Renormalization of gauge theories in the background-field approach
Andrei O. Barvinsky, Diego Blas, Mario Herrero-Valea, Sergey M. Sibiryakov, Christian F. Steinwachs
aa r X i v : . [ h e p - t h ] F e b CERN-TH-2017-099 , INR-TH-2017-010 , FR-PHENO-2017-011
Renormalization of gauge theories in thebackground-field approach
Andrei O. Barvinsky , , Diego Blas , Mario Herrero-Valea ,Sergey M. Sibiryakov , , , Christian F. Steinwachs Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991, Russia Department of Physics, Tomsk State University, Lenin Ave. 36, Tomsk 634050, Russia Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland LPPC, Institute of Physics, EPFL, CH-1015 Lausanne, Switzerland Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary Prospect, 7a, 117312 Moscow, Russia Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg,Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
Abstract
Using the background-field method we demonstrate the Becchi–Rouet–Stora–Tyu-tin (BRST) structure of counterterms in a broad class of gauge theories. Put simply,we show that gauge invariance is preserved by renormalization in local gauge fieldtheories whenever they admit a sensible background-field formulation and anomaly-free path integral measure. This class encompasses Yang–Mills theories (with possiblyAbelian subgroups) and relativistic gravity, including both renormalizable and non-renormalizable (effective) theories. Our results also hold for non-relativistic modelssuch as Yang–Mills theories with anisotropic scaling or Hoˇrava gravity. They strengthenand generalize the existing results in the literature concerning the renormalizationof gauge systems. Locality of the BRST construction is emphasized throughout thederivation. We illustrate our general approach with several explicit examples. ontents d + 1) dimensions . . . . . . . . . . . . . . . 183.4 General relativity as effective field theory in (3 + 1) dimensions . . . . . . . . 20 O ( N ) model: Explicitone-loop calculation 317 Conclusions and discussion 36A Derivation of Slavnov-Taylor and Ward identities 38B Homology of the operator Ωδ/δφ
40C Quadratic form for perturbations in the O ( N ) model 41 A central question in the perturbative quantization of gauge field theories is to what extentthe gauge symmetry is preserved by renormalization. Intuition tells that in the absence ofanomalies, i.e. when the measure in the path integral is gauge invariant, the counterterms1equired to cancel the ultraviolet divergences should be gauge invariant as well. A rigor-ous proof of this assertion, however, is highly non-trivial due to the breaking of the gaugesymmetry required to quantize gauge theories (gauge-fixing procedure). The original gaugeinvariance still survives in the Becchi–Rouet–Stora–Tyutin (BRST) [1, 2] structure of thegauge-fixed action Σ which remains invariant under infinitesimal variations generated bythe nilpotent BRST operator. At tree level, Σ is a sum of a BRST exact part responsi-ble for the gauge fixing and the classical gauge invariant action which depends only on thephysical fields (gauge fields and matter) and is independent of the Faddeev–Popov ghosts.The physical content of gauge invariance will be retained if this BRST structure persistsunder renormalization. In particular, it will guarantee that the partition function obeysSlavnov-Taylor identities at all orders of the perturbative expansion.In addition, to preserve the key properties of quantum field theory, the BRST structuremust be compatible with locality. Namely, starting from a gauge theory with a local La-grangian, both the BRST-exact and the gauge-invariant parts of the renormalized actionmust be given by integrals of local Lagrange densities.In the textbook examples of renormalizable relativistic theories, such as quantum elec-trodynamics or Yang–Mills (YM) theory, the previous properties can be proven by “bruteforce”: one first writes down all possible counterterms allowed by power counting and thensolves the equations for their coefficients following from the Slavnov–Taylor identities. Thelast step required to bring the renormalized action into the BRST form is a field redefinition.Positive canonical dimensions of the fields and the absence of any coupling constants withnegative dimensionality imply in these simple cases that the field redefinition must have theform of a multiplicative wavefunction renormalization, whose coefficient is easy to find, seee.g. [3].In general the situation is much more involved. This is the case, for example, in non-renormalizable theories (understood as effective field theories, EFTs) where one encounterscoupling constants of negative dimension. In these cases an explicit solution of the Slavnov–Taylor identities appears infeasible. Even if such solution were available, the field redefinitionbringing it to the BRST form could be nonlinear and arbitrarily complicated, rendering abrute-force search for it hopeless. The same is true for renormalizable higher-derivative grav-ity [4] where the canonical dimension of the metric is zero, implying that its renormalizationcan be, and actually is, nonlinear. To study the consistency of the BRST structure withrenormalization in this type of theories one needs more powerful methods. That is represented as a sum of terms depending on fields and their derivatives at a point. This sumcan, in principle, be infinite provided terms with higher number of derivatives are treated perturbatively, asit happens in effective field theories.
2t is well-known that the classification of possible counterterms arising in general gaugetheories requires computing the cohomology of an extended BRST operator [5, 6, 7, 8] (seealso [3]). To be compatible with the BRST structure, the latter must consist of local gauge-invariant functionals of physical fields only. This was indeed demonstrated in [9, 10]for the EFT consisting of general relativity coupled to YM with semisimple gauge groupextended by arbitrary gauge invariant higher-order operators. These references use theadvanced mathematical apparatus of local cohomology theory. Notably, for gauge groupswith Abelian factors they still leave room for non-trivial cohomologies different from gaugeinvariant functionals which, if generated by divergences, would imply deformations of theoriginal gauge symmetry. Additional arguments must be invoked to forbid the appearanceof such counterterms in the studied cases [8].The purpose of our work is to address the BRST structure of renormalized actions ingeneral gauge field theories admitting background field gauges. Our motivation is twofold.First, we will provide a new, and we believe simpler, derivation of the results concerningthe renormalization of Einstein–YM theories and strengthen them for the case of theorieswith Abelian subgroups. Second, our analysis covers a broader class of gauge theories notconsidered in the classic papers [9, 10]. This includes, in particular, the higher-derivativegravity and gauge/gravity theories without relativistic invariance.Non-relativistic gauge theories play a prominent role in condensed matter physics [11,12, 13] (see also references therein), investigations of non-relativistic Weyl invariance andholography [14, 15], and may be relevant for particle model building [16, 17] (see [18] fora summary of extra motivations and results in non-relativistic gauge theories). Further-more, abandoning relativistic invariance (while keeping the gauge group of time-dependentspatial diffeomorphisms) allows one to construct power-counting renormalizable models ofgravity in arbitrary spacetime dimensions including the phenomenologically interesting caseof (3 + 1) dimensions [19]. The renormalizability beyond power counting was established in[20] for a large subset of these gravity models, the so-called projectable Hoˇrava gravities. Itwas assumed in [20] that renormalization preserves gauge invariance, which was explicitlydemonstrated only at one loop. One of the goals of the present paper is to demonstrate thevalidity of this assumption to all loop orders and thereby complete the proof of renormaliz-ability of projectable Hoˇrava gravity.Our approach is based on the background field method [21, 22] (see also [23, 24]), apowerful tool for calculating the quantum effective action in gauge theories and gravity. Themain virtue of this method is that it preserves the gauge invariance of the calculations even The requirement of locality is crucial. Refs. [6, 7] studying the BRST cohomology in general gaugetheories do not guarantee its locality and have to postulate it as an additional assumption. si-multaneous gauge transformations of the variables in the path integral (“quantum fields”)and the background fields. We denote this transformation “background-gauge transforma-tion”. At the same time the quantum gauge transformations acting only on the quantumfields are broken by gauge fixing and the path integral is well defined (at least perturba-tively). The construction of background-covariant gauge fixing conditions is straightforwardin theories containing fields in linear representations of gauge groups with linear generators.These conditions imply that the background-gauge symmetry is preserved by renormalizationwhich serves as a strong selection criterion for possible counterterms. This method greatlysimplifies the renormalization of coupling constants in the one-loop approximation after thebackground fields are identified with the mean value of the quantum fields [25, 26, 27]. Inthis case the counterterms take a manifestly gauge invariant form.Beyond one-loop the situation becomes more complicated. The subtraction of subdi-vergences necessary to eliminate the nonlocal infinities requires counterterms where thequantum fields are distinct from the background fields. Background-gauge invariance isnot sufficient to completely fix the structure of such counterterms and the BRST structureassociated to the quantum gauge transformations must be exploited, as is done in the casesof gauges without background fields [28, 29, 30, 31, 32]. In practical calculations these coun-terterms can sometimes be avoided by subtle methods that have been developed for YMand relativistic gravity. However, these techniques generically feature nonlocal divergencesat intermediate steps of the calculations, that cancel only in the final quantities evaluatedon-shell [33, 34, 35, 36, 37]. The presence of nonlocal divergences makes these methods in-appropriate for a general analysis of renormalizability. More recently it has been advocated[38] that the use of a background gauge combined with the standard subtraction schemeprovides a valuable tool for such analysis (see also [39]). This reference uses the Batalin–Vilkovisky formalism [40, 41] to prove the existence of a canonical transformation bringingthe renormalized action to the BRST form. However, this requires introducing backgroundfield counterparts for all quantum fields of the theory including Faddeev–Popov ghosts and,moreover, the addition of Batalin–Vilkovisky antifields for all background fields. Such pro-liferation of objects makes the construction rather baroque and obscures the subtleties ofthe derivation.In this paper we adopt a different strategy and proceed along the lines of traditionalcohomology analysis. Our key finding is that the background-gauge invariance greatly facili-tates the computation of the local BRST cohomology. The latter reduces to cohomologies ofa few simpler nilpotent operators that are readily computed using elementary algebraic tech-4iques. The resulting constraints on the form of the renormalized action imply that, uponan appropriate field redefinition, it acquires the desired BRST form (a local gauge-invariantfunctional plus a BRST-exact piece). The argument does not involve any power-countingconsiderations. When available, such considerations lead to further refinements which wediscuss. We keep track of locality at all steps of the derivation.Our proof applies to theories characterized by the following properties: the gauge algebrais irreducible and closes off-shell; the gauge generators depend on the fields at most linearly;the structure functions are field independent. These conditions ensure that the theory admitsa convenient background-covariant gauge fixing. Besides, we assume the absence of anomaliesand locality of the leading ultraviolet divergences (ones that remain after subtraction ofsubdivergences). The latter requirement should not be confused with locality of the BRSTdecomposition, which is not postulated a priori, but is derived from the previous assumptions.The above class is quite broad. It encompasses renormalizable and non-renormalizable(effective) theories with Abelian and non-Abelian gauge groups, general relativity and higher-derivative gravity. Besides the standard relativistic versions of these theories, it also includestheir non-relativistic generalizations [42, 43, 19]. As a corollary of our general result weestablish for the first time the compatibility of the BRST structure with renormalization inprojectable Hoˇrava gravity [19] which completes the proof of its renormalizability. A notableexample that is not covered by our study is supergravity where the gauge algebra closes onlyon-shell .While various ingredients of our analysis have already appeared in the literature, to thebest of our knowledge, they have never been put together. To make the presentation self-contained we review these ingredients in the relevant sections. Several concrete examples aimto illustrate the physical content of the general result. For simplicity we focus throughoutthe paper on theories with bosonic gauge parameters.The paper is organized as follows. In Sec. 2 we describe our assumptions, introduce thebackground gauge fixing and formulate our main result (Sec. 2.4). In Sec. 3 we illustrate itsimplications on several examples. We discuss explicitly the standard renormalizable YM in(3 + 1) dimensions, relativistic higher-derivative gravity in (3 + 1) dimensions, projectableHoˇrava gravity in general dimensions and general relativity in (3+1) dimensions (understoodas an effective theory). In Sec. 4 we turn to the proof of our general result and derive theequations satisfied by the effective action as a consequence of the background and quantumgauge invariances. These equations are used to analyze the structure of the divergent coun-terterms in Sec. 5, which is the central part of the paper. Here we formulate the cohomology For N = 1 supergravity in four spacetime dimensions, the off-shell closure of the algebra can be achievedby introduction of auxiliary fields, but then the generators become nonlinear in the fields [44]. O ( N ) vector model in (1 + 1)spacetime dimensions written as an Abelian gauge theory. This example is interesting asit features nonlinear wavefunction renormalization, being at the same time simple enoughto admit an explicit treatment. We verify at one loop that the counterterms in this theoryhave the structure determined by the general argument. We conclude in Sec. 7. Appendix Acontains the derivation of the Slavnov–Taylor and Ward identities for the partition function.In Appendix B we prove a lemma about the cohomology of an operator appearing in ouranalysis. Some formulae used in the computation of the effective action of the O ( N ) modelare summarized in Appendix C. We consider a theory with local gauge and matter fields ϕ a , where a is a collective notationfor all indices and the coordinates. The theory is described by the action S [ ϕ ] whichis an integral of a local Lagrangian density L ( ϕ ). The latter is expanded as a sum ofterms depending on the fields ϕ a and their finite-order derivatives at a given point . Theaction S [ ϕ ] is invariant under gauge transformations with local bosonic parameters ε α . Thetransformations are assumed to have at most linear dependence on the fields, δ ε ϕ a = R aα ( ϕ ) ε α , R aα ( ϕ ) = P aα + R abα ϕ b , R aα ( ϕ ) δS [ ϕ ] δϕ a = 0 . (2.1)We further assume that the gauge algebra closes off-shell, (cid:2) δ ε , δ η (cid:3) ϕ a = δ ς ϕ a , (2.2)where ς α = C αβγ ε β η γ , (2.3)and C αβγ are field-independent structure functions. The closure condition implies the rela-tions, R abα P bβ − R abβ P bα = P aγ C γαβ , (2.4a) Throughout the text the dependence of local functions on the fields and their finite-order derivativeswill be denoted by round brackets, while square brackets will denote the functional dependence of integralquantities with local or nonlocal integrands. abα R bcβ − R abβ R bcα = R acγ C γαβ . (2.4b)Besides, C αβγ obey the Jacobi identities, C αβ [ γ C βλµ ] = 0 , (2.5)where the square brackets mean anisymmetrization over the respective indices.Next, we require that the set of gauge generators R aα ( ϕ ) is locally complete and irre-ducible . These properties are defined as follows:(i) Local completeness [41, 45]: Any local operator X aα ( ϕ ) satisfying the equation δSδϕ a X aα = 0 , (2.6)is represented as a linear combination of the gauge generators and equations of motion, X aα = R aβ Y βα + δSδϕ b I [ ba ] α , (2.7)where Y βα and I [ ba ] α are local and I [ ba ] α is antisymmetric in its indices. The localitycondition means that Y βα and I [ ba ] α are non-zero only if the coordinates correspondingto β and α or a , b and α coincide.(ii) Irreducibility [40]: Let ϕ a be a solution of the equations of motion, so that δSδϕ a ( ϕ ) = 0 . (2.8)If a gauge parameter ε α satisfies the relations R aα ( ϕ ) ε α = 0 , (2.9)then ε α = 0. In other words, gauge transformations act non-trivially on on-shellconfigurations.The class of theories described above is quite broad. It includes, in particular, relativisticAbelian and non-Abelian gauge theories together with their extensions by higher-derivativeoperators, general relativity and relativistic higher-derivative gravity, e.g. [4]. Besides, itcontains non-relativistic generalizations of these theories. Some examples are discussed inSec. 3 and in Sec. 6. As we mentioned, a notable exception from this class is supergravity,both due to the fermionic nature of the gauge parameter and openness of the gauge algebra.For the sake of clarity, we focus in what follows on theories where all fields ϕ a are bosonic.The inclusion of fermionic matter fields is straightforward, but would complicate the formulaeby additional ( −
1) factors. 7 .2 Background gauge
To quantize the theory we need to fix the gauge. We introduce the background fields φ a andchoose the gauge fixing function χ α ( ϕ, φ ) in such a way that it transforms covariantly undersimultaneous local gauge transformation of ϕ a and φ a with the same parameter ε but theirown generators R aα ( ϕ ) and R aα ( φ ) respectively, δ ε ϕ a = R aα ( ϕ ) ε α , δ ε φ a = R aα ( φ ) ε α . (2.10)Covariance of χ α under the transformations (2.10) implies, δ ε χ α ≡ δχ α δϕ a δ ε ϕ a + δχ α δφ a δ ε φ a = − C αβγ χ β ε γ . (2.11)We will refer to (2.10) as “background-gauge transformations” and to χ α ( ϕ, φ ) as “back-ground-covariant gauge conditions”. We further choose χ α to be linear in the difference( ϕ a − φ a ), χ α ( ϕ, φ ) = χ αa ( φ ) ( ϕ a − φ a ) . (2.12)The gauge-fixing function is assumed to be local in space-time, i.e. it depends only on thevalues of the fields and their derivatives of finite order at a point.The gauge fixing is implemented by the BRST procedure [1, 2] (see also [3]). Labellinganticommuting ghosts ω α , antighosts ¯ ω α and the Lagrange multiplier b α with the condensedgauge index α , we define the standard action of the BRST operator ss ϕ a = R aα ( ϕ ) ω α , (2.13a) s ω α = 12 C αβγ ω β ω γ , (2.13b) s ¯ ω α = b α , (2.13c) s b α = 0 . (2.13d)The closure conditions (2.4), (2.5) imply that s is nilpotent. The background fields φ a areinvariant under the action of s . Next, we introduce two sets of anticommuting auxiliary fields γ a , Ω a and a commuting field ζ α . They are also invariant under the BRST transformationsgenerated by s . We define the gauge fermion as Ψ [ ϕ, ω, ¯ ω, b, φ, γ, ζ ] = ¯ ω α (cid:18) χ αa ( φ )( ϕ a − φ a ) − O αβ ( φ ) b β (cid:19) − γ a ( ϕ a − φ a ) + ζ α ω α . (2.14) For bosonic gauge algebras that we consider in this paper, it is sufficient to introduce backgroundcounterparts to bosonic fields only, even if the theory contains fermionic matter. O αβ ( φ ) is an invertible local operator that can, in general, depend on the backgroundfields and transforms covariantly under the background-gauge transformations. Finally, weconstruct the gauge-fixed action, Σ [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] = S [ ϕ ] + Q Ψ , (2.15)with Q = s + Ω a δδφ a . (2.16)Following [28, 29, 30, 31, 32] we have extended the usual BRST operator in such a way thatit controls not only the field BRST transformations but also the variation of the gauge-fixingterm under the changes of φ . Clearly, Q is nilpotent due to the anticommuting nature of Ω a . Explicitly, the action (2.15) reads, Σ [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] = S [ ϕ ] + b α χ αa ( φ ) ( ϕ a − φ a ) − O αβ ( φ ) b a b β − ¯ ω α χ αa ( φ ) R aβ ( ϕ ) ω β + γ a R aα ( ϕ ) ω a + 12 ζ α C αβγ ω β ω γ + Ω c ¯ ω α (cid:20) δχ αb δφ c ( ϕ − φ ) b − χ αc − δO αβ δφ c b β (cid:21) + Ω c γ c . (2.17)One recognizes the gauge fixing part (second and third terms in the first line) and theFaddeev–Popov action for the ghost-antighost pair (last term in the first line). The secondline collects the dependence on the auxiliary fields γ a , ζ α and Ω a . Notice that γ a and ζ α couple as sources to the BRST variations of ϕ a and ω α respectively.In view of the nilpotency of Q the gauge-fixed action is BRST-invariant, Q Σ = 0 . (2.18)This equation will be used below to derive the Slavnov–Taylor identities constraining theultraviolet divergences. Besides, for background-covariant gauges of the above type, Ψ and Σ have an additional symmetry: they are invariant under background-gauge transforma-tions (2.10), δ ε Ψ = 0 , δ ε Σ = 0 , (2.19)if simultaneously with ϕ a and φ a we transform all fields in the appropriate linear represen-tations: δ ε γ a = − γ b R baα ε α , δ ε ω α = − C αβγ ω β ε γ , δ ε ζ α = ζ β C βαγ ε γ , δ ε Ω α = R abα Ω b ε α , (2.20) This dependence is, in fact, inevitable in gravity (see Sec. 3). Gaussian integration over the Lagrange multiplier b α gives a familiar gauge breaking term χ α O − αβ χ β with the weighting factor inverse to O αβ . ω α and b α . Note that for theories with diffeomorphism invariance ω α transforms as a contravariant vector, whereas ¯ ω α , b α , γ a , ζ α are vector/tensor densities.Finally, the action (2.17) possesses a global U (1) symmetry corresponding to the ghostnumber with the following assignment of charges:gh( ϕ ) = gh( φ ) = gh( b ) = 0 , gh( ω ) = gh( Ω ) = +1 , gh(¯ ω ) = gh( γ ) = − , gh( ζ ) = − . (2.21)Using (2.17) as the tree-level action and introducing sources coupled to the “quantum”fields ( ϕ, ω, ¯ ω, b ) we write the “bare” generating functional for connected graphs, W [ J, ¯ ξ, ξ, y, φ, γ, ζ , Ω ] = − ~ log Z dΦ exp (cid:20) − ~ (cid:0) Σ + J a ( ϕ a − φ a ) + ¯ ξ α ω α + ξ α ¯ ω α + y α b α (cid:1)(cid:21) . (2.22)Here we have collectively denoted all quantum fields by Φ in the integration measure andexplicitly included the Planck constant ~ as a counting parameter for the order of the loopexpansion . We impose two more conditions on the theory. First, we postulate the absence of gaugeanomalies, i.e. the existence of a regularization prescription that preserves the gauge invari-ance of the functional integration measure. This is achieved by dimensional regularizationin many cases.Second, we require that a variant of the standard subtraction scheme (e.g. minimalsubtraction) [46] eliminates all nonlocal divergences. Let us expand on this point. In thestandard scheme the counterterms are constructed inductively in the number of loops L or, equivalently, in the powers of ~ . Let us assume that at order O ( ~ L − ) we have alreadyconstructed the renormalized action Σ L − = Σ + L − X l =1 ~ l Σ Cl , (2.23)where Σ is the tree-level action (2.17) and Σ Cl are divergent local counterterms. This actionis such that the generating functional W L − defined by the formula analogous to (2.22) withthe replacement Σ Σ L − produces Green’s functions that are finite at ( L −
1) loops. Throughout the paper we work with Euclidean field theory and use the corresponding definition of thegenerating functional. As the operator O αβ in (2.17) is usually chosen positive-definite, the convergence ofthe path integral requires that the integration in b α runs along the imaginary axis. This subtlety does notaffect our analysis. as functional derivatives of the generating functionalwith respect to the sources , h ϕ a i − φ a = δWδJ a , h ω α i = δWδ ¯ ξ α , h ¯ ω α i = δWδξ α , h b α i = δWδy α , (2.24)and define the effective action Γ as the Legendre transform of W , Γ (cid:2) h ϕ i , h ω i , h ¯ ω i , h b i , φ, γ, ζ , Ω (cid:3) = W − J a ( h ϕ a i − φ a ) − ¯ ξ α h ω α i − ξ α h ¯ ω α i − y α h b α i . (2.25)Clearly, it satisfies, δΓδ h ϕ a i = − J a , δΓδ h ω α i = ¯ ξ α , δΓδ h ¯ ω α i = ξ α , δΓδ h b α i = − y α , (2.26)The ( L − Γ L − = Σ + ∞ X l =1 ~ l Γ ( l ) L − , (2.27)where Γ ( l ) L − is the contribution of diagrams with l loops. By the assumption of the inductionstep, all terms Γ ( l ) L − with l ≤ L − L -th term, Γ ( L ) L − , ∞ ≡ Γ L, ∞ [ h ϕ i , h ω i , h ¯ ω i , h b i , φ, γ, ζ , Ω ] (2.28)is local. Then the counterterm Σ CL is identified with − Γ L, ∞ where the mean fields arereplaced by the quantum fields, Σ L [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω] = Σ L − − ~ L Γ L, ∞ [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] . (2.29)According to the standard theorems [46] (see [47] for the generalization to theories withoutLorentz invariance), this subtraction removes the L -loop divergences, as well as all subdiver-gences in ( L + 1)-loop diagrams.In relativistic gauge theories with Lorentz-covariant gauge fixing, this guarantees that theremaining divergence of order O ( ~ L +1 ) in the effective action Γ L is local and the subtractioncan be repeated at the ( L + 1)-th loop order. The situation is less straightforward in theabsence of Lorentz invariance [20] and the locality of the remaining divergences must beverified in every given theory. It was shown to hold for non-relativistic YM theories withanisotropic (Lifshitz) scaling and projectable Hoˇrava gravity [20]. In the present paper wepostulate it as a property of the class of theories under study.To avoid cluttered notations, we will omit the averaging symbols on the arguments ofthe effective action Γ in what follows. These should not be confused with the background fields φ a . We fix the sign of the derivatives with respect to the anticommuting variables by placing the differentialon the left, df = dθf ′ ( θ ). .4 Proposition: BRST structure of the renormalized action We will show that a slight modification of the subtraction prescription by the inclusion ofadditional local terms of order O ( ~ L +1 ) on the r.h.s. of (2.29) leads to a renormalized action Σ L that preserves the BRST structure. More precisely, our result is formulated as follows.Let us denote the fields coupled to the external sources J, ¯ ξ by ˜ ϕ, ˜ ω and consider localfield reparameterizations of the form,˜ ϕ a = ˜ ϕ aL ( ϕ, ω, φ, ˆ γ, ζ , Ω ) ˜ ω α = ˜ ω αL ( ϕ, ω, φ, ˆ γ, ζ , Ω ) , (2.30)where we have introduced the combinationˆ γ a = γ a − ¯ ω α χ αa ( φ ) (2.31)that will play an important role below. Upon the field redefinition the L -th order generatingfunctional reads , W L [ J, ¯ ξ, ξ, y, φ, γ, ζ , Ω ] = − ~ log Z dΦ exp (cid:20) − ~ (cid:18) Σ L + J a ( ˜ ϕ aL − φ a ) + ¯ ξ α ˜ ω αL + ξ α ¯ ω α + y α b α (cid:19)(cid:21) . (2.32)We will demonstrate the existence of a field redefinition (2.30) such that Σ L takes the form, Σ L [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] = S L [ ϕ ] + Q Ψ L [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] , (2.33)where S L [ ϕ ] is a gauge invariant local functional and the BRST operator Q has been definedin (2.16). The gauge fermion Ψ L is a local functional with ghost number ( −
1) which isinvariant under background-gauge transformations (2.10), (2.20) and has the form, Ψ L = ˆ Ψ L [ ϕ, ω, φ, ˆ γ, ζ , Ω ] −
12 ¯ ω α O αβ ( φ ) b β , (2.34)where ˆ Ψ L = − ˆ γ a ( ϕ a − φ a ) + ζ α ω α + O ( ~ ) . (2.35)Further, the reparameterization (2.30) itself is generated by the gauge fermion,˜ ϕ aL = φ a − δΨ L δγ a , ˜ ω αL = δΨ L δζ α . (2.36)Together with (2.35) this implies that at tree level ˜ ϕ, ˜ ω coincide with ϕ, ω and the gaugefermion Ψ L coincides with the expression (2.14). Thus, we recover (2.22) at tree level. We disregard the functional Jacobian | δ ˜ Φ/δΦ | which gives an ultralocal contribution to the action. Suchcontributions vanish in dimensional regularization. J a , ¯ ξ α now couple tocomposite local operators that in general depend not only on the quantum fields, but alsoon the external backgrounds φ a , γ a , ζ α , Ω a . Nevertheless, this is not problematic due to theproperty (2.35), (2.36) that ensures linearity of the coupling at leading order in ~ .We will see in Sec. 3 and Sec. 6 that in many interesting theories, that are typicallyrenormalizable, power counting considerations strongly restrict the dependence of the renor-malized gauge fermion ˆ Ψ L on the auxiliary fields. Namely, in these cases ˆ Ψ L is independentof Ω a and can depend on ˆ γ a , ζ α only linearly,ˆ Ψ L = − ˆ γ a U aL ( ϕ, φ ) + ζ α ω β V αLβ ( ϕ, φ ) , (2.37)with U aL = ϕ a − φ a + L X l =1 ~ l u al ( ϕ, φ ) , (2.38a) V αLβ = δ αβ + L X l =1 ~ l v αlβ ( ϕ, φ ) . (2.38b)Correspondingly, the field redefinition (2.36) bringing the counterterms into the BRST-invariant form simplifies to˜ ϕ aL = φ a + U aL ( ϕ, φ ) , ˜ ω αL = V αLβ ( ϕ, φ ) ω β . (2.39)In this case it does not involve the auxiliary sources γ a , ζ α , Ω a . In this section we illustrate the notions and results described above on several gauge the-ories and discuss restrictions imposed on the structure of divergences by power countingin renormalizable cases. Together with a few well-known examples we consider the case ofprojectable Hoˇrava gravity whose BRST structure is studied here for the first time. Readersinterested in the general proof can skip this section and proceed directly to Sec. 4. (3 + 1) dimensions
As a first example, we consider the standard YM theory in (3 + 1) spacetime dimensions.It has been already studied using an approach similar to ours in [28, 29, 30, 31, 32]. Let us13tart by expanding the condensed notations , ϕ a A iµ ( x ) , ε α ε i ( x ) , (3.1a) R abα f ijk δ νµ δ ( x − x ) δ ( x − x ) , P aα δ ij ∂ µ δ ( x − x ) , (3.1b) C αβγ f ijk δ ( x − x ) δ ( x − x ) , (3.1c)where A iµ ( x ) is the usual Yang–Mills field, i is the color index and f ijk are the totallyantisymmetric coordinate independent structure constants of the gauge group. We nextintroduce the background field B iµ ( x ) and the gauge-fixing function, χ α ∂ µ ( A iµ − B iµ ) + f ijk B jµ ( A kµ − B kµ ) ≡ D µ ( B ) ( A iµ − B iµ ) . (3.2)Introducing the Faddeev–Popov ghosts ω i ( x ), antighosts ¯ ω i ( x ), the Lagrange multiplier b i ( x )and the BRST sources γ a γ iµ ( x ) , ζ α ζ i ( x ) , Ω a Ω iµ ( x ) , (3.3)we obtain the gauge-fixed action, Σ = Z d x (cid:20) g F iµν F iµν + b i D µ ( B ) ( A iµ − B iµ ) − α b i b i + D µ ( B ) ¯ ω i D ( A ) µ ω i + γ iµ D µ ( A ) ω i + 12 ζ i f ijk ω j ω k + Ω iµ D µ ( A ) ¯ ω i + Ω iµ γ iµ (cid:21) . (3.4)The constant g is the gauge coupling and α is the gauge-fixing parameter. The field strengthand covariant derivatives are defined in the standard way, F iµν = ∂ µ A iν − ∂ ν A iµ + f ijk A jµ A kν , (3.5a) D ( A ) µ ω i = ∂ µ ω i + f ijk A jµ ω k , (3.5b)and similarly for D ( A ) µ ¯ ω i . The B -covariant derivative D ( B ) µ ¯ ω i is given by an expressionanalogous to (3.2). Clearly, the action (3.4) is invariant under gauge rotations of all fieldsaccompanied by simultaneous gauge transformations of A iµ and B iµ : these are precisely thebackground-gauge transformations introduced in Sec. 2.2.An important property of the YM theory is renormalizability. Its key prerequisite arerestrictions imposed on divergences by power counting. The scaling transformations, x µ a − x µ , A iµ a A iµ , (3.6) Where no confusion is possible, we keep a condensed notation for space-time coordinates as x and theirdelta functions as δ ( x ). a is an arbitrary positive constant, leave the classical YM action invariant. We willsay that A iµ has scaling dimension (+1), whereas the dimension of x µ is ( − A iµ ] = [ B iµ ] = [ ω i ] = [¯ ω i ] = 1 , [ b i ] = [ γ iµ ] = [ ζ i ] = [ Ω iµ ] = 2 . (3.7)The textbook analysis of divergent Feynman diagrams shows that the scaling dimensions oflocal counterterms needed to cancel the divergences do not exceed 4. Comparing with theBRST form (2.33) and taking into account that the generalized BRST operator Q increasesthe scaling dimension by 1, we conclude that the dimensions of local operators entering intothe renormalized gauge fermion ˆ Ψ do not exceed 3. Recalling further that the ghost numberof ˆ Ψ is ( −
1) we write down the most general expression compatible with these requirements,ˆ Ψ = Z d x (cid:0) − ˆ γ iµ U iµ ( A, B ) + ζ i ω j V ij (cid:1) , (3.8)where V ij are dimensionless constants, while U iµ depends on A iµ and B iµ at most linearly. Weobserve that ˆ Ψ does not depend on Ω iµ and is linear in γ iµ and ζ i . As discussed in Sec. 2.4,this implies that the field redefinition needed to bring the counterterms into the BRST formis independent of the auxiliary BRST sources, see (2.39). Positive dimensions of the YMfield and ghosts further constrain this reparameterization to be linear. (3 + 1) dimensions The fields describing relativistic gravitational theories are identified as follows: ϕ a g µν ( x ) , ε α ε µ ( x ) , (3.9)where g µν ( x ) is the spacetime metric and ε µ ( x ) is a vector field generating infinitesimaldiffeomorphisms. The gauge transformations read, δ ε g µν = ε λ ∂ λ g µν + g µλ ∂ ν ε λ + g νλ ∂ µ ε λ = ∇ ( g ) µ ε ν + ∇ ( g ) ν ε µ , (3.10)where in the last equality we have lowered the indices using the metric g µν and introducedthe covariant derivative ∇ ( g ) constructed using this metric. The expression (3.10) implies, R abα δ ρµ (cid:2) δ σν (cid:0) ∂ λ δ ( x − x ) (cid:1) δ ( x − x ) + δ σλ δ ( x − x ) ∂ ν δ ( x − x ) (cid:3) + δ ρν δ σλ δ ( x − x ) ∂ µ δ ( x − x ) , (3.11a) P aα = 0 , C αβγ δ µλ δ ( x − x ) ∂ ν δ ( x − x ) − δ µν (cid:0) ∂ λ δ ( x − x ) (cid:1) δ ( x − x ) . (3.11b)15e focus on the theory in (3 + 1) dimensions including up to 4-th order derivatives of themetric. The classical action reads S = Z d x p | g | (cid:20) f R µν R µν + 1 f R − κ R + Λ κ (cid:21) , (3.12)where | g | is the determinant of the metric, R µν is the corresponding Ricci tensor and R ≡ R µν g µν is the Ricci scalar; f , f , κ and Λ are coupling constants. The quantum propertiesof this theory were first analyzed in [4]. The fact that the action contains fourth derivativesof the metric entails well-known problems with the physical interpretation of the theory [48].However, this issue is irrelevant for our purposes.Introducing the background metric g µν ( x ) we consider the gauge fixing function, χ α χ µ = g µλ g νρ (cid:3) ( g ) ∇ ( g ) ν ( g λρ − g λρ ) , (3.13)where ∇ ( g ) and (cid:3) ( g ) stand for the covariant derivatives and d’Alembertian constructed fromthe background metric . Introducing the fields of the BRST sector, ω α ω µ ( x ) , ¯ ω α ¯ ω µ ( x ) , b α b µ ( x ) , γ a γ µν ( x ) , ζ α ζ µ ( x ) , Ω a Ω µν ( x ) , (3.14)and the operator O αβ , O αβ
7→ − α g µν p | g | (cid:3) ( g ) δ ( x − x ) , (3.15)we arrive at the gauge-fixed action, Σ = S [ g µν ] + Z d x (cid:26) b µ χ µ + α b µ g µν p | g | (cid:3) ( g ) b ν + (cid:0) ∇ ( g ) ν ¯ ω µ (cid:1) g µλ g νρ (cid:3) ( g ) (cid:0) ∇ ( g ) λ ω ρ + ∇ ( g ) ρ ω λ (cid:1) + γ µν (cid:0) ∇ ( g ) µ ω ν + ∇ ( g ) ν ω µ (cid:1) + ζ µ ω λ ∂ λ ω µ + Ω µν γ µν + Ω µν ¯ ω λ (cid:20) δχ λ δ g µν + α δδ g µν (cid:18) g λρ p | g | (cid:3) ( g ) (cid:19) b ρ (cid:21)(cid:27) . (3.16)We have not expanded the variational derivatives in the last term as the correspondingexpressions are rather lengthy and not informative. The background-gauge transformationscorrespond to diffeomorphisms, x µ x µ + ε µ ( x ) , (3.17)under which g µν , g µν , ω µ , Ω µν transform as tensors, whereas ¯ ω µ , b µ , γ µν , ζ µ transform asvector/tensor densities. For example, δ ε ω µ = ε λ ∂ λ ω µ − ω λ ∂ λ ε µ , (3.18a) δ ε ¯ ω µ = ε λ ∂ λ ¯ ω µ + ¯ ω λ ∂ µ ε λ + ¯ ω µ ∂ λ ε λ , (3.18b)16nd similarly for the rest of the fields. It is straightforward to see that this is a symmetryof the action (3.16). The fact that b µ is a covariant vector density explains the unusualplacement of p | g | in the denominator of the operator (3.15).The four-derivative terms in the classical action (3.12) are invariant under rescaling x µ a − x µ , with the metric g µν kept intact. The same is true for the BRST-exact part of (3.16) if weassign the following scaling dimensions,[ g µν ] = [ g µν ] = [ ω µ ] = [¯ ω µ ] = 0 , [ b µ ] = [ Ω µν ] = 1 , [ γ µν ] = [ ζ µ ] = 3 . (3.19)As in the case of YM, it can be shown [4] that the power-counting restricts the scalingdimensions of counterterms in the Lagrangian to be less than or equal to 4. This againconstrains the dependence of the gauge fermion on the auxiliary fields. We observe thatthe BRST transformations increase the scaling dimension of all fields by 1. This impliesthat ˆ Ψ should contain local operators of dimensions not higher than 3. Besides, their ghostnumber must be equal to ( − Ψ = Z d x (cid:0) − ˆ γ µν U µν ( g, g ) + ζ µ ω ν V µν ( g, g ) (cid:1) , (3.20)where U µν , V µν are dimensionless functions of the quantum and background metric fieldsthat transform covariantly under background diffeomorphisms. We observe that, similarlyto YM, ˆ Ψ is linear in the BRST sources γ and ζ . However, since the scaling dimensionof both metrics is zero, the coefficients in (3.20) can depend nonlinearly on g µν and g µν .This implies that the field redefinition (2.39) required to restore the BRST structure of therenormalized action is genuinely nonlinear, cf. [4].It is worth noting that the original proof of renormalizability of the theory (3.12) pre-sented in [4] is tied to specific gauges where the structure of divergences is particularly simpledue to some special features of the action. For more general gauges, Ref. [4] took the co-homological structure of divergences as an assumption. Our results provide a proof of thisstructure for a general background gauge and, in this respect, complement the analysis of [4]. The choice of gauge (3.13) is important for the argument. It ensures that the propagators of the metricperturbations and ghosts fall off as the fourth power of momentum and as a consequence the degree ofdivergence of Feynman diagrams is consistent with the naive power counting. In this case it is due to the presence of derivatives acting on the transformed field, rather than thenon-zero dimension of ghosts as it happens for YM. .3 Projectable Hoˇrava gravity in ( d + 1) dimensions Consider a ( d + 1)-dimensional spacetime with Arnowitt–Deser–Misner (ADM) decomposi-tion of the metric, d s = N d t + g ij (d x i + N i d t )(d x j + N j d t ) , (3.21)where the indices i, j = 1 , . . . , d denote spatial directions ; they are raised and lowered usingthe spatial metric g ij . Let us impose the so-called “projectability” constraint that the lapse N is not dynamical and fix N = 1. This constraint is compatible with a subgroup of time-dependent diffeomorphisms along spatial directions. Thus, the fields and gauge parametersare identified as follows, ϕ a g ij ( t, x ) , N i ( t, x ) , ε α ε i ( t, x ) . (3.22)The gauge generators and the structure constants are given by the corresponding reductionof Eqs. (3.11). The classical action is taken in the form [19], S = 12 κ Z d t d d x p | g | (cid:0) K ij K ij − λK + V ( g ij ) (cid:1) , (3.23)where K ij = 12 ( ˙ g ij − ∇ ( g ) i N j − ∇ ( g ) j N i ) (3.24)is the extrinsic curvature on the constant-time slices and K ≡ K ij g ij is its trace. Heredot denotes derivative with respect to time and covariant derivatives ∇ ( g ) are constructedusing the spatial metric g ij ; κ and λ are coupling constants. The potential V contains alllocal terms invariant under spatial diffeomorphisms that can be constructed from the spatialmetric g ij using no more than 2 d spatial derivatives; generically, it is a finite polynomial of the d -dimensional Riemann tensor and its covariant derivatives. Clearly, the action (3.23) doesnot possess Lorentz symmetry. On the other hand, its highest-derivative part is invariantunder anisotropic (Lifshitz) scaling transformations, x a − x , t a − d t , (3.25)with the scaling dimensions of the fields,[ g ij ] = 0 , [ N i ] = d − . (3.26)Note that different components of the gauge fields (the components of the ADM metric(3.21) in this case) have different dimensions which is a common situation in theories withLifshitz scaling. We do not use color YM indices in this subsection, so there should be no confusion with the notationsof Sec. 3.1.
18 background-gauge fixing procedure compatible with the scaling symmetry (3.25) wasconstructed in [20]. We introduce the background fields g ij ( t, x ), N i ( t, x ) and the combina-tions h ij = g ij − g ij , n i = N i − N i . (3.27)Then the gauge-fixing function reads, χ α χ i = D t n i + α O ij g kl (cid:0) ∇ ( g ) k h lj − λ ∇ ( g ) j h kl (cid:1) , (3.28a)where D t n i = ˙ n i − N k ∇ ( g ) k n i + ∇ ( g ) k N i n k , (3.28b)and the operator O ij has the form , O ij = ( − d − ∇ k ( g ) . . . ∇ k d − ( g ) (cid:0) ∆ ( g ) g ij + ξ ∇ i ( g ) ∇ j ( g ) (cid:1) ∇ ( g ) k d − . . . ∇ ( g ) k . (3.28c)Here the covariant spatial Laplacian ∆ ( g ) and all covariant derivatives ∇ ( g ) are defined usingthe background metric g ij with their indices raised and lowered using the same metric; theconstants α and ξ are gauge parameters. This gauge fixing term satisfies all the requirementsformulated in Sec. 2.2: it is linear in the difference between the quantum and backgroundfields, and covariant under simultaneous gauge transformations of these fields.We now introduce the rest of objects entering in the BRST construction, ω α ω i ( t, x ) , ¯ ω α ¯ ω i ( t, x ) , b α b i ( t, x ) , O αβ α p | g | O ij δ ( t − t ′ ) δ ( x − x ′ ) , (3.29a) γ a (cid:8) γ ij ( t, x ) , γ i ( t, x ) (cid:9) , ζ α ζ i ( t, x ) , Ω a (cid:8) Ω ij ( t, x ) , Ω i ( t, x ) (cid:9) . (3.29b)The full gauge-fixed action is lengthy and we do not write it explicitly. Importantly, with anappropriate assignment of dimensions to the fields it is invariant under the scaling transfor-mations (3.25). By inspection of the gauge-fixing and the Faddeev–Popov ghost terms wefind, [ ω i ] = [¯ ω i ] = 0 , [ b i ] = 1 . (3.30a)The background fields inherit the dimensions from their dynamical counterparts,[ g ij ] = 0 , [ N i ] = d − . (3.30b)To determine the dimensions of the auxiliary fields γ ij , γ i , ζ i recall that they couple to theBRST variations s g ij , s N i , s ω i respectively. The latter have the same form as in relativisticgravity and thus contain one spatial derivative acting on the fields. This yields,[ s g ij ] = [ s ω i ] = 1 , [ s N i ] = d . (3.30c) Note that O ij corresponds to the operator denoted by O − ij in Ref. [20]. γ a s ϕ a , ζ α s ω α in the action requires[ γ i ] = d , [ γ ij ] = [ ζ i ] = 2 d − . (3.30d)Finally, the coupling Ω a γ a present in the action fixes the dimensions of Ω ij , Ω i ,[ Ω ij ] = 1 , [Ω i ] = d . (3.30e)The results of [20] imply that in this theory the ultraviolet divergences consist of localoperators with scaling dimensions not higher than 2 d . Thus, all assumptions of Sec. 2 aresatisfied and according to Sec. 2.4 the renormalizad action has the form (2.33). As the BRSTtransformations increase the dimensionality of the fields by unity, the renormalized gaugefermion ˆ Ψ appearing in (2.33) contains operators with dimensions less or equal to (2 d − −
1) reads,ˆ Ψ = Z d t d d x (cid:0) − ˆ γ ij U ij ( γ, g ) − ˆ γ i U i ( g, N, g , N ) + ζ i ω j V ij ( g, g ) (cid:1) , (3.31)where U ij , V ij are dimensionless functions of g ij , g ij , whereas U i can also linearly depend on N i , N i . Once more we observe that ˆ Ψ is independent of Ω ij , Ω i and depends linearly on therest of BRST sources.Establishing the BRST structure of counterterms in the projectable Hoˇrava gravity to-gether with the results of Ref. [20] completes the proof of renormalizability of this theory. (3 + 1) dimensions As an example of a non-renormalizable theory we consider Einstein’s general relativity in(3 + 1) dimensions. The field content and gauge transformations are the same as in Sec. 3.2.What differs is the structure of the classical action which now reads, S = 12 κ Z d x p | g | (2Λ − R + . . . ) , (3.32)where dots stand for an infinite sum of various local scalar operators constructed from theRiemann tensor and its derivatives. They can be ordered according to the total numberof derivatives n they contain . At each fixed order, the number of possible terms is finite(though it grows quickly with n ). In the spirit of effective field theory, the higher derivativecontributions are treated as corrections to the terms explicitly shown in (3.32). In particular,the graviton propagator is determined from the Einstein–Hilbert part and falls off as p − atlarge momenta p . Thus, the terms R µν R µν and R contain 4 derivatives ( n = 4), R µνλρ R λρστ R µνστ contains 6 derivatives( n = 6), etc. χ α χ µ = g µλ g νρ ∇ ( g ) ν ( g λρ − g λρ ) , (3.33)where g µν is the background metric. The rest of the gauge fixing construction proceeds incomplete analogy with Sec. 3.2. In the present case there are no power-counting argumentsconstraining the dependence of divergences on auxiliary fields. Still, the proposition for-mulated in Sec. 2.4 ensures that they are compatible with the BRST structure. The fieldrenormalization required to recover this structure is expected to have the general form (2.30)and involve ghosts and auxiliary fields in a nonlinear manner. We now derive the equations obeyed by the effective action Γ L defined in (2.25) correspondingto the generating functional of the form (2.32). We will omit the loop index L in this section.As shown in Appendix A, the closure of Σ under the action of the extended BRSToperator, Q Σ = 0, together with the absence of anomalies, implies the Slavnov–Tayloridentity for the partition function, (cid:20) − J a δδγ a + ¯ ξ α δδζ α + ξ α δδy α + Ω a δδφ a (cid:21) W = 0 . (4.1)Whereas the invariance of Σ and Ψ under background gauge transformations leads to theWard identities, (cid:20) − J a R abα δδJ b + C γβα ¯ ξ γ δδ ¯ ξ β − C βγα ξ γ δδξ β − C βγα y γ δδy β + R aα ( φ ) δδφ a − γ b R baα δδγ a + C βγα ζ β δδζ γ + R abα Ω b δδΩ a (cid:21) W = 0 . (4.2)Besides, the equations of motion for the Lagrange multiplier b α imply, (cid:20) χ αa δδJ a − O αβ δδy β − Ω a δO αβ δφ a δδξ β + y α (cid:21) W = 0 . (4.3)Let us stress that the derivation of Eqs. (4.2), (4.3) essentially relies on the property thatthe gauge generators and the gauge-fixing condition are linear in the quantum field.Turning to the effective action, we use the relations (2.24), (2.26) and rewrite the identities(4.1), (4.2) and (4.3) in the following form , δΓδγ a δΓδϕ a + δΓδζ α δΓδω α + b α δΓδ ¯ ω α + Ω a δΓδφ a = 0 , (4.4a) Recall that we omit averaging symbols on the mean fields. aα ( ϕ ) δΓδϕ a − C γβα ω β δΓδω γ + ¯ ω β C βγα δΓδ ¯ ω γ + b β C βγα δΓδb γ + R aα ( φ ) δΓδφ a − γ b R baα δΓδγ a + ζ β C βγα δΓδζ γ + R abα Ω b δΓδΩ a = 0 , (4.4b) χ αa ( ϕ a − φ a ) − O αβ b β − Ω a δO αβ δφ a ¯ ω β − δΓδb α = 0 . (4.4c)It is convenient to consider a reduced effective action ˆ Γ obtained from Γ by subtracting thegauge-fixing term and its derivatives with respect to the background fields,ˆ Γ = Γ − b α (cid:18) χ αa ( ϕ − φ ) a − O αβ b β (cid:19) − Ω a ¯ ω α (cid:18) δχ αb δφ a ( ϕ − φ ) b − χ αa − δO αβ δφ a b β (cid:19) . (4.5)Substituting this expression into (4.4c) yields that ˆ Γ is independent of b α , δ ˆ Γδb α = 0 . (4.6)Then the identity (4.4a) splits into two equations, χ αa δ ˆ Γδγ a + δ ˆ Γδ ¯ ω α = 0 , (4.7a) δ ˆ Γδγ a δ ˆ Γδϕ a + δ ˆ Γδζ α δ ˆ Γδω α + Ω a (cid:18) δ ˆ Γδφ a + ¯ ω α δχ αb δφ a δ ˆ Γδγ b (cid:19) = 0 . (4.7b)The first one implies that ˆ Γ depends on the antighost only through the combination (2.31),so that ˆ Γ = ˆ Γ [ ϕ, ω, φ, ˆ γ, ζ , Ω ] . (4.8)Next, we use the relation δδφ a (cid:12)(cid:12)(cid:12)(cid:12) ˆ γ = δδφ a (cid:12)(cid:12)(cid:12)(cid:12) γ + ¯ ω α δχ αb δφ a δδγ b , (4.9)where the index on the right of the vertical line means that the φ -derivative is taken at fixedˆ γ or γ . Consequently, Eq. (4.7b) takes the form, δ ˆ Γδ ˆ γ a δ ˆ Γδϕ a + δ ˆ Γδζ α δ ˆ Γδω α + Ω a δ ˆ Γδφ a = 0 . (4.10a)The Ward identities (4.4b) also simplify to, R aα ( ϕ ) δ ˆ Γδϕ a − C γβα ω β δ ˆ Γδω γ + R aα ( φ ) δ ˆ Γδφ a − ˆ γ b R baα δ ˆ Γδ ˆ γ a + ζ β C βγα δ ˆ Γδζ γ + R abα Ω b δ ˆ ΓδΩ a = 0 . (4.10b)Finally, ˆ Γ has zero ghost number, i.e. it is invariant under phase rotations of the fields ω ,ˆ γ , ζ , Ω with charges (2.21). Together with Eqs. (4.10) this will be used in the next sectionto constrain the structure of ultraviolet divergences.22learly, the identities (4.10) are satisfied by the reduced tree-level action ˆ Σ , which isrelated to (2.17) by a formula analogous to (4.5). Explicitly, we have,ˆ Σ = S [ ϕ ] + ˆ γ a R aα ( ϕ ) ω α + 12 ζ α C αβγ ω β ω γ . (4.11)Note that ˆ Σ does not have any explicit dependence on Ω a and φ a . Consequently, thelast term in (4.10a) and the third term in (4.10b) are absent in the corresponding identitiesfor ˆ Σ . We return to the renormalization procedure. Let us assume that at the order of ( L − W L − has the form(2.32)—(2.36). The first divergence of the effective action Γ L − appears at order ~ L and islocal, see Eqs. (2.27), (2.28). The standard procedure prescribes to subtract it from Σ L − inorder to obtain the action renormalized at L loops. Our task is to work out the structure ofthis divergence. To avoid cluttered notations we will omit the indices related to the inductionstep and will denote the relevant divergent part Γ L, ∞ simply as Γ ∞ .First, we observe that the transformation (4.5) involves only finite quantities, so that thedivergent parts of Γ and ˆ Γ coincide, Γ ∞ = ˆ Γ ∞ [ ϕ, ω, φ, ˆ γ, ζ , Ω ] . (5.1)Due to the linearity of the Ward identities (4.10b), they are obeyed separately by eachterm in the expansion of ˆ Γ in ~ ; in particular, they hold for the divergent part ˆ Γ ∞ . Next,we consider Eq. (4.10a). The first divergent contribution into it appears at the order ~ L .Equation (4.10a) at this order then implies Q + ˆ Γ ∞ = 0 , (5.2)where we have introduced an operator Q + that acts on a functional X of the fields ϕ , ω , φ ,ˆ γ , ζ , Ω as follows, Q + X = δ ˆ Σ δ ˆ γ a δXδϕ a + δ ˆ Σ δϕ a δXδ ˆ γ a + δ ˆ Σ δζ α δXδω α + δ ˆ Σ δω α δXδζ α + Ω a δXδφ a ≡ ( ˆ Σ , X ) + Ω a δXδφ a . (5.3) Only an implicit dependence of ˆ Σ on φ a through the combination (2.31) remains. Σ is the reduced tree-level action (4.11) and in the second line we defined the an-tibracket ( ˆ Σ , X ). A straightforward calculation using the structural relations (2.4), (2.5)shows that the latter is nilpotent, ( ˆ Σ , ( ˆ Σ , X )) = 0 , (5.4a)and anticommutes with the operator Ω δ/δφ ,( ˆ Σ , Ω a δXδφ a ) = − Ω a δδφ a ( ˆ Σ , X ) . (5.4b)The properties (5.4) imply nilpotency of Q + . Note that using the explicit form of Σ andthe BRST transformations (2.13), Q + can be written as Q + X = ( s ϕ a ) δXδϕ a + ( s ω α ) δXδω α + Ω a δXδφ a (cid:12)(cid:12)(cid:12)(cid:12) ˆ γ + δ ˆ Σ δϕ a δXδ ˆ γ a + δ ˆ Σ δω α δXδζ α . (5.5)The first three terms here resemble the action of the operator Q introduced in Sec. 2.2.However, there are a few differences. Q is defined on functionals of all quantum fields ϕ, ω, ¯ ω, b and external backgrounds φ, γ, ζ , Ω . On the other hand, Q + acts on functionalsthat are restricted to the minimal sector of quantum fields ϕ, ω and, instead of γ , dependon the combination ˆ γ (see (2.31)) treated as a free variable.We now use Eq. (5.2) to determine the dependence of ˆ Γ ∞ on the background fields φ a . We expand ˆ Γ ∞ in powers of the auxiliary source Ω ,ˆ Γ ∞ = X k ˆ Γ ∞ , { k } , ˆ Γ ∞ , { k } = Ω a . . . Ω a k ˆ Γ ∞ , { k } , [ a ,...,a k ] [ ϕ, ω, φ, ˆ γ, ζ ] . (5.6)We assume that this sum is finite, k ≤ K , which will be justified shortly. Substituting (5.6)into (5.2) we obtain Ω a δ ˆ Γ ∞ , { K } δφ a = 0 , (5.7a) Ω a δ ˆ Γ ∞ , { k } δφ a + ( ˆ Σ , ˆ Γ ∞ , { k +1 } ) = 0 , ≤ k ≤ K − . (5.7b)As shown in Appendix B, the cohomology of the operator Ωδ/δφ on the space of localfunctionals vanishing at Ω = 0 is trivial. In other words, Eq. (5.7a) implies that ˆ Γ ∞ , { K } isrepresented as ˆ Γ ∞ , { K } = Ω a δδφ a Υ { K − } , (5.8)24here Υ { K − } is a local functional of ghost number ( −
1) invariant under background-gaugetransformations. Inserting this representation into (5.7b) for k = K − Ω a δδφ a (cid:0) ˆ Γ ∞ , { K − } − ( ˆ Σ , Υ { K − } ) (cid:1) = 0 , (5.9)where we have used the property (5.4b). Again, this impliesˆ Γ ∞ , { K − } = ( ˆ Σ , Υ { K − } ) + Ω a δδφ a Υ { K − } . (5.10)By continuing this reasoning and using the properties (5.4) we obtain a representation of thetype (5.10) for all ˆ Γ ∞ , { k } , 1 ≤ k ≤ K −
1. For k = 0 an additional contribution appears,ˆ Γ ∞ , { } = ( ˆ Σ , Υ { } ) + Γ , (5.11)where Γ [ ϕ, ω, ˆ γ, ζ ] is independent of Ω and the background field φ . Collecting all contribu-tions together we arrive atˆ Γ ∞ = Γ [ ϕ, ω, ˆ γ, ζ ] + K − X k =0 ( ˆ Σ , Υ { k } ) + K X k =1 Ω a δδφ a Υ { k − } = Γ [ ϕ, ω, ˆ γ, ζ ] + Q + Υ , (5.12)where in the second line we have defined Υ [ ϕ, ω, φ, ˆ γ, ζ , Ω ] = K − X k =0 Υ { k } . (5.13)We can now appreciate the power of the background-field approach. The pieces dependenton the background fields have separated into a Q + -exact contribution leaving behind the part Γ that depends only on the quantum fields. The original invariance under background-gaugetransformations implies that Γ is gauge-invariant on its own. More precisely, we write Γ = S [ ϕ ] + Λ [ ϕ, ω, ˆ γ, ζ ] , (5.14)where Λ vanishes at ω = 0. The ghost-independent part S [ ϕ ] cannot depend on ˆ γ or ζ asthe latter have negative ghost charges (see (2.21)), whereas the ghost number of Γ is zero.Then, due to the Ward identities (4.10b), the local functional S [ ϕ ] satisfies δ S δϕ a R aα ( ϕ ) = 0 . (5.15)We will see in Sec. 5.3 that in the subtraction procedure it combines with the classical action S [ ϕ ] and corresponds to the renormalization of the couplings in the classical gauge invariant25agrangian. The rest of the terms in (5.12), (5.14) generates a renormalization of the gaugefermion and the corresponding field redefinition.We still have to justify the assumption that the sum (5.6) can be truncated at finite k . We do it using the notion of derivative expansion. Being local, the functional ˆ Γ ∞ isa spacetime integral of a Lagrangian which can be written as a series of terms, each ofthem containing a finite number of derivatives. Let us introduce a formal book-keepingparameter l ∗ of dimension of length counting the number of derivatives in a given term, andconvert the derivative expansion into a Taylor series in l ∗ . We denote by ˆ Γ N ∞ the part of ˆ Γ ∞ containing all terms of order l n ∗ , n ≤ N , i.e. all terms with up to N derivatives. Now, Ω is ananticommuting local field. With a finite number of derivatives at disposal, one can constructonly a finite number of local operators out of it. Therefore ˆ Γ N ∞ is a finite polynomial in Ω .Next, we observe that ˆ Σ is also a local functional and hence contains derivatives in non-negative powers. Thus, it is represented as a series with non-negative powers of l ∗ , so thatthe antibracket ( ˆ Σ , ... ) acting on a given operator cannot decrease its order in l ∗ . Besides,the operator Ωδ/δφ does not contain l ∗ at all. We conclude that ˆ Γ N ∞ satisfies Eq. (5.2), upto corrections of order l N +1 ∗ , Q + ˆ Γ N ∞ = O ( l N +1 ∗ ) . (5.16)Splitting ˆ Γ N ∞ into monomials in Ω one can repeat the derivation leading to (5.12), up tocorrections of order O ( l N +1 ∗ ) on the r.h.s. As this representation holds for any N , we cansend the latter to infinity and recover (5.12) for the full divergent part ˆ Γ ∞ without anycorrections. It remains to fix the structure of the term Λ in (5.14). It satisfies the equation,( ˆ Σ , Λ ) = 0 . (5.17) In theories with Lifshitz scaling it would be natural to assign different weights to derivatives alongdifferent spacetime directions, cf. Sec. 3.3. However, the argument presented below does not depend onwhether one introduces such weighting or not, so for simplicity we treat all derivatives on equal footing. Note that its highest power is not directly related to N and can depend on the specifics of the theorysuch as number of internal indices and spacetime dimensions, power-counting considerations, etc. The onlyproperty which is important for us here is that this power is finite. In renormalizable theories with finite number of coupling constants the derivative expansion usuallyterminates at a finite order in N . R aα ( ϕ ) ω α δΛδϕ a + (cid:18) δSδϕ a + ˆ γ b R baα ω α (cid:19) δΛδ ˆ γ a + 12 C αβγ ω β ω γ δΛδω α + (cid:16) − ˆ γ a R aα ( ϕ ) + ζ β C βαγ ω γ (cid:17) δΛδζ α = 0 . (5.18)Besides, the invariance of Λ with respect to background-gauge transformations implies theWard identities (cf. (4.10b)), R aα ( ϕ ) δΛδϕ a − C γβα ω β δΛδω γ − ˆ γ b R baα δΛδ ˆ γ a + ζ β C βγα δΛδζ γ = 0 . (5.19)Multiplying the latter expression by ω α and subtracting it from (5.18), we arrive at theequation ( q + q ) Λ = 0 , (5.20)where the operators q , are defined as q Λ = δSδϕ a δΛδ ˆ γ a − ˆ γ a R aα ( ϕ ) δΛδζ α , (5.21a) q Λ = − C γαβ ω α ω β δΛδω γ . (5.21b)Both operators are nilpotent and anticommute with each other,( q ) = ( q ) = q q + q q = 0 . (5.22)The operator q is known in the mathematical literature as Koszul–Tate differential [45].Let us expand Λ in powers of the ghost fields ω α , Λ = ∞ X k =1 Λ { k } , Λ { k } = ω α . . . ω α k Λ { k } [ α ,...,α k ] [ ϕ, ˆ γ, ζ ] . (5.23)Note that the sum starts at k = 1 as, by definition, Λ vanishes at ω = 0. The conservation ofthe ghost number and the ghost charges (2.21) imply that each term Λ { k } in the expansionis a finite polynomial in ˆ γ and ζ that vanishes at ˆ γ = ζ = 0. Thus Λ { k } satisfies, Λ { k } (cid:12)(cid:12) ω =0 = Λ { k } (cid:12)(cid:12) ˆ γ = ζ =0 = 0 . (5.24)We now substitute (5.23) into (5.20) and obtain a chain of equations, q Λ { } = 0 , (5.25a) q Λ { k } + q Λ { k − } = 0 , k ≥ . (5.25b)27he Koszul–Tate differential q has trivial cohomology on functionals satisfying (5.24) if thegauge algebra obeys the conditions (i), (ii) from Sec. 2.1 [41]: q X = 0 , X (cid:12)(cid:12) ω =0 = X (cid:12)(cid:12) ˆ γ = ζ =0 = 0 = ⇒ X = q Y . (5.26)Moreover, under natural assumptions about the regularity of the equations of motion, thefunctional Y can be chosen to be local [45, 49], provided X itself is local. Finally, one canshow along the lines of [49] that there exists a choice of Y which inherits all linearly realizedsymmetries commuting with q . In particular, we can take Y [ ϕ, ω, ˆ γ, ζ ] to be invariant underbackground-gauge transformations if so is X .Thus we write, Λ { } = q Ξ { } , (5.27)where Ξ { } is local and background-gauge invariant. Substituting this into Eq. (5.25b) for k = 2 and interchanging the order of q and q we obtain, q (cid:0) Λ { } − q Ξ { } (cid:1) = 0 , (5.28)whence Λ { } = q Ξ { } + q Ξ { } . (5.29)Continuing by induction, we obtain analogous representations for all Λ { k } . Collected togetherthey give, Λ = ( q + q ) Ξ , Ξ = ∞ X k =1 Ξ { k } . (5.30)To make the last step, we notice that Ξ , due its invariance under background-gauge trans-formations, obeys a Ward identity analogous to (5.19). Combining this with (5.30) we get, Λ = ( ˆ Σ , Ξ ) . (5.31)This is our final expression for Λ .Putting together the contributions (5.12), (5.14), (5.31) and reintroducing the loop index L , we obtain the desired form of the L -loop divergence Γ L, ∞ = S L [ ϕ ] + Q + Υ L , (5.32)where Υ L = Υ L + Ξ L and we have used that Ξ L is independent of φ .28 .3 Subtraction and field redefinition We now define the L -th order renormalized action as (compare with (2.29)), Σ L [ ϕ, ω, ¯ ω, b, φ, γ, ζ , Ω ] = Σ L − − ~ L Γ L, ∞ [ ϕ, ω, φ, ˆ γ, ζ , Ω ] + O ( ~ L +1 ) , (5.33)where the last term on the r.h.s. stands for local operators multiplied by at least ~ L +1 thatwill be specified shortly. The presence of these operators does not spoil the key propertyof the subtraction prescription, namely that it removes all subdivergences at ( L + 1)-looporder. Thus, according to the assumption stated in Sec. 2.3, the ( L + 1)-loop divergence willbe local.We now show that Σ L can be brought to the form (2.33) by a reparameterization of thefields ϕ , ω . Substituting the expression (5.32) in (5.33) and expanding explicitly the operator Q + we obtain, Σ L = Σ + L − X l =1 ~ l Σ Cl − ~ L S L − ~ L δ Υ L δ ˆ γ a δ ˆ Σ δϕ a + ~ L δ Υ L δζ α δ ˆ Σ δω α − ~ L δ ˆ Σ δ ˆ γ a δ Υ L δϕ a − ~ L δ ˆ Σ δζ α δ Υ L δω α − ~ L Ω a δ Υ L δφ a (cid:12)(cid:12)(cid:12)(cid:12) ˆ γ + O ( ~ L +1 ) . (5.34)As before, the index ˆ γ on the partial derivative with respect to the background field in thelast significant term emphasizes that it is taken at fixed ˆ γ . The first two terms in the lastline have the form, − ~ L s ϕ a δ Υ L δϕ a − ~ L s ω α δ Υ L δω α . (5.35)This suggests to define the L -th order gauge fermion, Ψ L = Ψ L − − ~ L Υ L , (5.36a)and the L -th order counterterm Σ CL = − S L [ ϕ ] − sΥ L − Ω a δ Υ L δφ a (cid:12)(cid:12)(cid:12)(cid:12) γ = − S L [ ϕ ] − Q Υ L . (5.36b)To proceed, we notice that the expressions (2.17) and (4.11) imply δΣ δϕ a = δ ˆ Σ δϕ a + b α χ αa + Ω b ¯ ω α δχ αa δφ b . (5.37a) Strictly speaking, according to the standard scheme one should take φ a − δΨ L − /δγ a and δΨ L − /δζ α instead of ϕ a and ω α as arguments of Γ L, ∞ . However, due to the representation (2.35) valid for Ψ L − , thedifference produced by this replacement is of higher order in ~ . It is included in the O ( ~ L +1 ) term in (5.33). sΥ L = s ϕ a δ Υ L δϕ a + s ω α δ Υ L δω α − b α χ αa δ Υ L δ ˆ γ a . (5.37b)Finally, the φ -derivatives at fixed γ and ˆ γ are related by (4.9). Collecting all the previousexpressions together, we find that Eq. (5.34) simplifies to Σ L = Σ + L X l =1 ~ l Σ Cl − ~ L δ Υ L δγ a δΣ δϕ a + ~ L δ Υ L δζ α δΣ δω α + O ( ~ L +1 ) . (5.38)The first two terms already have the desired BRST form (2.33), Σ + L X l =1 ~ l Σ Cl = S [ ϕ ] − L X l =1 ~ l S L [ ϕ ] + Q Ψ L . (5.39)The remaining contributions are absorbed by a field redefinition, as we now demonstrate.First we perform the change of variables ϕ, ω ϕ ′ , ω ′ given by ϕ a = ϕ ′ a + ~ L δ Υ L δγ a ( ϕ ′ , ω ′ , . . . ) + O ( ~ L +1 ) , (5.40a) ω α = ω ′ α − ~ L δ Υ L δζ α ( ϕ ′ , ω ′ , . . . ) + O ( ~ L +1 ) , (5.40b)where we again allow for possible local contributions of higher order in ~ . Next, we Taylorexpands all quantities in the differences ( ϕ − ϕ ′ ), ( ω − ω ′ ). Then, the third and fourth termsin (5.38) are cancelled by the linear contribution in the series for Σ . Other terms generatedby the expansion are of higher powers in ~ . Notice that they are local. Thus, by properlyadjusting the O ( ~ L +1 ) contribution in (5.38) they can be cancelled as well.To complete the argument we need to verify that the operators coupled to sources in thepath integral have the right form (2.36) in terms of the new variables. This is done throughthe following chain of relations,˜ ϕ aL − ( ϕ, ω, . . . ) − φ a = − δΨ L − δγ a ( ϕ, ω, . . . )= ϕ a − φ a + L − X l =1 ~ l δ Υ l δγ a ( ϕ, ω, . . . )= ϕ ′ a − φ a + L − X l =1 ~ l δ Υ l δγ a ( ϕ, ω, . . . ) + ~ L δ Υ L δγ a ( ϕ ′ , ω ′ , . . . ) + O ( ~ L +1 )= ϕ ′ a − φ a + L X l =1 ~ l δ Υ l δγ a ( ϕ ′ , ω ′ , . . . )= − δΨ L δγ a ( ϕ ′ , ω ′ , . . . ) = ˜ ϕ aL ( ϕ ′ , ω ′ , . . . ) − φ a , (5.41)30here in passing to the fourth line we have assumed that the O ( ~ L +1 ) terms in (5.40a)are adjusted to absorb the (local) contributions produced by the change of variables in Υ l ,1 ≤ l ≤ L −
1. Exactly the same reasoning applies to δΨ L − /δζ α .In the last step, we erase primes on the new variables. Thus, we have found the choiceof variables in the path integral, such that Eqs. (2.32)—(2.36) are satisfied at the L -th looporder. This statement extends to all loops by induction. This completes the proof of theproposition formulated in Sec. 2.4 and is the main result of this work. (cid:4) O ( N ) model: Explicit one-loop calculation As an illustration of the above formalism we study one-loop counterterms in the (1+1)-dimensional O ( N )-invariant sigma model. In particular, we will see the necessity of a non-linear field renormalization to restore the BRST structure. We start with the action, S = 12 g Z d x ∂ µ n i ∂ µ n i , (6.1)where i = 1 , . . . , N ; g is the coupling constant and the scalar fields n i ( x ) are subject to theconstraint, n ≡ δ ij n i n j = 1 . (6.2)The latter can be solved by expressing n i = ϕ i p ϕ , (6.3)where the fields ϕ i ( x ) are unconstrained. The price to pay is the appearance of a gaugesymmetry corresponding to the pointwise rescaling of ϕ i , δ ε ϕ i ( x ) = ϕ i ( x ) ε ( x ) , (6.4)where ε ( x ) is an arbitrary function. Clearly, the transformation (6.4) leaves n i ( x ), and hencethe action, invariant. In terms of ϕ i the action reads, S [ ϕ ] = 12 g Z d x (cid:26) ϕ (cid:20) δ ij − ϕ i ϕ j ϕ (cid:21) ∂ µ ϕ i ∂ µ ϕ j (cid:27) . (6.5)The gauge generator is linear in the fields, R abα δ ij δ ( x − x ) δ ( x − x ) , P aα = 0 , (6.6)31o this model belongs to the class of theories subject to our renormalization procedure. Forthe sake of convenience we set the coupling constant g to one in what follows.The local background-covariant gauge condition χ α ( ϕ, φ ), the gauge fixing matrix O αβ ( φ )and its (nonlocal) inverse can be conveniently chosen in the form χ α ( ϕ, φ ) χ = (cid:3) (cid:18) φ i ( x ) φ ( x ) (cid:0) ϕ i ( x ) − φ i ( x ) (cid:1)(cid:19) = (cid:3) (cid:18) φ ( x ) · ϕ ( x ) φ ( x ) (cid:19) , (6.7a) O αβ ( φ ) O ( x, x ′ ) = − (cid:3) δ ( x − x ′ ) , O − αβ ( φ ) O − ( x, x ′ ) = − (cid:3) δ ( x − x ′ ) , (6.7b)where we have introduced the notation for the O ( N )-invariant scalar product, A · B = δ ij A i B i ≡ A i B i . (6.8)The corresponding anticommuting ghost ω α and antighost ¯ ω α , as well as the Lagrange mul-tiplier b α , are scalars with respect to the (1 + 1)-dimensional Lorentz transformations anddo not carry any O ( N ) indices, ω α ω ( x ), ¯ ω α ¯ ω ( x ), b α b ( x ). The theory is Abelian, C αβγ = 0, so that the BRST transform of the ghost field ω ( x ) vanishes and the source ζ α does not appear in the gauge-fixed action. Nevertheless, we have to keep the source ζ α inthe gauge fermion to fulfil the requirement (2.35). Therefore, the tree level reduced gaugefermion equals ˆ Ψ = − ˆ γ a ( ϕ a − φ a ) + ζ α ω α = Z d x (cid:0) − ˆ γ i ( ϕ i − φ i ) + ζ ω (cid:1) , (6.9a)ˆ γ i = γ i − φ i φ (cid:3) ¯ ω . (6.9b)The background field independent choice (6.7b) of O considerably simplifies the form ofthe BRST action (2.17) and moreover simplifies the result of integrating over the Lagrangemultiplier b α . The effect of this integration is the replacement of the b α -dependent termsby the gauge breaking term quadratic in the gauge condition, after which the BRST action(2.17) takes the form (in condensed notations) Σ [ ϕ, ω, ¯ ω, φ, γ, Ω ] = S [ ϕ ] + 12 χ α ( ϕ, φ ) O − αβ χ β ( ϕ, φ ) − ¯ ω α χ αa ( φ ) R aβ ( ϕ ) ω β + γ a R aα ( ϕ ) ω α + Ω a ¯ ω α δχ α ( ϕ, φ ) δφ a + Ω a γ a . (6.10)Explicitly, the previous action reads Σ [ ϕ, ω, ¯ ω, φ, γ, Ω ] = Z d x ( G ij ∂ µ ϕ i ∂ µ ϕ j − ϕ · φφ (cid:3) (cid:18) ϕ · φφ (cid:19) − ϕ · φφ ( (cid:3) ¯ ω ) ω + ( γ · ϕ ) ω + (cid:18) Ω · ϕφ − ϕ · φ ) ( Ω · φ )( φ ) (cid:19) (cid:3) ¯ ω + Ω · γ ) . (6.11) We disregard the one-loop functional determinant (Det O ) − / originating from this integration, becauseit is a trivial field-independent normalization constant. G ij denotes the metric of the target manifold, G ij = P ij ϕ , P ij = δ ij − ϕ i ϕ j ϕ , (6.12)and P ij is a projector along the directions orthogonal to ϕ j . All terms in the Lagrangianhave mass dimension 2 if the dimensions of the fields are chosen as,[ ϕ ] = [ φ ] = [ ω ] = [¯ ω ] = [ Ω ] = 0 , [ γ ] = 2 . (6.13)The theory is renormalizable, hence all divergences also have dimension 2. This implies thatthe renormalized gauge fermion ˆ Ψ should remain linear in ˆ γ i and independent of Ω i , as inother renormalizable examples encountered in Sec. 3. On the other hand, due to the zeromass dimension of the gauge fields, we expect that it will have nonlinear dependence on ϕ i and φ i . These expectations are confirmed below by an explicit calculation.The one-loop effective action of the model is given by the functional supertrace, Γ = 12 STr (cid:0) log F IJ (cid:1) , (6.14)where F IJ is the inverse propagator of the theory. The latter is given by the second ordermixed (left and right) functional derivatives of the action with respect to the full set ofboson-fermion fields of the theory Φ I ( x ) = ( ϕ i ( x ) , ω ( x ) , ¯ ω ( x )) F IJ δ ( x − x ′ ) = → δδΦ I ( x ) Σ [ ϕ, ω, ¯ ω, φ, γ, Ω ] ← δδΦ J ( x ′ ) . (6.15)This second order differential operator acting in the space of perturbations of the fields δΦ J has the form, F IJ = D IJ (cid:3) + 2 Γ µIJ ∂ µ + Π IJ . (6.16)The expressions for the matrix valued coefficients D IJ , Γ µIJ and Π IJ are given in Appendix C.The divergent part of (6.14) for a general operator of the form (6.16) is easily obtainedby the heat kernel method as a local functional of the operator coefficients [21, 23, 37]. First,the inverse propagator is converted into the form of a covariant d’Alembertian, F IJ = − ( D µ D µ ) IJ + P IJ , (6.17)built in terms of covariant derivatives D µ with some generic connection Γ µ = Γ Iµ J . Thesecovariant derivatives act in the linear space of fields Φ = Φ I ( x ) and field matrices X = X IJ ( x )as D µ Φ = ∂ µ Φ + Γ µ Φ, D µ X = ∂ µ X + [ Γ µ , X ] . (6.18) Recall that ˆ Ψ has ghost number ( − ζ in ˆ Ψ does not get renormalized since ζ does not appear in the action.
33n the case of a (1+1)-dimensional flat spacetime the one-loop divergence takes a particularlysimple form: it depends only on the potential term P of this operator12 STr log F (cid:12)(cid:12) ∞ = 14 π (2 − d ) Z d x str P . (6.19)Here str is the matrix supertrace over indices I ,str P = X I ( − ǫ I P II , (6.20)where ǫ I = 0 , I . We useddimensional regularization to capture the divergence in the limit d → F IJ = − D IK F KJ . Then F = − (cid:0) (cid:3) + 2 Γ µ ∂ µ + Π (cid:1) , ( Γ µ ) IJ = D IK Γ µKJ , Π IJ = D IK Π KJ , (6.21)where D IK is the inverse of the matrix D IJ , D IK D KJ = δ IJ . Next, the first-order derivativeterm of (6.21) is absorbed into the covariant derivative (6.18) with the connection Γ µ . Asa result, the operator (6.21) takes the form (6.17) with P = − Π + ∂ µ Γ µ + Γ µ Γ µ , so thatfinally the one-loop divergence reads Γ , ∞ = − π (2 − d ) Z d x str ( Π − Γ µ Γ µ ) , (6.22)where we have dropped the total derivative term ∂ µ (str Γ µ ). The matrices Γ µ and Π areevaluated in Appendix C. Substituting the corresponding expressions into (6.22) we obtain, ~ Γ , ∞ = − ~ π (2 − d ) Z d x ( N − G ij ∂ µ ϕ i ∂ µ ϕ j + ( φ ) ( ϕ · φ ) ( ϕ · ˆ γ ) ω + (cid:18) δ ij ϕ − φ i ϕ j ( ϕ · φ ) ϕ + φ i φ j ( ϕ · φ ) (cid:19) ∂ µ ϕ i ∂ µ ϕ j − (cid:18) δ ij ( ϕ · φ ) − φ i ϕ j ( ϕ · φ ) (cid:19) ∂ µ ϕ i ∂ µ φ j − (cid:20) ϕ ( ϕ · φ ) δ ik − φ ( ϕ · φ ) ϕ i φ k − ϕ ( ϕ · φ ) (cid:0) φ i ϕ k + ϕ i φ k (cid:1) + φ ( ϕ + φ )( ϕ · φ ) ϕ i ϕ k (cid:21) Ω k ˆ γ i ) . (6.23)If we set φ i = ϕ i , Ω i = γ i = ω = 0, only the first term in this expression will survivecorresponding to the well-known expression for the 1-loop divergence in the O ( N )-model(see e.g. [50]). We also disregard the ultralocal contribution of the transition from F IJ to F ,STr log F IJ = STr log F + STr log( − D IJ ) = STr log F + δ (0)( ... ) , which might be canceled by an appropriate local contribution of the measure in the path integral and anywayvanishes in dimensional regularization. Ω k and ˆ γ i . According toEq. (5.32), they originate from the action of the operator Ωδ/δφ on the one-loop ( L = 1)quantum dressing Υ of the gauge fermion in (5.36a). Clearly, we are in the situation whenthis dressing is independent of Ω and linear in ˆ γ , Υ = ˆ γ a u a ( ϕ, φ ) . Therefore, the terms bilinear in Ω k and ˆ γ i should be identified with Ω a ˆ γ b δ u b /δφ a , or ∂ u i ( ϕ, φ ) ∂φ k = 12 π (2 − d ) (cid:20) ϕ ϕ · φ δ ik − φ ( ϕ · φ ) ϕ i φ k − ϕ ( ϕ · φ ) (cid:0) φ i ϕ k + ϕ i φ k (cid:1) + φ ( ϕ + φ )( ϕ · φ ) ϕ i ϕ k (cid:21) . (6.24)One can check that a nontrivial integrability condition for this equation is satisfied, and thesolution reads u i ( ϕ, φ ) = − π (2 − d ) (cid:20) φ ( ϕ + φ )( ϕ · φ ) ϕ i − ϕ ( ϕ · φ ) φ i (cid:21) . (6.25)According to (2.37), (2.38a) this function generates the one-loop field renormalization, ϕ i ˜ ϕ i = ϕ i + ~ u i ( ϕ, φ ) . (6.26)Notice that this renormalization is essentially nonlinear. Still, it is covariant with respect tosimultaneous gauge transformations of both quantum and background fields, as it should be.It remains to be shown that the rest of the terms in (6.23) recover the correct BRSTstructure of the renormalized action after the field redefinition (6.26). We observe that thefirst term of (6.23) is the gauge invariant counterterm – proportional to the classical action, S = − N − π (2 − d ) Z d x G ij ∂ µ ϕ i ∂ µ ϕ j . (6.27)The second term bilinear in ˆ γ i and ω can be represented as the sum of two terms:ˆ γ a δR aα δϕ b u b ω α = Z d x (ˆ γ · u ) ω = 14 π (2 − d ) Z d x (cid:20) − φ ( ϕ + φ )( ϕ · φ ) (ˆ γ · ϕ ) + 2 ϕ ( ϕ · φ ) (ˆ γ · φ ) (cid:21) ω , (6.28a) − ˆ γ a δ u a δϕ b R bα ω α = − Z d x ˆ γ i ∂ u i ∂ϕ k ϕ k ω = 14 π (2 − d ) Z d x (cid:20) φ ( ϕ − φ )( ϕ · φ ) (ˆ γ · ϕ ) − ϕ ( ϕ · φ ) (ˆ γ · φ ) (cid:21) ω. (6.28b)Finally, the second line of (6.23) coincides with the change of the classical action under thefield reparametrization (6.26), δSδϕ a u a = − π (2 − d ) Z d x (cid:20) (cid:18) δ ij ϕ − φ i ϕ j ( ϕ · φ ) ϕ + φ i φ j ( ϕ · φ ) (cid:19) ∂ µ ϕ i ∂ µ ϕ j − (cid:18) δ ij ( ϕ · φ ) − φ i ϕ j ( ϕ · φ ) (cid:19) ∂ µ ϕ i ∂ µ φ j (cid:21) . (6.29)35ith the field reparametrization (6.26) we therefore have Σ (cid:12)(cid:12) ϕ → ϕ + ~ u = S + Q Ψ + ~ (cid:18) δSδϕ a u a + b α χ αa u a + ˆ γ a δR aα δϕ b u b ω α + Ω a δχ αb δφ a u b ¯ ω α (cid:19) + O ( ~ ) , (6.30) ~ Γ , ∞ (cid:12)(cid:12) ϕ → ϕ + ~ u = ~ (cid:18) S + δSδϕ a u a + Ω a ˆ γ b δ u b δφ a + ˆ γ a δR aα δϕ b u b ω α − ˆ γ a δ u a δϕ b R bα ω α (cid:19) + O ( ~ ) . (6.31)Thus, the renormalized action reads Σ ≡ (cid:2) Σ − ~ Γ , ∞ (cid:3) ϕ → ϕ + ~ u = S [ ϕ ] − ~ S [ ϕ ] + Q (cid:0) Ψ − ~ Υ (cid:1) + O ( ~ ) , (6.32)where in the expression for Q Υ we took into account the dependence of ˆ γ a = γ a − ¯ ω α χ αa ( φ )on ¯ ω and φ . This BRST structure of the one-loop renormalization is in full agreement with(2.33) — the renormalized gauge invariant action S [ ϕ ] = S [ ϕ ] − ~ S [ ϕ ] plus the BRSTexact term with the gauge fermion dressed by a local quantum correction inducing the fieldreparameterization. In this paper we have demonstrated the local BRST structure of renormalization in a wideclass of gauge field theories admitting background-covariant gauges. Simply stated, we haveshown that, for theories of this class, the renormalization procedure does not spoil gaugeinvariance. This class encompasses all standard Einstein–YM–Maxwell theories, whetherrenormalizable or not. In this way we reproduce the classical results concerning renormal-ization of Einstein–YM theories and strengthen them for the case of theories with Abeliansubgroups. Other representatives of the class covered by our analysis are non-relativisticYM–Maxwell theories and projectable Hoˇrava gravity. This offers the first demonstration ofthe BRST structure of projectable Hoˇrava gravity which completes the proof of its renor-malizability. The previous list of applications of our results is certainly not exhaustive.As suggested by the example considered in Sec. 6, they can be useful for studying various σ -models and other theories where gauge invariance is introduced as a tool to resolve thecomplicated structure of the field configuration space.Our argument makes essential use of the background fields φ . With a suitable choice of thegauge condition they allowed us to introduce an additional gauge invariance with respectto background gauge transformations. We then extended the BRST construction with anauxiliary anticommuting source Ω controlling the dependence of the gauge-fixing term on the36ackground fields. The counterterms generated by renormalization were shown to belong tothe local cohomology of the extended BRST operators on the space of functionals polynomialin Ω and the Faddeev–Popov ghosts. Our key observation is that the presence of linearlyrealized background-gauge invariance allows one to split the computation of this cohomologyinto several steps involving cohomologies of a few simpler operators. By completing thesesteps we have concluded that the counterterms split into a BRST exact piece and a localgauge invariant functional S [ ϕ ] depending only on the dynamical – “quantum”– fields whichrenormalizes the physical action of the system. Our derivation is self-contained and doesnot rely on any power counting considerations. We have discussed the simplifications thatappear if such considerations apply. Our results agree with those of [38] whenever theyoverlap.We have discussed in detail the local field redefinition bringing the renormalized actioninto the BRST form. This field redefinition, which in simple models has a multiplicativelinear nature, becomes essentially nonlinear in generic theories, as we illustrated with anexplicit example (Sec. 6). Despite this complication, it preserves a universal structure: Atany order in the loop expansion, the renormalized quantum fields are generated by Eq. (2.36)with the local generating functional Ψ . The latter is identical to the gauge fermion appearingin the exact part Q Ψ of the full BRST action Σ = S [ ϕ ] + Q Ψ dressed by loop corrections.This property provides a systematic algorithm to construct the field redefinition order byorder in perturbation theory. What one needs to do is just to determine Ψ from the part of thecounterterm containing the BRST sources and background fields. This procedure becomesparticularly efficient when there are additional constraints, e.g. due to power counting, thatprevent Ψ from depending on the BRST source Ω associated to background fields. In thatcase, our results imply that the Ω -dependent part of the counterterm has the form Ω δΨ/δφ (see the definition of Q in (2.16)). Therefore, Ψ can be found by simply integrating thecoefficient in front of Ω with respect to the background fields. In terms of renormalizedfields, the physical part S [ ϕ ] of the renormalized action becomes gauge invariant. Thus, thedivergences contained in S [ ϕ ] have the same structure as the terms in the tree-level actionand are absorbed by renormalization of the physical coupling constants.It is worth reviewing the various assumptions about the gauge algebra that entered intoour derivation. An essential assumption is the linearity of the gauge generators in the gaugefields which allows one to easily construct background-covariant gauge conditions. Moreover,the linearity of the resulting background-gauge covariance is crucial for its preservation at thequantum level. Another essential requirement is local completeness of the gauge generatorsexpressed by Eqs. (2.6), (2.7). This plays an important role in the homological analysis ofthe Koszul–Tate differential performed in [45, 49] and whose results we used in Sec. 5.2.37n the other hand, it appears likely that the irreducibility condition (ii) from Sec. 2.1 canbe relaxed at the price of considerably complicating the ghost sector. Indeed, the mainsteps in the proof in Sec. 5 would be unchanged, including the results of [45, 49] that arestraightforwardly generalized to the reducible case. Finally, we assumed the gauge algebrato close off-shell which allowed us to use the standard BRST construction for the gaugefixing. It would be interesting to extend our analysis to gauge theories with open algebras.The close connection between our approach and the Batalin–Vilkovisky generalization ofthe BRST formalism to open algebras [40, 41] makes the existence of such extension quiteplausible.Though we have not addressed this topic in the present paper, we believe that ourmethod can be efficiently applied to renormalization of composite operators in gauge theories.Another aspect of renormalization that has been left outside the scope of this paper is thatof quantum gauge anomalies. These are known to be related to BRST cohomologies withnon-vanishing ghost number. It would be interesting to see if the background-field approachalong the lines developed here can shed new light on this topic. We leave this study forfuture. Acknowledgments
We are indebted to Frank Ferrari, Elias Kiritsis and Igor Tyutin forstimulating discussions. We thank Ioseph Buchbinder and Marc Henneaux for valuablecomments on the first version of the paper. A.B. and C.S. are grateful for hospitality ofthe CERN Theoretical Physics Department where part of this work has been completed.This work was supported by the RFBR grant No.17-02-00651 (A.B. and S.S.), the TomskState University Competitiveness Improvement Program (A.B.), the Tomalla Foundation(M.H.-V.) and the Swiss National Science Foundation (S.S.).
A Derivation of Slavnov-Taylor and Ward identities
To obtain the Slavnov-Taylor identity (4.1), note that the total action including the sourceterm in the exponential of (2.32) can be written in a BRST invariant form. For this purposewe introduce the “doubly extended” BRST operator Q ext = s + Ω δδφ − J δδγ + ¯ ξ δδζ + ξ δδy , Q = 0 , (A.1)and notice that the source term in the non-minimal sector can also be rewritten as a BRST-exact expression, ξ ¯ ω + yb = (cid:18) s + ξ δδy (cid:19) y ¯ ω = Q ext ( y ¯ ω ) , (A.2)38here for brevity we omit the condensed indices of all quantities. Therefore, the total BRSTaction including all sources takes a compact form in terms of the extended gauge fermion Ψ ext , Σ ext = Σ − J δΨδγ + ¯ ξ δΨδζ + ξ ¯ ω + yb = S + Q ext Ψ ext , Ψ ext ≡ Ψ + y ¯ ω, (A.3)and the path integral for the generating functional (2.32) reads e − W/ ~ = Z dΦ e − Σ ext / ~ . (A.4)Clearly, Q ext e − Σ ext / ~ = 0 , (A.5)or (cid:18) − J δδγ + ¯ ξ δδζ + ξ δδy + Ω δδφ (cid:19) e − Σ ext / ~ = − s e − Σ ext / ~ , (A.6)whence (cid:18) − J δδγ + ¯ ξ δδζ + ξ δδy + Ω δδφ (cid:19) e − W/ ~ = − Z dΦ s e − Σ ext / ~ . (A.7)The path integral here has the form, Z dΦ (cid:0) s Φ I (cid:1) δδΦ I e − F [Φ] = − Z dΦ (cid:16) δδΦ I s Φ I ( Φ ) (cid:17) e − F [Φ] , (A.8)where the expression in brackets on the r.h.s. is the variation of the integration measure dΦ under the BRST variation of the fields. It vanishes according to the assumption ofanomaly-free regularization and we arrive at Eq. (4.1).For the derivation of the Ward identity (4.2), we introduce, together with the quantumfields Φ and background fields φ , also the collective notations for all the sources J = J a , ¯ ξ α , ξ α , y α , γ a , ζ α , Ω a . (A.9)Then, in view of our choice of background-covariant gauge conditions, Σ ext [ Φ, φ, J ] is in-variant with respect to the background-gauge transformations (2.10), (2.20) supplementedby δ ε J a = − J b R baα ε α , δ ε ¯ ξ α = ¯ ξ β C βαγ ε γ , δ ε ξ α = − C αβγ ξ β ε γ , δ ε y α = − C αβγ y β ε γ . (A.10) Note that the invariance of the source term − J a δΨ/δγ a = J a ( ˜ ϕ L − φ ) a relies on the homogeneity of thelinear transformation law for ( ˜ ϕ L − φ ) a contragredient to the transformation of J a in (2.20).
39e have, δ ε Σ ext = (cid:18) δ ε Φ δδΦ + δ ε φ δδφ + δ ε J δδ J (cid:19) Σ ext = 0 . (A.11)Next, we perform the change of integration variables Φ → Φ + δ ε Φ in the path integral (A.4).If, as we did before, we disregard the gauge variation of the integration measure, we obtainthe following integral identity, Z dΦ δ ε Φ δΣ ext δΦ e − Σ ext / ~ = 0 . (A.12)On account of Eq. (A.11), its l.h.s. equals Z dΦ (cid:18) δ ε φ δδφ + δ ε J δδ J (cid:19) e − Σ ext / ~ = (cid:18) δ ε φ δδφ + δ ε J δδ J (cid:19) e − W/ ~ , (A.13)because the operator δ ε φ δ/δφ + δ ε J δ/δ J is independent of the integration fields Φ and canbe commuted with the integration sign. Therefore, (cid:18) δ ε φ δδφ + δ ε J δδ J (cid:19) W = 0 , (A.14)which in view of the expressions (2.10), (2.20) and (A.10) for δ ε φ and δ ε J is just the expres-sion (4.2). B Homology of the operator
Ωδ/δφ
In this Appendix we prove the statement used in Sec. 5.1 that the cohomology of the operator
Ωδ/δφ on the space of local functionals vanishing at Ω = 0 is trivial. Lemma:
Let X [ ϕ, φ, Ω, . . . ] be a local functional of the gauge fields ϕ a , background fields φ a , anticommuting BRST sources Ω a and, possibly, other fields represented by dots. Assumethat X vanishes at Ω a = 0, X (cid:12)(cid:12) Ω =0 = 0 , (B.1)that it is invariant under background-gauge transformations and satisfies the equation Ω a δXδφ a = 0 . (B.2)Then there exists a local functional Y , invariant under background-gauge transformations,such that X = Ω a δYδφ a . (B.3)40 roof: The functional Y is constructed explicitly as, Y = ( φ a − ϕ a ) δδΩ a Z d zz X [ ϕ, ϕ + z ( φ − ϕ ) , zΩ, . . . ] , (B.4)where the arguments of X represented by dots are left untouched. According to the as-sumption (B.1), this expression indeed provides a regular functional. Notice that if X islocal, so is (B.4). Moreover, Y inherits background gauge invariance from X due to thelinearity of background-gauge transformations. It remains to demonstrate (B.3). Using theanticommutator Ω a δδφ a ( φ b − ϕ b ) δδΩ b + ( φ b − ϕ b ) δδΩ b Ω a δδφ a = Ω a δδΩ a + ( φ a − ϕ a ) δδφ a (B.5)we find Ω a δδφ a Y = Z d zz (cid:18) z dd z X [ ϕ, ϕ + z ( φ − ϕ ) , zΩ, . . . ] (cid:19) = X [ ϕ, φ, Ω, . . . ] , (B.6)where we again used (B.1). (cid:4) C Quadratic form for perturbations in the O ( N ) model Here we summarize the expressions for the coefficients of the operator (6.16) appearing inthe quadratic action for the perturbations δΦ I = ( δϕ i , δω, δ ¯ ω ) of the O ( N ) model. We writethese coefficients as matrices with 3 × D IJ = A ij B i C − B j − C , (C.1)with the following boson-boson A ij , boson-fermion B i and fermion-fermion C entries A ij = − G ij − φ i φ j ( φ ) = − ϕ (cid:20) δ ij − ϕ i ϕ j ϕ + ϕ φ φ i φ j φ (cid:21) , (C.2) B i = φ i φ ω + Ω i φ − Ω · φ )( φ ) φ i , C = ϕ · φφ . (C.3)The other two matrix coefficients have the form, Γ µIJ = Γ µij − ∂ µ B j − ∂ µ C , Π IJ = Π ij ˆ γ i − ˆ γ j − (cid:3) B j − (cid:3) C , (C.4)41 µij = −
12 ( G ji,k + G ki,j − G jk,i ) ∂ µ ϕ k − φ i φ ∂ µ (cid:18) φ j φ (cid:19) , (C.5) Π ij = − (cid:18) G ik,jl − G kl,ij (cid:19) ∂ µ ϕ k ∂ µ ϕ l − G ki,j (cid:3) ϕ k − φ i φ (cid:3) (cid:18) φ j φ (cid:19) , (C.6)where G ij,k ≡ ∂G ij ∂ϕ k = − δ ij ϕ k + δ ik ϕ j + δ jk ϕ i ( ϕ ) + 4 ϕ i ϕ j ϕ k ( ϕ ) ,G ij,kl ≡ ∂ G ij ∂ϕ k ∂ϕ l = − δ ij δ kl + δ ik δ jl + δ jk δ il ( ϕ ) − ϕ i ϕ j ϕ k ϕ l ( ϕ ) + 4( ϕ ) (cid:16) δ ij ϕ k ϕ l + δ ik ϕ j ϕ l + δ il ϕ j ϕ k + δ jk ϕ i ϕ l + δ jl ϕ i ϕ k + δ kl ϕ i ϕ j (cid:17) . For the computation of the one-loop divergence of the effective action, we need thematrices Π and Γ µ defined in (6.21). This, in turn, requires the inverse D IJ of the matrix(C.1), which reads D IJ = A ij − A ik B k /C − B k A kj /C B k A kl B l /C − /C /C , (C.7)where B j and C are given by (C.3) and A ij = − ϕ (cid:20) δ ij − ( ϕ i φ j + φ i ϕ j ) ϕ · φ + φ ϕ ( ϕ + φ )( ϕ · φ ) ϕ i ϕ j (cid:21) (C.8)is the inverse of the matrix A ij defined by (C.2), A il A lj = δ ji . Using these expressions weobtain, Γ µ = A il Γ µlk ∂ µ B k − B l A lm Γ µmk ) /C ∂ µ C/C
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