 4 (3.1.34) and e ik(x y) d 4 k/(2π) 4 = δ 4 (x y). At the one-loop level, there are two diagrams which contribute; both come from the first term in the Ward identity because the 9 To get the correct signs, we complete squares in the action as b a c a + β a c a + b a γ a = (b a + β 1 a ) (c a + 1 γ a ) β 1 a γ a. Then c a (x)b b (y) = ( i γ b (y) )( i β )lnz a(x) i ( 1 )δb aδ4 (x y). One can also use canonical quantization to derive this result.

22 22 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES second term is independent of, and has already been used up in the tree graph relation. One must show in diagramatic notation that the following identity holds y y x + y µ y x = 0 (3.1.35) Note that this is an identity between connected graphs. The vertex on the left-hand side in the second graph comes from expanding ( µ D µ c a )(y). It is now clear that the sum of both graphs cancels: the y cancels the propagator on the far left, and the two vertices on the left-hand side of both loops are equal. The reader may check that all signs and factors i are such that the identity holds. This concludes the check of the local Ward identity for connected graphs at the tree and one-loop level. Next we check the Ward identity in (3.1.27). This is an integrated Ward identity for proper graphs. By differentiating with respect to A ν b (y), A ρ c (z) and c d (w), and then setting all fields to zero, one finds the following identity y x x w z [ ( 3ˆΓ/A ) ( a µ (x)a b ν(y)a c ρ(z) K a µ (x) ) c d (w) ˆΓ y x x z w z x x y w ( + 2ˆΓ/A ) ( a µ (x)a b ν(y) K a µ (x) ( + 2ˆΓ/A ) ( a µ (x)a c ρ(z) K a µ (x) ) c d (w) A c ρ(z) ˆΓ ) ] c d (w) A b ν(y) ˆΓ d 4 x = 0 (3.1.36) There are no other terms because ghost number is conserved, and the tadpole graphs Γ/A a µ vanish after all remaining fields have been set to zero. Making a Fourier transform of the coordinates y, z, w (using that the Green functions only depend on the differences of the coordinates by translational invariance), we find the corresponding relation in momentum space with momenta p, q and r for the gauge fields. Energy-momentum conservation yields p + q + r = 0. If we then take all terms at tree

23 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 23 level (all terms of order = 0), we find the following identity 10 gf abc (η µν (p q) ρ + η νρ (q r) µ + η ρµ (r p) ν )(p µ δ a d) +(η µν q 2 q µ q ν )δ ab (gf a cdδ ρ µ) +(η µρ r 2 r µ r ρ )δ ac (gf a bdδ ν µ) = 0 (3.1.37) It is easy to check that this identity is satisfied, by replacing p by q r. Thus we have checked the ˆΓˆΓ equation at the tree level. However, it also holds at any loop level. As another application of the Ward identity for proper graphs we prove the transversality of the selfenergy of the gauge fields. In this case we differentiate the integrated Ward identity in (3.1.27) w.r.t. A b ν(y) and c d (w). Since 2 KA ˆΓ, ˆΓ/A, A ˆΓ/c, L ˆΓ, ˆΓ/c (3.1.38) all vanish after setting all remaining fields to zero due to ghost number conservation or Lorentz invariance, there is only one term left ( 2ˆΓ A a µ(x)a b ν(y) ) ( 2ˆΓ K a µ (x)c d (w) ) d 4 x = 0 (3.1.39) or in graphical notation: A b ν A a µ y x ( K µ a c d ) d 4 x = 0 (3.1.40) x w To go over to momentum space we multiply by e ip(y w) and integrate over d 4 (y w). Setting e ip(y w) e ip(x y) e ip(x w) and d 4 xd 4 (y w) = d 4 (x y)d 4 (x w), we obtain 10 Using ˆΓ = 1 a 2 Aµ (x)( η µν µ ν )A a ν (x)d 4 x +... one finds 2 ˆΓ A a µ (x)a b ν = ( η µν (y) µ ν )δ ab δ 4 (x y) where δ 4 (x y) = e iq(x y) d4 q (2π 4 ). Similarly, the other two point Green functions contain one delta function of the spactime coordinates, and the three point functions contain two such delta functions. For example, ˆΓ A c ρ (z) with ˆΓ = K µ a(v)gf a cda c µ (v)c d (v)d 4 v +... K a µ (x) c d (w) gives ( gf a cd)δ ρ µδ 4 (x w)δ 4 (x z), where δ 4 (x w) = e ip(x w) d4 p (2π) 4 and δ 4 (x z) = e ir(x z) d4 r (2π) 4. Integration over d 4 x in (3.1.36) yields the energy momentum conservation law p + q + r = 0, and the result is (3.1.37).

24 24 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES in (3.1.40) the product of two proper graphs between which momentum p flows. The second factor can only be proportional to the momentum p µ which flows through the graph 11 with external K and c, and the first factor yields the proper selfenergy of the gauge fields. Then one finds indeed transversality, even off-shell k µ A a µ(k)a b ν( k) = 0 (3.1.41) We shall encounter many further Ward identites for proper graphs; they are all due to differentiating the local or integrated Ward identity. We have thus obtained two Ward identities for ˆΓ = Γ S(fix) [ ( ) ( ) ˆΓ/A a µ (x) K aµ (x) ˆΓ ( ) ( )] + ˆΓ/c a (x) L a (x) ˆΓ d 4 x = 0 ( x µ K aµ (x) ) ˆΓ = 0 b a (x) (I) (II) The first Ward identity involves an integration over spacetime, whereas the second one is local. The first one is quadratic in ˆΓ, while the second one is linear in ˆΓ. It is not possible to solve the quadratic equation; if one could, one would have the result for the sum of all proper Feynman diagrams. However, we shall soon derive an equation for the divergences in proper Feynman graphs which is linear in ˆΓ and can be solved perturbatively. It is clear that (I) by itself gives no information on the b a dependence of ˆΓ; this information is provided by (II). The need for (II) is not so surprising, since we used so far only the nilpotency of BRST transformations of A c µ and c a as expressed by L(extra), but the information that two BRST transformations of b a are proportional to the b a field equation should also be provided, and this is what (II) does. One can use other field equations to derive further Ward identities, but they involve new nonlinear objects for which one must introduce new external sources. These new 11 In noncovariant gauges such as n A a = 0, the proper two-point function with external K a µ and c b contains also terms proportional to n µ, and then transversality no longer holds.

25 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 25 external sources lead to new Z factors, and because the new nonlinear objects are in general not BRST invariant, 12 one would need even further new external sources for their BRST variations. There is no net gain with this approach. The two Ward identities we have derived hold for the regularized but not yet renormalized theory. (Note that we assumed that we were using such a regularization scheme that the BRST Ward identities were satisfied. One must assume that one is using a regularization scheme because without regularization all path integral manipulations have no meaning.) We use dimensional regularization because the BRST symmetry of the action we have been using is also valid in n dimensions, so that all relations we derived are finite. We must now deduce corresponding Ward identities for the renormalized effective action Γ ren. We shall prove that the regularized and renormalized effective actions are equal Γ ren (A µ a,ren, b a ren, c ren a, K a µ,ren, L a ren, µ, u, ɛ,, ξ ren ) = Γ(A µ a, b a, c a, K a µ, L a, g, ɛ,, ξ) (3.1.42) where the relation between the unrenormalized dimensionful coupling constant g and the dimensionless renormalized coupling constant u is given by g = Z 1 Z 3/2 3 µ 1 2 (4 n) u, (see (3.1.2)). We shall comment on this relation shortly. The renormalized effective action Γ ren is computed with the quantum action plus counter terms all in terms of renormalized fields and parameters, while the renormalized effective action Γ is computed using the quantum action in terms of nonrenormalized fields and without counterterms. In both cases one uses the same regularization scheme with the same regularization parameter ɛ. In practice this is dimensional regularization with ɛ n 4. Then both Γ ren and Γ are first evaluated in n dimensions. We recall the definition of counter terms in a multiplicative renormalizable model S(A µ a, b a, c a, K a µ, L a, g) = S ren + ΔS ren 12 For example, the ghost field equation reads < D µ µ b a (y) + β a (y) + >= 0 but D µ µ b a is not BRST invariant.

26 26 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES S ren = S(A µ a,ren, b a ren, c a ren, K µ,ren a L a ren, u) ΔS ren S S ren (3.1.43) where S = S(qu) + S(extra). More explicitly, we renormalize, in addition to previous renormalizations, also the external sources K a µ and L a K a µ = Z K K a µ,ren, L a = Z L L a ren (3.1.44) and then we define while S ren = S qu (A a,ren µ, b ren a, c a ren, u) [ + K µ,ren a ( µ c a ren + uf a bca b,ren µ c c ren ren ) + L 1 ] a 2 uf a bcc b c ren c ren d 4 x ΔL ren = 1 4 (Z 3 1)( µ A ν a,ren ν A µ a,ren ) 2 + (3.1.45) 1 2 (Z 1 1)uf a bc( µ A a,ren ν ν A a,ren µ )A b,ren c,ren µ A ν ( ) ZL Z 1 Z gh /Z 3/2 ren 3 1 L 1 a 2 uf a bcc b c ren c ren (3.1.46) The regularized unrenormalized effective action depends on the regulating parameter ɛ Γ(A a µ, b a, c a, K µ a, L a, g, ɛ,, ξ) (3.1.47) and is computed using the propagators and vertices of the unrenormalized action S(cl) + S(fix) + S(gh) + S(extra) (3.1.48) while the renormalized effective action Γ ren (A a,ren µ, b ren a, c a ren, K µ,ren a, L ren a, u, µ, ɛ,, ξ ren ) (3.1.49) is computed using S ren (qu) + S ren (extra) + ΔS ren (3.1.50)

27 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 27 In Γ, the limit ɛ to zero while keeping unrenormalized quantitites fixed does not exist, but in Γ ren keeping renormalized quantities fixed this limit exists if we have renormalized properly. (Of course, the limit of vanishing ɛ exists also in Γ if one varies A a µ etc. in such a way as to keep A a µ,ren fixed.) For nonvanishing ɛ, we have the fundamental identity (3.1.42) Γ = Γ ren (3.1.51) Some physicists consider this equality evident since S = S ren + ΔS ren, but note that one uses J a µ A µ a +... in one case and J a µ A µ a,ren +... (and not J a µ Z 3 A µ a,ren +...) in the other case to define Γ and Γ ren. We give a proof of Γ = Γ ren in appendix A to this chapter, starting with S = S ren + ΔS ren as input, and performing the Legendre transformations in both the unrenormalized and the renormalized case to arrive at the relation between Γ and Γ ren. Since S(fix) = S ren (fix), as follows from 1 2ξ (µ A µ a ) 2 = 1 2ξ ren ( µ A µ a,ren ) 2 due to Z 3 = Z ξ, we also have where ˆΓ ren should ˆΓ = ˆΓ ren (3.1.52) be a finite functional (i.e., without divergences) if the theory can be shown to be renormalizable. So, differentiating ˆΓ ren w.r.t. its variables (A µ a,ren,... L a ren ) should yield again a finite result. This at once shows that the Z factors in (I) and (II) due to rescaling must cancel. For example, rewriting (II) in terms of renormalized objects ( 1 x µ ZK K µ,ren a (x) 1 Zgh and using that ˆΓ ren is finite, implies 13 ) b ren a (x) ˆΓ ren = 0 (3.1.53) Z K = Z gh (3.1.54) 13 We assume here that we can use multiplicative renormalizability, and the results show that this leads indeed to a finite theory. The Z-factors are due to minimal subtraction (keeping only the pole terms in the divergences due to dimensional regularization). If the theory is first made finite by using these Z factors, a further rescaling by additional finite Z factors will keep the theory finite.

28 28 3. RENORMALIZATION OF UNBROKEN GAUGE THEORIES because K a µ,ren ˆΓ ren and b ren a ˆΓ ren are both finite. These Z factors are by assumption the Z factors which are needed to make all (n 1) loop proper graphs finite; we shall denote them by Z (n 1) where confusion might arise. (In principle, there is the more general solution Z K = αz gh with α a constant, but since for u tending to zero all Z s tend to one, α must be unity as well.) Similarly, from (I), we see that Z 3 Z K = Z gh Z L (3.1.55) or, combining with Z K = Z gh, Z L = Z 3 (3.1.56) We could also have renormalized the Ward identity for Γ in (3.1.23), instead of the Ward identity for ˆΓ and the effective antighost field equation, and then we would have found (3.1.55) and (3.1.56) simultaneously. Hence, we can only hope to prove renormalizability if we assume from the beginning that Z ξ = Z 3, Z K = Z gh and Z L = Z 3. This leaves only three Z factors in pure Yang-Mills theory to absorb infinities, and hence there should not be more than three independent divergences in the proper graphs. The renormalized Ward identities now read [ ( ) ( ˆΓ ren a,ren /A µ ( x µ K a µ,ren K µ,ren a (x) b ren a (x) ) ( ) ( )] ren ˆΓ + ˆΓ ren a ren /c ren ˆΓ d 4 x = 0 L ren a ) ˆΓ ren = 0 (3.1.57) The local Ward identity states that ˆΓ ren depends on b a ren and K a µ,ren only in the combination µ b a ren K a µ,ren (3.1.58) ren (For example, if ˆΓ = ( µ b ren a K µ,ren a ) F d 4 x for some b- and K-independent function F, the local Ward identity yields µ F + µ F = 0.) The integrated Ward identity restricts the dependence on the other variables even further, but its complete

29 1. WARD IDENTITIES FOR DIVERGENCES IN PROPER GRAPHS 29 solution is out of the question. equation which we now derive. Assume by induction that However, the divergences satisfy a much simpler (i) the theory has been renormalized up to and including (n 1) loops. This means that the terms in ˆΓ ren of order n 1 and less are finite. (ii) the equalities Z 3 = Z ξ, Z K = Z gh, Z L = Z 3 hold for the terms in the Z factors which are of order n 1 or less. Then we shall prove that after a further rescaling which removes the n-loop divergences, the same is true to order n. We start the induction at n 1 = 0, i.e. at the classical level without loops; here (i) and (ii) are obviously satisfied. Since the divergences in n-loop graphs 14 are proportional to n, whereas the terms of order n 1 and less in ˆΓ ren are finite by assumption we can decompose ˆΓ ren as ˆΓ ren = ˆΓ ren,(n) div + ˆΓ ren,(n) finite + ˆΓ ren,(n 1) finite + + Ŝren. Thus the divergences in the integrated Ward identity can only be present in the first or in the second factor of the first term, or in the first or in the second factor of the second term in (3.1.57). In each case we have a product of a term with ˆΓ ren ( n ) with ˆΓ ren ( = 0). Since ˆΓ ren ( = 0) = Ŝren, we find then the following equation for the divergences of the n-loop part of the effective action [ Ŝren /A µ a,ren K a µ,ren Ŝren /K a µ,ren + Ŝren /c a ren L ren Ŝren ren /L a a c a ren A µ a,ren ] d 4 x ˆΓ ren,(n) div = 0 (3.1.59) Recall that the letters ren denote (n 1) loop renormalizability and Ŝ denotes S(class) + S(ghost) + S(extra). From now on we will drop the subscripts ren. The 14 The divergences may, of course, also contain finite parts in addition to divergent parts. One can unambiguously define the divergences as the coefficients of pole terms ɛ n with ɛ = n 4 in dimensional regularization. However, this is not necessary; finite terms in the Z factors are allowed (sometimes called recalibrations). These finite terms should still be such that (ii) is satisfied and they can be fixed by suitable renormalization conditions.