Lifshitz Anomalies, Ward Identities and Split Dimensional Regularization
PPrepared for submission to JHEP
Lifshitz Anomalies, Ward Identities and SplitDimensional Regularization
Igal Arav, Yaron Oz, Avia Raviv-Moshe
Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, 55 HaimLevanon street, Tel-Aviv, 69978, Israel
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We analyze the structure of the stress-energy tensor correlation functions inLifshitz field theories and construct the corresponding anomalous Ward identities. We de-velop a framework for calculating the anomaly coefficients that employs a split dimensionalregularization and the pole residues. We demonstrate the procedure by calculating the freescalar Lifshitz scale anomalies in 2 + 1 spacetime dimensions. We find that the analysis ofthe regularization dependent trivial terms requires a curved spacetime description withouta foliation structure. We discuss potential ambiguities in Lifshitz scale anomaly definitions.
Keywords:
Anomalies in Field and String Theories, Space-Time Symmetries
ArXiv ePrint: a r X i v : . [ h e p - t h ] M a r ontents z = 2 Lifshitz Scalar Field and its Scale Anomalies 28 z = 2 Free Lifshitz Scalar Field in General Spacetime Dimensions 285.2 The Lifshitz Anomaly of the Free Scalar 305.2.1 Results for the Two Point Function 325.2.2 Results for the Three Point Function and the Anomaly 34 A.1 Notations and Conventions in Curved Spacetime 37A.2 Notations and Conventions for the Relativistic Scalar Case 38A.3 Notations and Conventions for the Lifshitz z = 2 Scalar Case 39– i – Non-Relativistic Curved Spacetime with Multiple Time Directions 40
B.1 Curved Spacetime Definitions 40B.2 Symmetries Over Curved Spacetime 43B.3 Action of the Free z = 2 Scalar 44 C Relativistic Scalar Field – Feynman Rules, Vertexes and Integrals 46
C.1 Feynman Rules and Diagrams 46C.2 Massless Integrals 48
D Lifshitz z = 2 Scalar Field – Feynman Rules, Vertexes and Integrals 49E Variations of the Anomaly and Trivial Term Densities 55
E.1 The Two Derivatives Sector 55E.2 The Four Derivatives Sector 57
Lifshitz scale symmetry is a symmetry under which space and time scale differently, t → λ z t, x i → λx i , i = 1 , . . . , d , (1.1)where z is the dynamical critical exponents which represents the anisotropy between spaceand time, and d is the number of spatial dimensions. Non-relativistic field theories thatexhibit a Lifshitz scale symmetry have attracted much attention in recent years. In natureLifshitz scaling is a property of certain low energy systems of condensed matter, whichexhibit quantum criticality (see for example [1]). The study of holographic Lifshitz systemshas been initiated e.g. in [2, 3]Similary to the relativistic trace anomalies (for brief reviews see e.g. [4, 5]), non-relativistic quantum field theories may exhibit Lifshitz scale anomalies, where the classicalLifshitz symmetries are broken by the quantum corrections. A cohomological analysis pro-vides a general framework to determine the possible structures of Lifshitz scale anomalies[6–9] (see also [10, 11] for a z = 2 example). Naturally, it is of importance to calculate theanomaly coefficients, which carry information about the quantum field theory. A heat ker-nel calculation of anomaly coefficients of a Lifshitz scalar in 2 + 1 spacetime dimensions hasbeen performed in [11], and in 3 + 1 dimensions using a different regularization scheme in[12]. The aim of this work is to develop a general scheme for such field theory calculations.In order to do that we will first analyze the general structure of correlation functions ofthe stress-energy tensor in Lifshitz field theories and analyze the corresponding anomalousWard identities. We will encounter a subtle ambiguity in the definition of the anomalycoefficients and will clarify its meaning. Next, we will develop a general framework forcalculating the anomaly coefficients. It consists of two elements: a split dimensional regu-larization [13, 14] where space (momentum) and time (frequency) integrals are regulated– 1 –eparately, and pole residue calculations [15] that allow to extract the anomaly coefficientswithout a full calculation of the correlation functions. In order to demonstrate the powerof the latter we will calculate using the pole residues the trace anomaly coefficients of arelativistic scalar in two and four spacetime dimensions and obtain the known results. Wewill then apply the complete framework to calculate the anomaly coefficients of a Lifshitzscalar in 2 + 1 spacetime dimensions. The results agree with the heat kernel calculation[11]. Following the discussion in [7], we will show that the analysis of the regularization de-pendent trivial terms arising from this calculation requires a curved spacetime descriptionthat violates the Frobenius condition (and therefore has no foliation structure).This paper is organized as follows. In section 2 we present the Ward identities corre-sponding to the symmetries of the field theories we consider, in terms of the expectationvalue of the conserved currents in curved spacetime, and in terms of the flat space cor-relation functions of the stress-energy tensor. We also discuss a possible ambiguity inthe Lifshitz anomaly coefficients. In section 3 we explain the method by which we usesplit dimensional regularization to extract the Lifshitz anomaly coefficients from the flatspace field theory correlation functions. In both sections we start by reviewing the caseof a conformal field theory and then discuss the Lifshitz case. In section 4 we apply theaforementioned method of extracting scale anomaly coefficients from the pole residues ofcorrelation functions in dimensional regularization to calculate the trace anomaly coeffi-cients for a free relativistic scalar, both in two and four spacetime dimensions. In section 5we use this method in its split dimensional regularization version to calculate the anomalycoefficients for a z = 2 free Lifshitz scalar in 2 + 1 dimensions. We conclude in section 6. In this section we present the form of the Ward identities corresponding to the symmetriesof the field theories we consider, both in terms of the expectation values of the conservedcurrents over curved spacetime, and of the flat space correlation functions of these currents.These include the anomalous scaling symmetry as well as the other symmetries that areassumed to be non-anomalous. The anomalous Ward identities will later be used to extractthe anomaly coefficients from the flat space correlation functions. We start by reviewingthe conformal case, and then discuss the Lifshitz (non-relativistic) case.
Consider a conformal field theory in d dimensions. The theory can be coupled to a curvedspacetime manifold equipped with a background metric g µν (or alternatively vielbein struc-ture e aµ ). Suppose this theory is described by the classical action S ( g µν , { φ } ), where { φ } stands for the dynamic fields of the theory. This action reduces to the flat space action ofthe theory when g µν → η µν , and is invariant under the following symmetries:1. Diffeomorphisms, given by: δ Dξ g µν = ∇ µ ξ ν + ∇ ν ξ µ , (2.1)– 2 –r in vielbein notation (here we also include local Lorentz transformations): δ Dξ e aµ = ξ ν ∇ ν e aµ + ∇ µ ξ ν e aν , δ Lα e aµ = − α ab e bµ . (2.2)2. Weyl transformations, given by: δ Wσ g µν = 2 σg µν , δ Wσ e aµ = σe aµ . (2.3)The stress-energy tensor is defined as the variation of the action with respect to themetric: T µν = 2 (cid:112) | g | δSδg µν , (2.4)where g ≡ det g µν . The stress-energy tensor satisfies classical Ward identities correspondingto the above symmetries. From diffeomorphism invariance (and local Lorentz invariance)we have the conservation and symmetry of the stress-energy tensor: ∇ µ T µν = 0 , T µν = T νµ , (2.5)and from Weyl invariance we get the traceless property: T µµ = 0 . (2.6)When conformal anomalies are present, the expectation value of the stress-energy tensorno longer satisfies identity (2.6). Instead it is modified: (cid:104) T µµ (cid:105) = A , (2.7)where A ( g µν ) is a local scalar function of the metric. The possible form of A is restrictedby the Wess-Zumino consistency condition to a linear combination of possible expressions(see e.g. [5, 16–19] for details): the Euler density of the background manifold E d (A-type anomaly), the various Weyl invariant densities of the manifold (B-type anomalies)and other terms that can be cancelled by adding local counterterms to the curved spaceeffective action of the theory (trivial terms).In d = 2 dimensions, the only such expression is the Euler density E = R (where R is the Ricci scalar of the manifold), so that: A = βR. (2.8)In d = 4 dimensions, these expressions consist of the Euler density E (A-type anomaly),the Weyl tensor squared W (B-type anomaly) and (cid:3) R (trivial term): A = β a W + β b E + β c (cid:3) R. (2.9)While these expressions are universal, the value of their coefficients depends on the contentof the theory. Note that the coefficients of the trivial terms may depend on the regular-– 3 –zation scheme, but those of the anomalies are regularization independent. One methodto obtain the value of these coefficients is to extract them from the flat space n -pointcorrelation functions of the stress-energy tensor.For later reference, for a free scalar in d = 2 dimensions, the anomaly coefficient hasbeen determined to be (see e.g. [20]): β = − π . (2.10)For a free scalar in d = 4 dimensions, the coefficients have been determined to be (see[4, 21–25] and references therein): β a = 32 12880 π , β b = −
12 12880 π , β c = 12880 π . (2.11) We denote by W ( g µν ) the effective action of the conformal field theory on curved spacetime,defined by: e i W ( g µν ) ≡ (cid:90) Dφ e i S ( g µν ) , (2.12)where φ stands for the dynamic fields in the theory. Two types of correlation functions canbe defined for the stress-energy tensor: First, the connected Feynman n -point correlationfunctions, given by the expectation value : (cid:104)O ( x ) . . . O n ( x n ) (cid:105) F ≡ Z (cid:90) Dφ O ( x ) . . . O n ( x n ) e i S − Non-connected terms , (2.13)where O , . . . , O n are local functions of the dynamic and the background fields, and Z isthe flat space partition function. Alternatively, one may define the correlation functions asvariations of the effective action W with respect to the metric: (cid:104) T µ ν ( x ) . . . T µ n ν n ( x n ) (cid:105) W ≡ ( − i ) n − n (cid:112) | g ( x ) | . . . (cid:112) | g ( x n ) | δ n Wδg µ ν ( x ) ...δg µ n ν n ( x n ) . (2.14)We will refer to these as the variational correlation functions. Since the stress-energy tensoritself depends on the metric, these two types of correlation function do not coincide. Oneobtains the following relations between the two types of two point functions and three pointfunctions in flat spacetime (see [24]): (cid:104) T µν ( x ) T ρσ ( y ) (cid:105) W = (cid:104) T µν ( x ) T ρσ ( y ) (cid:105) F , (2.15) The coefficient β c here corresponds to a dimensional regularization scheme. The values cited here ofthe coefficients β, β a , β b , β c correspond to our conventions for the definitions of the stress-energy tensor andthe Riemann tensor. In the following sections we may omit the F subscript when referring to Feynman correlation functions. In deriving these formulas, one point functions have been dropped as they correspond to massless“tadpole” diagrams and therefore vanish in the flat space limit in a dimensional regularization scheme. – 4 – T µν ( x ) T ρσ ( y ) T αβ ( z ) (cid:105) W = (cid:104) T µν ( x ) T ρσ ( y ) T αβ ( z ) (cid:105) F − i (cid:20)(cid:28) δ Sδg µν ( x ) δg ρσ ( y ) δSδg αβ ( z ) (cid:29) F + (cid:28) δ Sδg µν ( x ) δg αβ ( z ) δSδg ρσ ( y ) (cid:29) F + (cid:28) δ Sδg ρσ ( y ) δg αβ ( z ) δSδg µν ( x ) (cid:29) F (cid:21) . (2.16)The Ward identities satisfied by the variational correlation functions can be obtainedby taking derivatives of the curved spacetime Ward identities (2.5)–(2.6) (or (2.7) in thepresence of conformal anomalies) with respect to the background metric. These identitieshave been derived in [24–26] for Euclidean signature. We repeat them here modified toour use of Lorentzian signature. For diffeomorphism invariance, one obtains the followingidentities for two and three point correlation functions: ∂ xν (cid:104) T µν ( x ) T ρσ ( y ) (cid:105) W = 0 , (2.17) ∂ xν (cid:104) T µν ( x ) T ρσ ( y ) T αβ ( z ) (cid:105) W + i (cid:104) (cid:104) T ρσ ( x ) T αβ ( z ) (cid:105) W ∂ µ δ ( x − y ) + (cid:104) T αβ ( x ) T ρσ ( y ) (cid:105) W ∂ µ δ ( x − z ) (cid:105) − i (cid:104) η µρ (cid:104) T νσ ( x ) T αβ ( z ) (cid:105) W + η µσ (cid:104) T νρ ( x ) T αβ ( z ) (cid:105) W (cid:105) ∂ ν δ ( x − y ) − i (cid:104) η µα (cid:104) T νβ ( x ) T ρσ ( y ) (cid:105) W + η µβ (cid:104) T να ( x ) T ρσ ( y ) (cid:105) W (cid:105) ∂ ν δ ( x − z ) = 0 . (2.18)For the (anomalous) Weyl invariance, one obtains the following identities: η µν (cid:104) T µν ( x ) T ρσ ( y ) (cid:105) W = − i δ A ( x ) δg ρσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) flat , (2.19) η µν (cid:104) T µν ( x ) T ρσ ( y ) T αβ ( z ) (cid:105) W − i (cid:104) T ρσ ( y ) T αβ ( z ) (cid:105) W [ δ ( x − y ) + δ ( x − z )]+ iη ρσ η µν (cid:104) T µν ( x ) T αβ ( z ) (cid:105) W δ ( x − y ) + iη αβ η µν (cid:104) T µν ( x ) T ρσ ( y ) (cid:105) W δ ( x − z )= − δ A ( x ) δg ρσ ( y ) δg αβ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) flat . (2.20)Using identities (2.19)–(2.20), one can extract the conformal anomaly coefficients from theflat space correlation functions. Consider a non-relativistic field theory in d + 1 spacetime dimensions with a Lifshitz scalesymmetry of the form (1.1). Assume this theory can be coupled to a curved spacetimemanifold equipped with a metric g µν (or alternatively vielbein structure e aµ ) and a 1-form t α corresponding to the time direction at each point (or the normalized n α : n α n α = − . Suppose this theory is described by the classical action S ( g µν , t α , { φ } ),where { φ } are the dynamic fields of the theory (or alternatively S ( e aµ , t α , { φ } ) in vielbein As in [7] we do not assume that t µ satisfies the Frobenius condition. This description is therefore moregeneral than one using an ADM-like decomposition. – 5 –ormalism). Further assume that this action can be defined such that it reduces to the flatspace action of the theory when g µν → δ µν = diag( − , , , . . . ) , t α → (1 , , . . . ), and isinvariant under the following symmetries:1. Time-direction preserving diffeomorphisms (TPD). These are the curved spacetimegeneralization of space rotations. As explained in [6, 7], in our covariant notationthese take the form of standard diffeomorphisms, given by: δ Dξ g µν = ∇ µ ξ ν + ∇ ν ξ µ , δ Dξ t α = L ξ t α = ξ β ∇ β t α + ∇ α ξ β t β , (2.21)or in vielbein notation (here we also include local Lorentz transformations): δ Dξ e aµ = ξ ν ∇ ν e aµ + ∇ µ ξ ν e aν , δ Dξ t a = ξ ν ∇ ν t a ,δ Lα e aµ = − α ab e bµ , δ Lα t a = − α ab t b . (2.22)2. Anisotropic Weyl transformations. These are the local generalization of Lifshitz scal-ing, given by: δ Wσ t α = 0 , δ Wσ ( g αβ t α t β ) = − σz ( g αβ t α t β ) ,δ Wσ P αβ = 2 σP αβ , δ Wσ n α = zσn α , δ Wσ n α = − zσn α , (2.23)where P µν = g µν + n µ n ν is the spatial projector, or alternatively using the vielbeins: δ Wσ ( n a e aµ ) = zσn a e aµ , δ Wσ ( P ab e bµ ) = σP ab e bµ ,δ Wσ t b = − zσt b , δ Wσ n b = 0 . (2.24)We define various field theory currents as the variation of the action with respect tothe background fields. The stress-energy tensor can be defined in two possible ways, eitherusing the metric or vielbein descriptions: T µν ( g ) ≡ (cid:112) | g | δSδg µν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t α , T ( e ) µa ≡ e δSδe aµ (cid:12)(cid:12)(cid:12)(cid:12) t a . (2.25)We also define the variation of the action with respect to the time direction 1-form: J α ≡ (cid:112) | g | δSδt α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g µν = 1 e e bα δSδt b (cid:12)(cid:12)(cid:12)(cid:12) e aµ , (2.26)as well as its normalized version: ˆ J α ≡ (cid:113) | g µν t µ t ν | J α , (2.27)Note that unlike the relativistic case, these two definitions of the stress-energy tensor do See appendix A for our notations and conventions. – 6 –ot coincide. Instead they are related by the following identity: T µν ( e ) = T µν ( g ) + J µ t ν . (2.28)From the symmetries and these definitions, one can obtain the classical Ward identitiesthese currents satisfy over curved spacetime (see [6, 7]). From TPD invariance follow theidentities: ∇ µ T µ ( g ) ν = ˆ J µ ∇ ν n µ − ∇ µ ( ˆ J µ n ν ) , (2.29)or equivalently in terms of T µν ( e ) : T ( e )[ µν ] = ˆ J [ µ n ν ] , ∇ µ T ( e ) µν = ˆ J µ ∇ ν n µ . (2.30)These identities imply that in the flat spacetime limit, T µν ( g ) is symmetric but not conserved,while T µν ( e ) is conserved but not symmetric. The latter then corresponds to the conservedenergy and momentum currents of the flat space theory, and we therefore choose to use itand the vielbein description in the following sections .From anisotropic Weyl symmetry, one obtains the following Ward identity: D ≡ D µν T ( g ) µν = D µν T ( e ) µν = 0 , (2.31)where D µν ≡ P µν − z n µ n ν . Note that similarly to the relativistic case, assuming that aWeyl-invariant coupling of the theory to curved spacetime exists is equivalent to assumingthat the flat space stress-energy tensor can be improved to satisfy equation (2.31) (andstill remain conserved).In a field theory for which TPD invariance is not anomalous, the expectation value ofthe stress-energy tensor satisfies the Ward identities (2.30). However, when the Lifshitzscaling symmetry is anomalous, its expectation value no longer satisfies identity (2.31).Instead it is modified: (cid:104) D (cid:105) ≡ D µν (cid:104) T ( e ) µν (cid:105) = A , (2.32)where A ( e aµ , t a ) is a TPD invariant and local function of the background fields. Thepossible form of A is restricted by the Wess-Zumino consistency condition to a linearcombination of possible expressions (analogous to E , W and (cid:3) R of the (3+1)-dimensionalconformal case), and was obtained for several cases in [6–8, 10, 11]. While these expressionsare universal, the value of their coefficients depends on the content of the theory. Likethe conformal case, the coefficients of the trivial terms may depend on the regularizationscheme, but those of the anomalies are regularization independent. Our goal in this work isto extract these coefficients from the flat space n -point correlation functions of the stress-energy tensor.For the case of a Lifshitz field theory in 2 + 1 dimensions with z = 2, which is the case In the following sections we omit the ( e ) subscript for brevity, so that T µν ( e ) will be denoted simply as T µν . – 7 –e study in section 5, it was shown in [7] that generally (when the Frobenius condition isnot assumed) the possible expressions in A consist of an infinite set of linearly independentanomalies and trivial terms, with increasing number of derivatives. However, the onlyanomalies that possibly contribute to the three point correlation functions of the stress-energy tensor (and therefore relevant to the calculations we perform here) are given by thefollowing expressions: A (2 , , = Tr( K S ) − K S , (2.33)in the two derivatives sector, and: A (0 , , = K A L n K A + K A K S L n K A , A (0 , , = ˜ K αβS ( a α ¯ ∇ β K A + ¯ ∇ α ¯ ∇ β K A ) , A (0 , , = (cid:16) (cid:98) R + ¯ ∇ α a α (cid:17) , (2.34)in the four derivatives sector. The various geometrical structures in these expressions aredefined as follows: • ( K S ) µν ≡ L n P µν is a generalization of the extrinsic curvature of the foliation in-duced by n µ to the non-Frobenius case. We also define K S ≡ ( K S ) µµ , Tr( K S ) ≡ ( K S ) µν ( K S ) µν and ( ˜ K S ) αβ ≡ ¯ (cid:15) ( αγ K Sβ ) γ , where ¯ (cid:15) µν ≡ n α (cid:15) αµν . • ( K A ) µν ≡ P µ (cid:48) µ P ν (cid:48) ν ∇ [ µ (cid:48) n ν (cid:48) ] is antisymmetric and vanishes when n µ satisfies the Frobe-nius condition. We also define K A ≡ K Aµν ¯ (cid:15) µν . • a µ ≡ L n n µ is the acceleration vector associated with n µ . • ¯ ∇ µ is a space projected covariant derivative. • (cid:98) R is a generalization of the Ricci scalar of the foliation induced by n µ to the non-Frobenius case.For further discussion and more detailed definitions see [7] and appendix B. The possibletrivial terms relevant to our calculations are listed in appendix E. Let us denote by W ( e aµ , t a ) the effective action of the Lifshitz field theory on curvedspacetime, defined by: e i W ( e aµ ,t a ) = (cid:90) Dφ e i S ( e aµ ,t α , { φ } ) , (2.35)where φ again stands for the dynamic fields in the theory. We define two types of correlationfunctions for the stress-energy tensor. First, the connected Feynman n -point correlationfunctions, defined similarly to the relativistic case (2.13), where O , . . . , O n are local func-tions of the dynamic and the background fields, and Z is the flat space partition function.Alternatively, we define the correlation functions given by variations of the effective action– 8 – with respect to the background fields: (cid:104) T µ a ( x ) . . . T µ n a n ( x n ) (cid:105) W ≡ ( − i ) n − e ( x ) . . . e ( x n ) δ n Wδe a µ ( x ) . . . δe a n µ n ( x n ) . (2.36)We will refer to these as the variational correlation functions. Like the relativistic case,these two definitions for the correlation functions do not coincide, since the curved spacefield theory currents depend on the background fields. Instead these two types of correlationfunctions are related via relations, that are obtained by differentiating equation (2.35). Forthe two and three point functions, one obtains: (cid:104) T µa ( x ) T ρb ( y ) (cid:105) W = − ie ( x ) e ( y ) (cid:28) δ Sδe aµ ( x ) δe bρ ( y ) (cid:29) F + (cid:104) T µa ( x ) T ρb ( y ) (cid:105) F , (2.37) (cid:104) T µa ( x ) T ρb ( y ) T αc ( z ) (cid:105) W = − e ( x ) e ( y ) e ( z ) (cid:28) δ Sδe aµ ( x ) δe bρ ( y ) δe cα ( z ) (cid:29) F − ie ( x ) e ( y ) (cid:28) δ Sδe aµ ( x ) δe bρ ( y ) T αc ( z ) (cid:29) F − ie ( x ) e ( z ) (cid:28) δ Sδe aµ ( x ) δe cα ( z ) T ρb ( y ) (cid:29) F − ie ( y ) e ( z ) (cid:28) δ Sδe bρ ( y ) δe cα ( z ) T µa ( x ) (cid:29) F + (cid:104) T µa ( x ) T ρb ( y ) T αc ( z ) (cid:105) F . (2.38)In the flat space limit, these relations reduce to the following: (cid:104) T µa ( x ) T ρb ( y ) (cid:105) W = (cid:104) T µa ( x ) T ρb ( y ) (cid:105) F , (2.39) (cid:104) T µa ( x ) T ρb ( y ) T αc ( z ) (cid:105) W = (cid:104) T µa ( x ) T ρb ( y ) T αc ( z ) (cid:105) F − i (cid:28) δ Sδe aµ ( x ) δe bρ ( y ) T αc ( z ) (cid:29) F − i (cid:28) δ Sδe aµ ( x ) δe cα ( z ) T ρb ( y ) (cid:29) F − i (cid:28) δ Sδe bρ ( y ) δe cα ( z ) T µa ( x ) (cid:29) F . (2.40)These relations make it clear that, while the flat space Feynman correlation functionsdepend only on the field theory currents (given as the first derivative of the action withrespect to the background fields), the variational correlation functions depend on higherderivatives of the action as well. The implication is that, if there is more than one wayto couple the field theory to curved spacetime while preserving all symmetries discussedin subsection 2.2.1 and leaving the flat space currents the same, then any such couplingwould produce the same Feynman correlation functions but different variational correlationfunctions. As we will show, this leads to a possible ambiguity in the relation between theflat space Feynman correlation function and the consistent anomalies obtained from theWess-Zumino analysis, which does not occur in the relativistic case.One method to derive Ward identities directly for the flat space Feynman correlationfunctions is via a change of variables in the path integral. Suppose that the operator δ implements an infinitesimal transformation that corresponds to a symmetry of the theory,and transforms both the dynamic and the background fields so that δS = 0. Let δ dyn Note that the one point functions correspond to “tadpole” diagrams with no dimensionful parameter,and therefore vanish in the flat space limit. – 9 –enote an operation that transforms only the dynamic fields, leaving the background fieldsunchanged, and similarly let δ bg transform only the background fields, so that the followingis satisfied: δ dyn φ = δφ, δ dyn e aµ = 0 , δ dyn t a = 0 ,δ bg φ = 0 , δ bg e aµ = δe aµ , δ bg t a = δt a , (2.41)and δ = δ dyn + δ bg . These operators are explicitly given by: δ dyn = (cid:90) δφ δδφ (cid:12)(cid:12)(cid:12)(cid:12) e aµ ,t a , δ bg = (cid:90) δe aµ δδe aµ (cid:12)(cid:12)(cid:12)(cid:12) φ,t a + δt a δδt a (cid:12)(cid:12)(cid:12)(cid:12) φ,e aµ . (2.42)By performing a change of variables φ → ˜ φ = φ + δφ in the path integral (2.13), andassuming for now that that the symmetry corresponding to δ is non-anomalous (so thatthe change of integration measure does not contribute in the context of the dimensionalregularization scheme we are using here), we obtain: (cid:90) Dφ δ dyn (cid:2) O ( x ) . . . O n ( x n ) e i S (cid:3) = 0 . (2.43)Noting that δ dyn S = − δ bg S , the following identity can then be derived: (cid:104) ( δ bg S ) O ( x ) . . . O n ( x n ) (cid:105) F + i (cid:104) δ dyn O ( x ) . . . O n ( x n ) (cid:105) + . . . + i (cid:104)O ( x ) . . . δ dyn O n ( x n ) (cid:105) = 0 . (2.44)Using TPD (as given in (2.22)) in identity (2.44) and choosing ξ σ ( w ) = δ σa δ ( w − x ), we getthe following Ward identities for the flat space two and three point Feynman correlationfunctions of the stress-energy tensor: ( I (2) D ) ρab ( x, y ) ≡ (cid:104) ( ∂ µ T µa ( x )) T ρb ( y ) (cid:105) F = 0 , (2.45)( I (3) D ) ραabc ( x, y, z ) ≡ (cid:104) ( ∂ µ T µa ( x )) T ρb ( y ) T αc ( z ) (cid:105) F − i (cid:10) ( δ D dyn T ρb )( x, y ) T αc ( z ) (cid:11) F − i (cid:10) T ρb ( y )( δ D dyn T αc )( x, z ) (cid:11) F = 0 , (2.46)where for a scalar field φ , for example, we have ( δ D dyn φ )( x, w ) = δ ( w − x ) ∂ a φ ( w ). Similarly,when applying identity (2.44) to the anisotropic Weyl transformation (2.24) and setting σ ( w ) = δ ( w − x ), we obtain the following identities for the flat space correlation functions:( I (2) W ) ρb ( x, y ) ≡ (cid:104) D ( x ) T ρb ( y ) (cid:105) F = 0 , (2.47)( I (3) W ) ραbc ( x, y, z ) ≡ (cid:104) D ( x ) T ρb ( y ) T αc ( z ) (cid:105) F + i (cid:10) ( δ W dyn T ρb )( x, y ) T αc ( z ) (cid:11) F + i (cid:10) T ρb ( y )( δ W dyn T αc )( x, z ) (cid:11) F = 0 , (2.48)where for a Lifshitz scalar φ we have ( δ W dyn φ )( x, w ) = (cid:0) z − d (cid:1) δ ( w − x ) φ ( w ). In the presence See appendix A.3 for details on our notations for the Ward identities. Here and in the following sections, “Lifshitz scalar” will refer to a scalar φ with a second order timederivative kinetic term in the action as used in section 5, so that it has a Lifshitz dimension of [ φ ] = d − z . – 10 –f Lifshitz scaling anomalies, expressions (2.47)–(2.48) will not vanish, and in general willbe equal to some linear combinations of contact terms instead.An alternative method to derive the Ward identities for the variational correlationfunctions is by taking derivatives of the curved spacetime Ward identities (2.30) and (2.31)(or (2.32) in the presence of Lifshitz anomalies) with respect to the background fields. Thismethod allows one to directly relate the flat space correlation functions to derivatives ofthe consistent anomalies as derived in [7], and extract the anomaly coefficients. Taking thefirst and second derivatives of identity (2.32) in the flat space limit, we obtain the followingWard identities for the two and three point variational correlation functions:( W (2) W ) ρb ( x, y ) ≡ D aµ (cid:104) T µa ( x ) T ρb ( y ) (cid:105) W = − i δ A ( x ) δe bρ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) flat , (2.49)( W (3) W ) ραbc ( x, y, z ) ≡ D aµ (cid:104) T µa ( x ) T ρb ( y ) T αc ( z ) (cid:105) W − i ( δ ρµ D ab − δ ρb D aµ ) δ ( x − y ) (cid:104) T µa ( x ) T αc ( z ) (cid:105) W − i ( δ αµ D ac − δ αc D aµ ) δ ( x − z ) (cid:104) T µa ( x ) T ρb ( y ) (cid:105) W = − δ A ( x ) δe bρ ( y ) δe cα ( z ) (cid:12)(cid:12)(cid:12)(cid:12) flat . (2.50)Using the relations (2.39)–(2.40) in these expressions, W (2) W , W (3) W can be written as linearcombinations of the expressions in the Ward identities (2.47)–(2.48) and other expressionsthat vanish in the absence of Lifshitz anomalies:( W (2) W ) ρb ( x, y ) = ( I (2) W ) ρb ( x, y ) , (2.51)( W (3) W ) ραbc ( x, y, z ) = ( I (3) W ) ραbc ( x, y, z ) − i J ραbc ( x, y, z ) − i J αρcb ( x, z, y )+ iδ ρb δ ( x − y )( I (2) W ) αc ( x, z ) + iδ αc δ ( x − z )( I (2) W ) ρb ( x, y ) − i K ραbc ( x, y, z ) , (2.52)where J ραbc ( x, y, z ) and K ραbc ( x, y, z ) are given by: J ραbc ( x, y, z ) ≡ D aµ (cid:28) δ Sδe aµ ( x ) δe bρ ( y ) T αc ( z ) (cid:29) F + D ab δ ( x − y ) (cid:104) T ρa ( x ) T αc ( z ) (cid:105) F + (cid:10) ( δ W dyn T ρb )( x, y ) T αc ( z ) (cid:11) F , (2.53) K ραbc ( x, y, z ) ≡ (cid:28) D ( x ) δ Sδe bρ ( y ) δe cα ( z ) (cid:29) F . (2.54)When no anomalies are present, the expressions I (2) W , I (3) W and K are expected to vanishdue to identity (2.44). The expression J can also be shown to vanish in this case, by notingthat: (cid:10) ( δ W bg + δ W dyn − δ W ) T ρb ( x, y ) T αc ( z ) (cid:11) F = 0 , (2.55) Like the Ward identities (2.47)–(2.50), this identity may acquire contact terms on its RHS after arenormalization procedure. – 11 –nd using the following expression for ( δ W bg T ρb )( x, y ) (where again σ ( w ) = δ ( w − x )):( δ W bg T ρb )( x, y ) = D aµ δT ρb ( y ) δe aµ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) flat = D aµ δ Sδe aµ ( x ) δe bρ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) flat − ( d + z ) δ ( x − y ) T ρb ( y ) , (2.56)and the anisotropic Weyl scaling properties of T ρb :( δ W T ρb )( x, y ) = − ( d + z ) δ ( x − y ) T ρb ( y ) − D ab δ ( x − y ) T ρa ( y ) . (2.57)This scaling property can be derived by using the definition of the stress-energy tensor(2.25), expressing the operator δ W in terms of the anisotropic Weyl transformations of thebackground and dynamic fields as in (2.42), exchanging the order of derivatives and usingthe anisotropic Weyl invariance of the classical action S . As mentioned earlier, unlike the relativistic case, the relation between the flat space Feyn-man correlation functions and the Lifshitz anomaly coefficients (that is, the coefficients onthe RHS of equation (2.32) of the various anomalous terms as obtained from the Wess-Zumino consistency conditions) may be ambiguous in some cases. This is due to the Weylinvariant coupling of the theory to curved spacetime being non-unique.As an example, consider any Lifshitz invariant theory for which d = z , that containsa real Lifshitz scalar φ , so that the field φ is dimensionless. In the absence of anomalies,one expects the two point function (cid:10) D ( x ) φ ( y ) (cid:11) F to vanish. When Lifshitz anomalies arepresent, it will instead be equal to some contact term, and from dimensional analysis wehave: (cid:10) D ( x ) φ ( y ) (cid:11) F = ic δ ( x − y ) , (2.58)where c is some constant.Suppose that A ( e aµ , t a ) is any local functional of the background fields with thefollowing 2 properties:1. A is second order in the background fields, that is A | flat = δ A δe aµ (cid:12)(cid:12)(cid:12) flat = δ A δt a (cid:12)(cid:12)(cid:12) flat = 0,but δ A δe aµ δe bρ (cid:12)(cid:12)(cid:12) flat (cid:54) = 0.2. A is a Weyl invariant density of dimension d + z , that is: δ Wσ A = − ( d + z ) σ A . (2.59)Next, consider adding to the curved spacetime classical action S ( e aµ , t a ) a term of theform: S ≡ (cid:90) d d +1 w e A ( w ) φ ( w ) . (2.60) This derivation assumes that the anisotropic Weyl transformation of the dynamic fields δ W φ does notexplicitly depend on the vielbeins e aµ . This is indeed the case whenever the dynamic fields transformcovariantly under anisotropic Weyl transformations, i.e. δ W φ = sσφ for some s . – 12 –he new action ˜ S = S + S is still invariant under TPD and anisotropic Weyl transfor-mations, and it coincides with S in flat spacetime. Moreover, the flat spacetime conservedcurrents derived from ˜ S (including the stress-energy tensor) are the same as those derivedfrom S . As a result, the anomalous Ward identity expressions I (2) W , I (3) W (as defined in(2.47)–(2.48)) remain unchanged in flat spacetime. However, expression W (3) W does change.Due to the assumed properties of A , the term S satisfies: D aµ δ S δe aµ ( x ) δe bρ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) flat = 0 , (2.61)and therefore does not contribute to J (as defined in (2.53)) either. The only contributionof S to the anomalous identity W (3) W is from the expression K (as defined in (2.54)). Itfollows that the change in W (3) W due to the S term is given by:(∆ W (3) W ) ραbc ( x, y, z ) = − i (cid:28) D ( x ) δ S δe bρ ( y ) δe cα ( z ) (cid:29) F = − i (cid:90) d d +1 w δ A ( w ) δe bρ ( y ) δe cα ( z ) (cid:10) D ( x ) φ ( w ) (cid:11) F = − i (cid:90) d d +1 w δ A ( w ) δe bρ ( y ) δe cα ( z ) icδ ( x − w )= c δ A ( x ) δe bρ ( y ) δe cα ( z ) . (2.62)If we choose A = a ˜ A , where ˜ A is a possible B-type anomaly of the theory and a is aconstant, we conclude that S contributes an additional − ac to the coefficient of the ˜ A anomaly, without changing the action or the current operators of the flat space theory. The freedom to add such a term to the action and thereby change the correspondinganomaly coefficient was previously pointed out in [27] for a purely spatial anomaly of afree Lifshitz scalar in d = z = 2, but it is in fact more general. It could be applied, forexample, to any of the consistent anomalies in the d = z = 2 case (which are all B-type,see [6, 7, 10, 11]), for any theory that contains a Lifshitz scalar. This freedom may suggestthat, unlike the relativistic case, knowing the flat space action, the flat space currentsand their Feynman correlation functions is not enough in these cases to determine theconsistent anomaly coefficients – one needs to specify the full curved spacetime action, andby choosing different couplings of the theory to curved spacetime one may obtain any valuefor any of the B-type anomaly coefficients.Stated differently, knowing all the Feynman correlation functions of the flat space fieldtheory is not sufficient in order to determine the curved spacetime action. In order toconstruct the full curved spacetime action we have to know all the variational correlationsfunctions and the ambiguity is in the relation between these two types of correlation func-tions, the variational and Feynman. This ambiguity can be avoided if we add anotheringredient to the discussion. The φ field in (2.60) is a Log-correlated field and is therefore In fact, the same argument can be made using the more general term S ≡ (cid:82) d d +1 w e A ( w ) φ n ( w ). – 13 –ll defined when we take the large volume limit. Consistency of the quantum field theoryat infinite volume forbids such an operator in the correlation functions. The ambiguitymay still have consequences for field theories on a finite volume spacetime, or with othermodifications of the IR physics that take care of the Log divergence. We leave this forfuture studies.In the following sections, we use split dimensional regularization to calculate the Lif-shitz anomaly coefficients for a free Lifshitz scalar in 2 + 1 dimensions and z = 2. Theabove discussion implies that a specific coupling of the theory to curved spacetime needsto be specified. However, since we are performing the calculation for a free theory in aninfinite volume and no physical IR regulator, we will use the minimal coupling (which isWeyl invariant in this case) and will not allow for Weyl invariant couplings of the form(2.60). In this section we present the split dimensional regularization scheme we employ and explainthe method by which we use it to extract the Lifshitz anomaly coefficients from the fieldtheory correlation functions. We start by reviewing the relativistic conformal case forreference, and then explain the non-relativistic Lifshitz case.
In the standard relativistic dimensional regularization scheme, one defines the theory andcalculates various quantities in a general dimension d , and then analytically continues theobtained expressions to dimension d = d phys − ε , where d phys is the physical dimension.Suppose that I ( n ) ( d, p i , m ) is some n -point correlation function written in momentum spaceand calculated to one-loop order in perturbation theory using the corresponding 1PI Feyn-man diagrams, where p µi ( i = 1 , . . . , n ) are external momenta and m is an IR mass regulator.Generally after analytic continuation of the dimension, I ( n ) will take the form: I ( n ) ( ε, p i , m ) = 1 ε f ( ε, p i , m ) , (3.1)where f ( ε, p i , m ) is an expression which is regular around ε = 0, and is a linear combinationof terms of the form: g ( ε, p i , m ) p µ i p µ i . . . η ν ν η ν ν . . . , (3.2) We thank Z. Komargodski for this comment. Beyond one-loop order, one has to first cancel the possible subdivergences using the appropriate coun-terterms. The pole in ε can then be of higher order. In the following sections we focus on a free theory,and therefore on the one-loop case. – 14 –here g ( ε, p i , m ) is a scalar expression. Expanding around ε = 0 we have: I ( n ) ( ε, p i , m ) = 1 ε f (0 , p i , m ) + ∂∂ε f (0 , p i , m ) + O ( ε ) ≡ ε I (res)( n ) ( p i , m ) + I (ren)( n ) ( p i , m ) + O ( ε ) , (3.3)where I (res)( n ) is the residue of the ε pole, and I (ren)( n ) ≡ lim ε → (cid:104) I ( n ) − ε I (res)( n ) (cid:105) is the renor-malized correlation function. It is a well known property of relativistic field theories thatthe residue I (res)( n ) is always a polynomial in the external momenta and the mass regulator(see for example [28–30]). This can be shown by taking derivatives of I ( n ) with respect tothe external momenta (and mass regulator) enough times so that the corresponding Feyn-man diagram no longer diverges. As long as there are no IR divergences in the physicaldimension, one can safely take the limit m → I (ren)( n ) to obtain the physical renor-malized correlation function (the correlation functions of the stress-energy tensor in thecases studied here are indeed free of IR divergences, even in d = 2 dimensions, as will beexplained in subsection 4.1.2).A useful property of scale anomalies is that, in some cases, one can calculate themfrom the ε pole residue alone: Suppose the theory has a symmetry (such as a scalingsymmetry) that is not explicitly broken by the dimensional regularization scheme itself,with a corresponding Ward identity of the form: T ( ε ) [ I k ] = 0 , (3.4)where { I k } is a set of correlation functions and T ( ε ) is a linear operator that takes ex-pressions of the form (3.2) to expressions of the same form, and may or may not dependon the dimension. Since the symmetry is not broken by dimensional regularization, theunrenormalized correlation functions satisfy identity (3.4). Therefore we can deduce:1 ε T ( ε ) (cid:104) I (res) k (cid:105) = − T ( ε ) (cid:104) I (ren) k (cid:105) + O ( ε ) . (3.5)The anomalous Ward identity is then given by: T (0) (cid:104) I (ren) k (cid:105) = − lim ε → (cid:18) ε T ( ε ) (cid:104) I (res) k (cid:105)(cid:19) . (3.6)Since the LHS of equation (3.6) is finite, we can immediately draw two conclusions fromit: 1. T ( ε ) (cid:104) I ( res ) k (cid:105) ∼ O ( ε ). We use a minimal subtraction renormalization scheme. If no IR divergences occur, I ( n ) is regular when m →
0, and I (res)( n ) is polynomial in m and thereforealso regular. Therefore I (ren)( n ) is regular in this limit too, and the order of taking the limit m → – 15 –. If T does not depend on ε then T (cid:104) I ( res ) k (cid:105) = 0 and there is no anomaly. For exam-ple, the operator T corresponding to diffeomorphism invariance does not explicitlyintroduce new factors that depend on d , and therefore as long as the dimensionalregularization itself does not break this symmetry, it will not be anomalous. Equation (3.6) allows one to calculate the anomalous Ward identity from the ε pole residue.This is useful, since there is no need to calculate the full correlation functions in order toextract their divergent part – it can be obtained simply by expanding the Feynman diagramintegrand in powers of the external momenta. Suppose the integrand is h ( p i , q, m ), where q is the internal loop momentum and h has a mass dimension l and therefore satisfies: h ( λp i , λq, λm ) = λ l h ( p i , q, m ) . (3.7)Rescaling q and m by a factor of 1 /λ where λ →
0, we get: h (cid:16) p i , qλ , mλ (cid:17) = λ − l h ( λp i , q, m ) . (3.8)We next expand in powers of λ around λ = 0 to obtain: h (cid:16) p i , qλ , mλ (cid:17) = λ − l (cid:34) k (cid:88) k =0 λ k k ! p µ i p µ i . . . p µ k i k ∂ k h∂p µ i ∂p µ i . . . ∂p µ k i k (0 , q, m ) + O ( λ k +1 ) (cid:35) = k (cid:88) k =0 k ! p µ i p µ i . . . p µ k i k ∂ k h∂p µ i ∂p µ i . . . ∂p µ k i k (cid:16) , qλ , mλ (cid:17) + O ( λ k − l +1 ) . (3.9)Defining: ˜ h ( p i , q, m ) ≡ k (cid:88) k =0 k ! p µ i p µ i . . . p µ k i k ( h ( k ) ) i ...i k µ ...µ k ( q, m ) , (3.10)where ( h ( k ) ) i ...i k µ ...µ k ( q, m ) ≡ ∂ k h∂p µ i ∂p µ i . . . ∂p µ k i k (0 , q, m ) , (3.11)we conclude that the integral over h ( p i , q, m ) − ˜ h ( p i , q, m ) has a divergence degree of d + l − k −
1. If we choose k = d phys + l , the integral over h ( p i , q, m ) − ˜ h ( p i , q, m ) convergesin d = d phys dimensions, and therefore the integrals over h ( p i , q, m ) and ˜ h ( p i , q, m ) havethe same ε pole.Thus in order to calculate the pole residue, the only integrals left to evaluate are theones over the expressions ( h ( k ) ) i ...i k µ ...µ k ( q, m ). Each of these expressions is a linear combina- Formally, this expansion is done after performing a Wick rotation to Euclidean signature, however oneobtains the same results by performing the expansion first and Wick rotating only in the last step whenevaluating the integrals (3.15). Note that m always appears alongside q in these 1PI diagrams, and therefore scaling q and m togetherhere still gives the correct divergence degree in q . – 16 –ion of terms of the form: η ν ν η ν ν . . . ( q ) a q µ q µ . . . q µ b [ q − m + i(cid:15) ] s . (3.12)When performing the integration over q we may use the Lorentz symmetry of the integralto make the standard replacement: q µ q µ . . . q µ b ⇒ , b = 2 n − n (cid:81) j =0 1 d +2 j G ( n ) µ µ ...µ n , b = 2 n , (3.13)where G ( n ) µ µ ...µ n is a sum of all possible unique products of n metric factors, i.e.: G ( n ) µ µ ...µ n = η µ µ η µ µ . . . η µ n − µ n + All possible permutations . (3.14)Using (3.13), the integral over ˜ h ( p i , q, m ) can be written in terms of the following knownintegral (see e.g. [30–32]): X ( d, r, s, m ) ≡ (cid:90) d d q (2 π ) d (cid:0) q (cid:1) r [ q − m + i(cid:15) ] s = i ( − r − s (4 π ) d/ Γ ( r + d/
2) Γ ( s − r − d/ d/
2) Γ ( s ) (cid:0) m (cid:1) r − s + d/ , (3.15)where k = l − r + 2 s ≤ d phys + l (so that d phys + 2 r − s ≥ m r − s + d phys . Since termswith 2 r − s + d phys > m → r − s + d phys = 0 (those of order k = d phys + l ) in order to obtain the ε pole residue. For these terms, the integral (3.15) has the following pole: X ( d, r, s, m ) = i ( − r − s ε π ) d phys / Γ( r + d phys / d phys / s ) + O (1) . (3.16)Using this procedure of expanding the Feynman diagram integrand in external momenta,computing the pole residue via equations (3.13)–(3.16) and applying formula (3.6), onecan calculate the anomalous Ward identities corresponding to scale symmetry withoutcalculating the full correlation functions. This will be especially useful in the Lifshitz case,where the Feynman diagrams are more difficult to fully evaluate. In the non-relativistic case, in analogy to the relativistic case, we use a split dimensionalregularization scheme (first suggested in [13, 14] for regularizing gauge theories in theCoulomb gauge). We follow the general scheme defined in [15], in the context of non-relativistic field theories. We start by defining the theory in a general number of time– 17 –imensions d t and space dimensions d s (while keeping the critical dynamical exponent z constant).In flat spacetime, the theory is defined on a manifold M = M t × M s , where M t isa d t -dimensional time manifold and M s is a d s -dimensional space manifold, such that isinvariant both under rotations in the time manifold and in the space manifold separately.Spacetime coordinates will be denoted by x µ = ( x ˆ µ , x ¯ µ ) where ˆ µ = 1 , . . . , d t are timeindexes, ¯ µ = 1 , . . . , d s are space indexes and µ = 1 , . . . , d t + d s are spacetime indexes. Wedefine a flat metric ˆ δ ˆ µ ˆ ν = diag( − , . . . , −
1) on M t , and ¯ δ ¯ µ ¯ ν = diag(1 , . . . ,
1) on M s . Wealso define the time projector on M as ˆ δ µν = diag(ˆ δ ˆ µ ˆ ν ,
0) and similarly the space projectoras ¯ δ µν = diag(0 , ¯ δ ¯ µ ¯ ν ), so that δ µν = ˆ δ µν + ¯ δ µν . Given a vector v µ on M , we denote its timeprojection by ˆ v µ ≡ ˆ δ µν v ν , and its space projection by ¯ v µ ≡ ¯ δ µν v ν . Similarly to the relativistic case, we calculate various expressions for general d t and d s values, and then analytically continue them to non-integer dimensions d t = 1 − ε t and d s = d phys s − ε s , where d phys s is the number of physical space dimensions. Supposethat I ( n ) ( d t , d s , p i , m ) is some n -point correlation function calculated to one-loop order inperturbation theory using the corresponding 1PI Feynman diagrams, where p µi ( i = 1 , . . . n )are external momenta and m is an IR mass regulator. Generally after analytic continuationof the dimensions, I ( n ) will take the form (see [15]): I ( n ) ( ε t , ε s , p i , m ) = 1 ε lif f ( ε t , ε s , p i , m ) , (3.17)where ε lif ≡ zε t + ε s and f ( ε t , ε s , p i , m ) is an expression which is regular around ε t = ε s = 0and is a linear combination of terms of the form: g ( ε t , ε s , p i , m )ˆ p µ i ˆ p µ i . . . ¯ p ν j ¯ p ν j . . . ˆ δ ρ ρ ˆ δ ρ ρ . . . ¯ δ σ σ ¯ δ σ σ . . . , (3.18)where g ( ε t , ε s , p i , m ) is a scalar expression (with respect to time and space rotations).Since there are two different dimensional regularization parameters in this case, unlikethe relativistic case, in order to renormalize the expression one must choose a particularway in which one takes the limit ( ε t , ε s ) → (0 , ε ( ε t , ε s ) to be some coordinate on the two dimensionalregularization parameter space such that ˜ ε (0 ,
0) = 0 and such that the transformation( ε t , ε s ) → ( ε lif , ˜ ε ) is regular and invertible around ε t = ε s = 0. Expanding f ( ε t , ε s , p i , m )in ε lif while keeping ˜ ε constant, we have: I ( n ) ( ε lif , ˜ ε, p i , m ) = 1 ε lif f (0 , ˜ ε, p i , m ) + ∂f∂ε lif (cid:12)(cid:12)(cid:12)(cid:12) ˜ ε (0 , ˜ ε, p i , m ) + O ( ε lif )= 1 ε lif f (0 , ˜ ε, p i , m ) + ∂f∂ε lif (cid:12)(cid:12)(cid:12)(cid:12) ˜ ε (0 , , p i , m ) + O ( ε lif ) + O (˜ ε ) ≡ ε lif I (res)( n ) (˜ ε, p i , m ) + I (ren)( n ) ( p i , m ) + O ( ε lif ) + O (˜ ε ) , (3.19) See Appendix A for our conventions and notations. – 18 –here I (res)( n ) is the residue of the ε lif pole, and I (ren)( n ) ≡ lim ( ε lif , ˜ ε ) → (cid:104) I ( n ) − ε lif I (res)( n ) (cid:105) is therenormalized correlation function. Like the relativistic case, the residue I (res)( n ) (˜ ε ) is a poly-nomial in the external momenta and the mass regulator for any value of ˜ ε (see [15]), andtherefore represents contact terms in coordinate space. This can be shown by taking deriva-tives of I ( n ) with respect to the external momenta (and mass regulator) enough times sothat the corresponding Feynman diagram integral no longer diverges. As long as there areno IR divergences in the physical dimension, one can safely take the limit m → I (ren)( n ) to obtain the physical renormalized correlation function (the correlation functions of thestress-energy tensor in the case studied here are indeed free of IR divergences, as will beexplained in section 5).As mentioned earlier, the renormalized function I (ren)( n ) depends on the choice of theparameter ˜ ε that is kept constant as we take the limit ( ε t , ε s ) → (0 , ε (cid:48) ( ε t , ε s ), that corresponds to the renormalized correlationfunction (cid:16) I (ren)( n ) (cid:17) (cid:48) . Then we have the following relation between I (ren)( n ) and (cid:16) I (ren)( n ) (cid:17) (cid:48) : I (ren)( n ) = ∂f∂ε lif (cid:12)(cid:12)(cid:12)(cid:12) ˜ ε =const (0 , , p i , m )= ∂f∂ε lif (cid:12)(cid:12)(cid:12)(cid:12) ˜ ε (cid:48) =const (0 , , p i , m ) + ∂f∂ ˜ ε (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ε lif =const (0 , , p i , m ) ∂ ˜ ε (cid:48) ∂ε lif (cid:12)(cid:12)(cid:12)(cid:12) ˜ ε =const (0 , (cid:16) I (ren)( n ) (cid:17) (cid:48) − α ∂f∂ ˜ ε (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ε lif =const (0 , , p i , m ) , (3.20)where α ≡ − ∂ ˜ ε (cid:48) ∂ε lif (cid:12)(cid:12)(cid:12) ˜ ε (0 , f (0 , ˜ ε (cid:48) , p i , m ) is a polynomial in p i and m forany value of ˜ ε (cid:48) , so is ∂f∂ ˜ ε (cid:48) (cid:12)(cid:12)(cid:12) ε lif (0 , , p i , m ). Therefore I (ren)( n ) and (cid:16) I (ren)( n ) (cid:17) (cid:48) differ from eachother by contact terms, as expected from a change in the renormalization scheme. Alsonote that the possible change in the renormalized expressions as a result of the choice of˜ ε is completely described by the single parameter α . In order to account for all possiblechoices, we leave α as a free parameter in our calculations, and use the following choice of˜ ε : ˜ ε = ε t + α zα ε s , (3.21)which is chosen such that moving from ˜ ε to ˜ ε (cid:48) = ε t we get the factor ∂ ˜ ε (cid:48) ∂ε lif (cid:12)(cid:12)(cid:12) ˜ ε = ∂ε t ∂ε lif (cid:12)(cid:12)(cid:12) ˜ ε = − α .The inverse transformation from ( ε lif , ˜ ε ) to ( ε t , ε s ) is given by: ε t = − αε lif + (1 + zα )˜ ε,ε s = (1 + zα )( ε lif − z ˜ ε ) . (3.22)As in the relativistic case, we can calculate the Lifshitz scale anomaly coefficients fromthe ε lif pole residue. Suppose the theory has a symmetry (such as a Lifshitz scale symmetry)that is not explicitly broken by the split dimensional regularization scheme itself, with a– 19 –orresponding Ward identity of the form: T ( ε lif , ˜ ε )[ I k ] = 0 , (3.23)where { I k } is a set of correlation functions and T ( ε lif , ˜ ε ) is a linear operator that takes ex-pressions of the form (3.18) to expressions of the same form, and may or may not dependexplicitly on the time and space dimensions. Since the symmetry is not broken by the reg-ularization, the unrenormalized correlation functions satisfy identity (3.23). We thereforehave from (3.19): 1 ε lif T ( ε lif , ˜ ε ) (cid:104) I (res) k (cid:105) = − T ( ε lif , ˜ ε ) (cid:104) I (ren) k (cid:105) + O ( ε lif ) + O (˜ ε ) . (3.24)The anomalous Ward identity is then given by: T (0 , (cid:104) I (ren) k (cid:105) = − lim ( ε lif , ˜ ε ) → (cid:18) ε lif T ( ε lif , ˜ ε ) (cid:104) I (res) k (˜ ε ) (cid:105)(cid:19) . (3.25)Since the LHS of equation (3.25) is finite, we can again draw the following conclusions:1. T ( ε lif , ˜ ε ) (cid:104) I (res) k (˜ ε ) (cid:105) ∼ O ( ε lif ).2. If T does not depend on ε lif and ˜ ε then T (cid:104) I (res) k (˜ ε ) (cid:105) = 0, and there is no anomaly.Therefore, as long as TPD invariance is not explicitly broken by the split dimensionalregularization scheme (as is the case with the free scalar we consider in the followingsections), we don’t expect it to be anomalous. It is important to consider the consequences of changing the choice of the parameter ˜ ε (and thereby the renormalization) on the anomalous Ward identity (3.25). By changing ourchoice from ˜ ε (cid:48) to ˜ ε , we know from (3.20) that the change in the anomalous Ward identityis given by: T (0 , (cid:104) I (ren) k (cid:105) = T (0 , (cid:20)(cid:16) I (ren)( n ) (cid:17) (cid:48) (cid:21) − α T (0 , (cid:20) ∂f∂ ˜ ε (cid:48) (0 , (cid:21) . (3.26)Since ∂f∂ ˜ ε (cid:48) (0 ,
0) is a local expression (a polynomial in the external momenta), the term T (0 , (cid:104) ∂f∂ ˜ ε (cid:48) (0 , (cid:105) represents a trivial solution of the WZ consistency condition (one thatcan be cancelled by a local counterterm). We therefore expect only the coefficients of trivialterms to depend on α . This is consistent with the general expectation that only coefficientsof trivial terms can be regularization dependent. Equation (3.25) allows us to calculate the anomalous Ward identity from the ε lif poleresidue. This is especially useful in the Lifshitz case, since the denominators of the prop-agators are generally polynomials of degree 2 z , and the Feynman diagram integrals are Like in the relativistic case, the operator T that corresponds to conservation of the stress-energy tensordoes not introduce any factors that depend on d t or d s . – 20 –herefore more difficult to fully evaluate than they are in the relativistic case. Their di-vergent parts, however, can again be obtained simply by expanding the Feynman diagramintegrands in powers of the external momenta. Suppose the integrand is h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ),where q is the internal loop momentum and h has a Lifshitz dimension l and thereforesatisfies: h ( λ z ˆ p i , λ ¯ p i , λ z ˆ q, λ ¯ q, λ z m ) = λ l h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) . (3.27)Rescaling ¯ q by a factor of λ , and ˆ q and m by a factor of λ z (where λ →
0) we get: h (cid:18) ˆ p i , ¯ p i , ˆ qλ z , ¯ qλ , mλ z (cid:19) = λ − l h ( λ z ˆ p i , λ ¯ p i , ˆ q, ¯ q, m ) . (3.28)We next expand h ( λ z ˆ p i , λ ¯ p i , ˆ q, ¯ q, m ) in powers of λ to obtain: h (cid:18) ˆ p i , ¯ p i , ˆ qλ z , ¯ qλ , mλ z (cid:19) = λ − l (cid:34) k (cid:88) k =0 λ k h ( k ) (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) + O ( λ k +1 ) (cid:35) = k (cid:88) k =0 h ( k ) (cid:18) ˆ p i , ¯ p i , ˆ qλ z , ¯ qλ , mλ z (cid:19) + O ( λ k − l +1 ) , (3.29)where h ( k ) (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) is a polynomial in the external momenta, given by: h ( k ) (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) ≡ (cid:88) rz + s = k r ! s ! ˆ p µ i . . . ˆ p µ r i r ¯ p ν j . . . ¯ p ν s j s ∂ r + s h∂ ˆ p µ i . . . ∂ ˆ p µ r i r ∂ ¯ p ν j . . . ∂ ¯ p ν s j s (0 , , ˆ q, ¯ q, m ) . (3.30)Defining: ˜ h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) ≡ k (cid:88) k =0 h ( k ) (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) , (3.31)we conclude that the integral over h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) − ˜ h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) has a divergence degreeof zd t + d s + l − k −
1. If we choose k = d phys s + z + l , the integral over h − ˜ h converges in d phys s + 1 dimensions, and therefore the integrals over h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) and ˜ h (ˆ p i , ¯ p i , ˆ q, ¯ q, m )have the same ε lif pole.Thus in order to calculate the pole residue, the only integrals left to evaluate are theones over the polynomial coefficients in the expressions ˜ h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ). Each of these is alinear combination of terms of the form:ˆ δ ρ ρ ˆ δ ρ ρ . . . ¯ δ σ ¯ δ σ . . . (ˆ q ) a (¯ q ) b ˆ q µ . . . ˆ q µ c ¯ q ν . . . ¯ q ν d [ˆ q + κ (¯ q ) z + m + i(cid:15) ] I . (3.32)When performing the integrations over ˆ q and ¯ q , we may use the time rotation and spacerotation symmetries to make the replacements given in equations (3.13)–(3.14) separately In cases where the integrand does not have a uniform Lifshitz dimension, one can always write it as asum of terms with uniform Lifshitz dimensions and calculate the pole residue of each of them separately. – 21 –or ˆ q products (using the ˆ δ µν metric) and for ¯ q products (using the ¯ δ µν metric). The integralover ˜ h (ˆ p i , ¯ p i , ˆ q, ¯ q, m ) can then be written in terms of the following known integral (see [15]): X ( d t , d s , r, s, I, z ) ≡ (cid:82) d dt ˆ q (2 π ) dt (cid:82) d ds ¯ q (2 π ) ds ( ˆ q ) r ( ¯ q ) s [ˆ q + κ (¯ q ) z + m + i(cid:15) ] I == i dt κ − (2 s + ds ) / z ( m ) r − I + s/z +( dt + ds/z ) / z (4 π ) ( dt + ds ) / Γ ( s + ds z ) Γ (cid:16) r + dt (cid:17) Γ (cid:16) dt (cid:17) Γ ( ds ) Γ( I ) Γ (cid:0) I − r − sz − zd t + d s z (cid:1) , (3.33)where k = l − zr − s + 2 zI ≤ d phys s + z + l (so that d phys s + z + 2 zr + 2 s − zI ≥ m r − I +2 s/z +1+ d phys s /z . Since terms with 2 r − I +2 s/z +1+ d phys s /z > m → r − I +2 s/z +1+ d phys s /z = 0(those of order k = d phys s + z + l ) in order to obtain the ε lif pole residue. For these terms,the integral (3.33) has the following pole: X ( d t , d s , r, s, I, z ) = i d t ε lif κ − (2 s + d s ) / z (4 π ) ( d t + d s ) / Γ (cid:0) s + d s z (cid:1) Γ (cid:0) r + d t (cid:1) Γ (cid:0) d t (cid:1) Γ (cid:0) d s (cid:1) Γ( I ) + O (1) . (3.34)Note that since r, s and I appear in the correlation functions as non-negative integers, it ispossible to relate the different poles X ( d t , d s , r, s, I, z ) appearing in the correlation functionof a given order for a fixed value of z and with various values of r, s, I using the recursiveproperty of Gamma functions Γ( n + 1) = n Γ( n ).In conclusion, this procedure of expanding the Feynman diagram integrand in externalmomenta, computing the pole residue via equations (3.13)–(3.14), (3.33)–(3.34) and apply-ing formula (3.25) enables us to calculate the anomalous Ward identities corresponding toLifshitz scale symmetry without calculating the full correlation functions. In the followingsections we use this procedure to calculate the anomaly coefficients for the case of a free z = 2 Lifshitz scalar in 2 + 1 dimensions. In this section we review the calculation of the conformal anomaly coefficients for therelativistic free and massless scalar field using dimensional regularization and the proceduredescribed in section 3. We include this example as a reference for the calculation of Lifshitzanomaly coefficients for the non-relativistic Lifshitz scalar given in section 5 using a similarprocedure. The calculation was performed both for two and four spacetime dimensions. Inboth cases the Weyl anomalies agree with the known results found in literature [4, 20–25].In order to calculate the conformal anomalies for the free scalar, one must first definethe flat space stress-energy tensor such that it satisfies the Ward identities (2.5)–(2.6) (i.e.it is conserved, symmetric and traceless). One way to do this is to define the theory overa curved spacetime manifold such that the action is invariant under both diffeomorphismsand Weyl transformations (see section 2), and derive the stress-energy tensor from thecurved spacetime action using the definition (2.4).– 22 –he conformal coupling of a free relativistic scalar field φ to curved spacetime is givenby the following action (see e.g. [23]): S = (cid:90) d d x (cid:112) | g | (cid:20) ∂ µ φ∂ µ φ − d −
24 (1 − d ) Rφ (cid:21) , (4.1)where d is the number of spacetime dimensions, and R is the Ricci scalar of the backgroundmanifold. This action in indeed diffeomorphism and Weyl-invariant. The improved stress-energy tensor calculated from this action using the definition in equation (2.4) is (see [5]): T µν = − ∂ µ φ∂ ν φ + 12 η µν ( ∂ ρ φ∂ ρ φ ) − d −
24 (1 − d ) (cid:0) ∂ µ ∂ ν − η µν ∂ (cid:1) φ . (4.2)Using the equations of motion, one can check that it indeed satisfies the conservation andtracelessness Ward identities: ∂ µ T µν = 0 , (4.3) T µµ = 0 , (4.4)for any number of spacetime dimensions d . We start with the calculation of the Weyl anomaly of a relativistic free scalar field in twospacetime dimensions from the two point correlation function of the stress-energy tensor.We demonstrate two ways of performing the calculation: First, by preforming the fullcalculation of the one-loop diagram using dimensional regularization. Second, using theprocedure of extracting only the divergent part of the diagram and using it to computethe anomaly coefficients, as described in section 3. The results agree with the known onesfrom the literature.
In the first way of calculating the anomaly, the two point correlation function of the stress-energy tensor is fully calculated using standard dimensional regularization. The Feynmanrules and diagram used for the calculation are given in appendix C.1.The full evaluation of the expression that corresponds to the diagram (C.2) was per-formed using the massless integral formulas given in appendix C.2. The final result for the The stress-energy tensor satisfies these identities as operator equations, taking into account the equa-tions of motion for φ . Its renormalized correlation functions, however, satisfy the corresponding identitiesonly up to local contact terms, as mentioned in sections 2 and 3. – 23 –wo point correlation function is given by the following expression: (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) = − (cid:20) p η µν η ρσ (cid:18) −
118 (3 + 7 ε ) (cid:19) I + p ( η µρ η νσ + η µσ η νρ ) (cid:18)
112 + ε (cid:19) I ++ p ( η µν p ρ p σ + η ρσ p µ p ν ) (cid:18)
118 (3 + 7 ε ) (cid:19) I − ε p µ p ν p ρ p σ I ++ p ( η µρ p ν p σ + η µσ p ρ p ν + η ρν p µ p σ + η νσ p ρ p µ ) (cid:18) −
136 (3 + 4 ε ) (cid:19) I (cid:21) , (4.5)where p is the external momentum of the diagram, ε is defined by d = 2 − ε and the basicintegral I is given by: I ( d, p ) = (cid:90) d d q (2 π ) d q p − q ) = ( − d/ − i Γ ( d/ −
1) Γ ( d/ −
1) Γ (2 − d/ π ) d/ Γ ( d − (cid:0) p (cid:1) d/ − . (4.6)One can expand the integral I around the physical dimension d phys = 2 to obtain: I ( d = 2 − ε, p ) = 1 ε (cid:18) iπ p − (cid:19) + O (1) . (4.7)It is easy to verify that the result in equation (4.5) indeed satisfies the conservation Wardidentity (2.17), which in Fourier space takes the form: p µ (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) = 0 . (4.8)As expected, this identity holds separately on the finite part and on the pole part of(4.5) and is not anomalous. In addition one can verify that, when tracing over the fullunrenormalized expression (4.5), the Ward identity corresponding to Weyl invariance issatisfied: η µν (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) = 0 . (4.9)However, this identity is only satisfied on the pole and finite parts of expression (4.5)together. Thus after performing renormalization of the correlation function we have fromequation (3.6): η µν (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) (ren) = − lim ε → (cid:18) ε η µν (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) (res) (cid:19) . (4.10) See equations (A.15) and (A.16) for our Fourier conventions. – 24 –he ε pole residue of expression (4.5) is given by: (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) (res) = i π (cid:20) p η µν η ρσ − p ( η µρ η νσ + η µσ η νρ ) + −
318 ( η µν p ρ p σ + η ρσ p µ p ν ) + 112 ( η µρ p ν p σ + η µσ p ρ p ν + η ρν p µ p σ + η νσ p ρ p µ ) (cid:21) . (4.11)Tracing over (4.11) (in d = 2 − ε dimensions) and using equation (4.10) we get: η µν (cid:104) T µν ( − p ) T ρσ ( p ) (cid:105) (ren) = i π (cid:0) p η ρσ − p ρ p σ (cid:1) , (4.12)which is the well known anomalous Ward identity in two dimensions (see e.g. [16, 20]). Using the procedure described in subsection 3.1, the pole residue of the two point corre-lation function of the stress-energy tensor can also be computed without evaluating thefull expression. Starting from the expression for the two point function given in (C.2), weexpand the integrand in powers of the external momentum and extract the terms propor-tional to m (where m is an IR mass regulator). The expression for the two point functioncontains only five different types of integrals in this case. Their relevant ε poles around twospacetime dimensions, obtained using expansion in the external momentum p , are given bythe following expressions: (cid:90) d d q (2 π ) d q − m + i(cid:15) ] 1 (cid:104) ( q − p ) − m + i(cid:15) (cid:105) = O (1) , (4.13) (cid:90) d d q (2 π ) d q µ [ q − m + i(cid:15) ] 1 (cid:104) ( q − p ) − m + i(cid:15) (cid:105) = O (1) , (4.14) I µν ( p ) ≡ (cid:90) d d q (2 π ) d q µ q ν [ q − m + i(cid:15) ] 1 (cid:104) ( q − p ) − m + i(cid:15) (cid:105) = I µν ( p = 0) + O (1) = − i π η µν Γ (cid:18) − d (cid:19) (cid:18) m (cid:19) − d + O (1) , (4.15) I µνρ ( p ) ≡ (cid:90) d d q (2 π ) d q µ q ν q ρ [ q − m + i(cid:15) ] 1 (cid:104) ( q − p ) − m + i(cid:15) (cid:105) = p α (cid:20) dI µνρ ( p ) dp α (cid:21) p =0 + O (1)= ( − i ) ( p ρ η µν + p µ η ρν + p ν η ρµ )16 π Γ (cid:18) − d (cid:19) (cid:18) m (cid:19) − d + O (1) , (4.16)– 25 – µνρσ ( p ) ≡ (cid:90) d d q (2 π ) d q µ q ν q ρ q σ [ q − m + i(cid:15) ] 1 (cid:104) ( q − p ) − m + i(cid:15) (cid:105) = 12! p α p β (cid:20) d I µνρσ ( p ) dp α dp β (cid:21) p =0 + O (1)= (cid:18) i π J µνρσ − i π K µνρσ (cid:19) Γ (cid:18) − d (cid:19) (cid:18) m (cid:19) − d + O (1) , (4.17)where: J µνρσ ≡ p ( η µν η ρσ + η µσ η ρν + η µρ η νσ ) , (4.18) K µνρσ ≡ p ρ p σ η µν + p ρ p ν η µσ + p ρ p µ η νσ + p σ p µ η νρ + p σ p ν η µρ + p µ p ν η ρσ . (4.19)Note that although some of these expressions (4.13)–(4.17) are individually IR diver-gent at d = 2, the total expression for the two point function of the stress-energy tensor(and indeed any correlation functions of the stress-energy tensor) is not IR divergent inthis case. This follows from the form of the action (4.1) and the improved stress-energytensor (4.2): The action at d = 2 contains only derivatives of the field φ , and only the O ( ε )part of it contains φ with no derivatives. It follows that the stress-energy tensor (and anyother operator constructed by taking variations of the action with respect to the metric)has the same structure – at d = 2 it contains only derivatives of φ . These operators aretherefore not Log correlated at d = 2, and their correlation functions do not diverge inthe IR. In terms of dimensionally regulated Feynman diagrams, for any propagator in the1-loop diagram with momentum q µ , the d = 2 part of each of its adjacent vertexes is oforder O ( q ). Thus the terms in the integrand of the diagram that contribute to the IRdivergence when q → O ( ε ), and therefore vanish as ε → We next turn to the example of a relativistic free scalar field in four spacetime dimensions.Again using dimensional regularization and the procedure described in subsection 3.1 with d = 4 − ε , the anomalous contributions to the Ward identities (2.19) and (2.20) can becomputed from the ε pole residues of the two and three point correlation functions of thestress-energy tensor.The details needed for the calculation of the correlation functions as defined in equation(2.14), including the Feynman rules for the vertexes and the expressions for the diagrams,are given in appendix C.1. The pole residues were extracted via a power expansion in theexternal momenta as described in subsection 3.1. Since these calculations involve a very– 26 –arge number of terms, they were performed using a computer script that was written forthis purpose.The pole residue of the two point correlation function, as obtained from the powerexpansion in the external momentum p , is given by: (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) (res) = − i π (cid:2) − p η µν η ρσ + p ( η µρ η νσ + η µσ η νρ )+ p ( η µν p ρ p σ + η ρσ p µ p ν ) + p µ p ν p ρ p σ − p ( η µρ p ν p σ + η µσ p ρ p ν + η ρν p µ p σ + η νσ p ρ p µ ) (cid:3) . (4.20)It can be easily checked that this expression satisfies the conservation identity (4.8). How-ever, tracing over the indexes µ, ν in (4.20) yields an expression that does not vanish for ageneral dimension d . Using equation (3.6), one obtains: η µν (cid:104) T µν ( − p ) T ρσ ( p ) (cid:105) (ren) = − i π p (cid:0) p ρ p σ − η ρσ p (cid:1) . (4.21)This is in agreement with the anomalous Ward identity (2.19), due to the trivial (cid:3) R termin the Weyl cohomology of the relativistic theory in four dimensions (see e.g. [24, 25]).The pole residue of the three point correlation function (cid:104) T µν ( k ) T ρσ ( q ) T αβ ( p ) (cid:105) (res) W canalso be calculated using expansion in the external momenta. However, the final result istoo long to be shown here. As expected, this pole residue and the pole residue of the twopoint function together satisfy the following conservation identity: k ν (cid:10) T µν ( k ) T ρσ ( q ) T αβ ( p ) (cid:11) W = ip µ (cid:10) T ρσ T αβ (cid:11) ( q ) + iq µ (cid:10) T ρσ T αβ (cid:11) ( p ) − ip ν (cid:2) η µβ (cid:104) T να T ρσ (cid:105) ( q ) + η µα (cid:10) T νβ T ρσ (cid:11) ( q ) (cid:3) − iq ν (cid:2) η µρ (cid:10) T νσ T αβ (cid:11) ( p ) + η µσ (cid:10) T νρ T αβ (cid:11) ( p ) (cid:3) , (4.22)which is the Fourier transformed version of the conservation Ward identity (2.18). Fromthese pole residues, one can use equation (3.6) to calculate the LHS of the Fourier trans-formed version of the anomalous Ward identity of the three point correlation function(2.20), given by: η µν (cid:68) T µν ( k ) T ρσ ( q ) T αβ ( p ) (cid:69) (ren) W − i (cid:68) T ρσ ( − q ) T αβ ( q ) (cid:69) (ren) − i (cid:68) T ρσ ( − p ) T αβ ( p ) (cid:69) (ren) = − (cid:104) δ (cid:16)(cid:112) | g |A (cid:17) ( q, p ) (cid:105) ρσαβ , (4.23)where (cid:104) δ (cid:16)(cid:112) | g |A (cid:17) ( q, p ) (cid:105) ρσαβ represents the Fourier transformed second variation of theanomalous contribution to the Ward identity (2.7) with respect to the metric (see appendixA.2 for notations). After calculating the second variation of the anomaly and trivial termdensities listed in equation (2.9) with respect to the background metric, the results can besubstituted into the RHS of the anomalous Ward identity (4.23).The coefficients β a , β b and β c can then be extracted by comparing the two sides of the– 27 –dentity. The values of the coefficients obtained using this procedure are the same as theones given in (2.11) and therefore agree with those found in the literature. z = 2 Lifshitz Scalar Field and its Scale Anomalies
In this section we study the Lifshitz anomaly of a Lifshitz z = 2 scalar field in 2 + 1 dimen-sions, using the method of split dimensional regularization and renomarlization describedin subsection 3.2, applied to the two and three point correlation functions of the stress-energy tensor. Up to second order in the background fields, we find one anomaly in the twoderivatives sector, and no anomalies in the four derivatives sector. This is in agreementwith the results previously found in [11] using a heat kernel calculation. The value of theanomaly coefficient also agrees with the result in [11]. As expected, only the coefficientsof the trivial terms which appear in the two and three point correlation functions dependon the regularization parameter α , defined in subsection 3.2 – they are all regularizationdependent, and can be removed by adding the appropriate counterterms to the effectiveaction. We also show that trivial terms that correspond to a curved background structurethat violates the Frobenius condition (that is, terms that vanish when the Frobenius con-dition is assumed) appear in the case we study with non-vanishing coefficients. Therefore,the curved spacetime description of these terms, and their cancellation via a counterterm,requires giving up the foliation structure of the background manifold (see [7] for furtherdiscussion). z = 2 Free Lifshitz Scalar Field in General Spacetime Dimensions
The Lifshitz anomalies of a free Lifshitz scalar field with a dynamical critical exponent z = 2 in 2 + 1 dimensions have been considered in several previous works [11, 27]. The flatspace action of the theory is given by: S = (cid:90) d xdt (cid:18)
12 ( ∂ t φ ) − κ ( ∇ φ ) (cid:19) . (5.1)In order to apply the method of split dimensional regularization and renormalizationdescribed in subsection 3.2, one must first couple the theory described by the action (5.1)to a curved background manifold with a general number of space dimensions d s and timedimensions d t . This coupling must be done in a way that preserves the curved spacetimesymmetries detailed in subsection 2.2.1, as well as local rotations of the time directions,and smoothly reduces back to (5.1) in the limit of flat spacetime and d s → , d t → S = (cid:90) d d t + d s x (cid:112) | g | (cid:26) (cid:104) L n ( i ) φ + ξ K ( i ) S φ (cid:105) − κ (cid:2) ¯ ∇ φ + ξ a µ ¯ ∇ µ φ + ξ a φ + ξ ¯ ∇ µ a µ φ (cid:3) (cid:111) , (5.2)– 28 –here d t and d s are the numbers of time and space dimensions respectively and { n ( i ) µ } ( i =1 , . . . , d t ) is an orthonormal set of 1-forms that represents the local d t time directions onthe manifold. The coefficients ξ , ξ , ξ , ξ are given by: ξ ≡ d s (cid:18) d lif − (cid:19) , ξ ≡ d t − d t ,ξ ≡ d t (cid:18) d lif − (cid:19) (cid:18) d t − d s (cid:19) , ξ ≡ d t (cid:18) d lif − (cid:19) , (5.3)where d lif ≡ d t + d s . The various background expressions and notations used in (5.2)(the derivatives L n ( i ) , ¯ ∇ µ and the expressions a µ , K ( i ) S ) are defined in appendix B as ageneralization of the definitions used in [6, 7] to the case of multiple time directions. Thisaction is indeed invariant under TPDs and anisotropic Weyl transformations, as well asunder local time rotations of the form n ( i ) µ → Λ ij ( x ) n ( j ) µ , where Λ ij ( x ) is any orthogonalmatrix in d t dimensions that depends on the spacetime coordinates x .The action (5.2) reduces in flat space to the following action: S = (cid:90) d d s + d t x (cid:18) −
12 ˆ ∂ µ φ ˆ ∂ µ φ − κ ¯ ∇ φ ¯ ∇ φ (cid:19) , (5.4)where, as mentioned in subsection 3.2, we use a flat spacetime metric of the form δ µν =diag(ˆ δ ˆ µ ˆ ν , ¯ δ ¯ µ ¯ ν ), with ˆ δ ˆ µ ˆ ν = diag( − , . . . , −
1) over the time dimensions and ¯ δ ¯ µ ¯ ν = diag(1 , . . . , x µ → λ − ˆ x µ , ¯ x µ → λ − ¯ x µ , φ → λ (2 d t + d s − / φ, (5.5)where λ is a parameter of the scaling transformation, and κ is dimensionless under thescaling transformation. The flat space stress-energy tensor, as derived from the action(5.2) by taking the variation of the action with respect to the vielbeins (according to thedefinition (2.25)), is given by: T µν = ˆ ∂ µ φ∂ ν φ + 2 κ ¯ ∂ µ ∂ ν φ ¯ ∇ φ − δ µν (cid:16) ˆ ∂ σ φ ˆ ∂ σ φ + κ ¯ ∇ φ ¯ ∇ φ (cid:17) − κ (cid:0) ∂ µ ¯ ∂ ν φ ¯ ∇ φ + ∂ µ φ ¯ ∇ ¯ ∂ ν φ + ¯ ∂ µ ∂ ν φ ¯ ∇ φ + ∂ ν φ ¯ ∇ ¯ ∂ µ φ (cid:1) + κ d t ((2 d t + d s ) − (cid:16) ˆ ∂ ν φ ¯ ∂ µ ¯ ∇ φ + φ ¯ ∂ µ ˆ ∂ ν ¯ ∇ φ (cid:17) + κ d t (cid:16) ( d s − d t ) (cid:16) ˆ ∂ ν ¯ ∂ µ φ ¯ ∇ φ + ¯ ∂ µ φ ¯ ∇ ˆ ∂ ν φ (cid:17)(cid:17) + κ ¯ δ µν (cid:0) ¯ ∇ φ ¯ ∇ φ + ¯ ∂ σ φ ¯ ∂ σ ¯ ∇ φ (cid:1) (5.6) − κ d t ˆ δ µν (cid:0) (2 d s −
4) ¯ ∂ σ φ ¯ ∂ σ ¯ ∇ φ + (2 d t + d s − φ ¯ ∇ φ (cid:1) − κ d t ˆ δ µν ( d s − d t ) ¯ ∇ φ ¯ ∇ φ + κ (cid:16) ¯ ∂ ν ˆ ∂ µ φ ¯ ∇ φ + ˆ ∂ µ φ ¯ ∇ ¯ ∂ ν φ (cid:17) – 29 – 12 d s (4 − (2 d t + d s )) (cid:16) ¯ ∂ ν φ ˆ ∂ µ φ + φ ˆ ∂ µ ¯ ∂ ν φ (cid:17) − d s ¯ δ µν (4 − (2 d t + d s )) (cid:16) ˆ ∂ σ φ ˆ ∂ σ φ + φ ˆ ∂ σ ˆ ∂ σ φ (cid:17) . Note that this expression is symmetric in its spatial components ( T ¯ µ ¯ ν = T ¯ ν ¯ µ ), and itstemporal components ( T ˆ µ ˆ ν = T ˆ ν ˆ µ ), indicating both time and space rotations invariance,but not in its combined space/time components ( T ˆ µ ¯ ν (cid:54) = T ¯ ν ˆ µ ), since there is no Lorentzinveariance. It is also regular in the limit d s → , d t →
1. These two properties are crucialfor the assumptions underlying the procedure outlined in subsection 3.2 to be satisfied. Using the equations of motion in flat space, given by: (cid:16) ˆ ∂ µ − κ ¯ ∇ (cid:17) φ = 0 , (5.7)it is easily verified that the stress-energy tensor (5.6) satisfies the following Ward identitiesfor any values of d t and d s (these are just the flat space versions of identities (2.30)–(2.31),generalized to d t > ∂ µ T µν = 0 , (5.8) D aµ T µa ≡ (cid:16) δ aµ + ¯ δ aµ (cid:17) T µa = 0 . (5.9)In the particular case of d s = 2, d t = 1, the stress-energy tensor (5.6) reduces to thefollowing expression: T µν = ˆ ∂ µ φ∂ ν φ + 2 κ ¯ ∂ µ ∂ ν φ ¯ ∇ φ − δ µν (cid:16) ˆ ∂ σ φ ˆ ∂ σ φ + κ ¯ ∇ φ ¯ ∇ φ (cid:17) − κ (cid:0) ¯ ∂ µ ¯ ∂ ν φ ¯ ∇ φ + ¯ ∂ µ φ ¯ ∇ ¯ ∂ ν φ + ¯ ∂ µ ∂ ν φ ¯ ∇ φ + ∂ ν φ ¯ ∇ ¯ ∂ µ φ (cid:1) + κ ¯ δ µν (cid:0) ¯ ∇ φ ¯ ∇ φ + ¯ ∂ σ φ ¯ ∂ σ ¯ ∇ φ (cid:1) . (5.10) In this subsection, we apply the procedure outlined in subsection 3.2 to calculate theLifshitz anomaly coefficients of the free Lifshitz z = 2 scalar field in 2 + 1 dimensions,using a split dimensional regularization scheme. The general steps of this calculation areas follows:1. We consider the Feynman diagrams contributing to the flat space variational two andthree point correlation functions of the stress-energy tensor, as given in (2.39)–(2.40),in a general number of time and space dimensions.2. For each of these diagrams, we extract the ε lif pole residue of the diagram by expand-ing the integrand in powers of the external momenta as explained in subsection 3.2,utilizing the formulas (3.13)–(3.14) and (3.33)–(3.34) (Note that we use the choice of˜ ε given in (3.21)). This stress-energy tensor is significantly different from the one found in [15], which is not symmetric inits spatial components, and contains a coefficient that diverges in the limit d t → – 30 –. We substitute the pole residues into the LHS of Fourier transformed versions of theWard identities (2.49)–(2.50), and then use formula (3.25) (taking the limit ( ε lif , ˜ ε ) →
0) to obtain the anomalous contribution to these Ward identities.4. We calculate the first and second order variation of each of the possible independentanomaly and trivial term densities corresponding to this case , as listed in [7], withrespect to the vielbeins in the flat space limit. Note that we do not consider any n -point functions with n > For the purpose of calculating the pole residues of the correlation functions of the stress-energy tensor, it is convenient to drop terms in (5.6) that carry coefficients proportionalto the parameter ε lif . These terms will only contribute expressions which are regular inthe limit ( ε lif , ˜ ε ) → T µν = ˆ ∂ µ φ ˆ ∂ ν φ − ˆ δ µν (cid:18) ˆ ∂ σ φ ˆ ∂ σ φ + κ d s d t ¯ ∇ φ ¯ ∇ φ + κd t d s − d t ∂ σ φ ¯ ∂ σ ¯ ∇ φ (cid:19) + ˆ ∂ µ φ ¯ ∂ ν φ + κ d t + d s d t ¯ ∂ µ ˆ ∂ ν φ ¯ ∇ φ − κ ˆ ∂ ν φ ¯ ∇ ¯ ∂ µ φ + κ d s − d t d t ¯ ∂ µ φ ¯ ∇ ˆ ∂ ν φ − κ ¯ ∂ µ φ ¯ ∇ ¯ ∂ ν φ − κ ¯ ∂ ν φ ¯ ∇ ¯ ∂ µ φ + ¯ δ µν (cid:18) −
12 ˆ ∂ σ φ ˆ ∂ σ φ + κ ∇ φ ¯ ∇ φ + κ ¯ ∂ σ φ ¯ ∂ σ ¯ ∇ φ (cid:19) + O ( ε lif ) . (5.11)The expression for the Feynman diagram vertex that corresponds to the stress-energytensor (5.11) is given in (D.3). All other Feynman rules needed for the calculation of therelevant Feynman correlation functions can be found in the appendix D.Note that although the correlation functions of the stress-energy tensor in this caseseem like they might be IR divergent, similarly to the d = 2 relativistic case as discussed insubsection 4.1.2 this is in fact not the case. The terms in the action (5.2) of order O ( ε )contain only derivatives of the field φ , and only the terms of order O ( ε lif ) and highercontain φ with no derivatives. The stress-energy tensor (5.11) and other operators definedas variations of the action with respect to the vielbeins (such as δ Sδe bα δe cγ ) therefore havethe same structure. These operators are therefore not Log correlated at 2 + 1 dimensions,and their correlation functions are not expected to diverge in the IR. In terms of thecorresponding Feynman diagrams regulated using split dimensional regularization, for anypropagator in the 1-loop diagram with spacetime momentum Q µ , the O ( ε ) part of each of Note that we do not assume the Frobenius condition on the background 1-form n µ here. When extracting these coefficients, one must be careful to take into account various dimensionallydependent identities that apply only in the physical dimensions of the theory, in this case d s = 2 , d t = 1.For example, for d t = 1 the following identity applies: ˆ P µ ˆ P ν = ˆ P ρ ˆ P ρ ˆ δ µν . – 31 –ts adjacent vertexes is of order O ( ˆ Q ) or O ( ¯ Q ). The terms in the integrand of the diagramthat contribute to the IR divergence when Q → O ( ε ), and therefore vanishas ε lif → (2 π ) d t + d s δ ( P + Q ) (cid:104) I (2) D (cid:105) ρab ( Q, P ) ≡ F T (cid:104) ( I (2) D ) ρab ( x, y ) (cid:105) , (5.12)(2 π ) d t + d s δ ( P + Q ) (cid:104) W (2) W ) (cid:105) ρb ( Q, P ) ≡ F T (cid:104) ( W (2) W ) ρb ( x, y ) (cid:105) , (5.13)(2 π ) d t + d s δ ( P + Q + K ) (cid:104) I (3) D (cid:105) ραabc ( Q, P, K ) ≡ F T (cid:104) ( I (3) D ) ραabc ( x, y, z ) (cid:105) , (5.14)(2 π ) d t + d s δ ( P + Q + K ) (cid:104) W (3) W (cid:105) ραbc ( Q, P, K ) ≡ F T (cid:104) ( W (3) W ) ραbc ( x, y, z ) (cid:105) , (5.15)where the expressions ( I (2) D ) ρab ( x, y ), ( W (2) W ) ρb ( x, y ), ( I (3) D ) ραabc ( x, y, z ) and ( W (3) W ) ραbc ( x, y, z )are defined in equations (2.45), (2.49), (2.46) and (2.50) respectively. The conventionsfor the Fourier transforms are given in (A.21) and (A.22). The results we present herefor the anomalous contributions to the Ward identities ( W (2) W ) ρb ( x, y ) and ( W (3) W ) ραbc ( x, y, z )are divided into two separate sectors according to the total number of derivatives: a twoderivatives sector ( n D = 2), and a four derivatives sector ( n D = 4). This is in accordancewith the definitions and the discussion in [6, 7] (higher derivative sectors do not appear inthe two and three point functions).At this point, we would like to reiterate that since we study the correlation functionsonly up to the three point function level, we are able to study only the anomalies andtrivial terms (as found in [7]) that appear in this level, that is, only the ones which are atmost second order in the background fields. The pole residues of the two point correlation function of the stress-energy tensor wereextracted using the method described in subsection 3.2. Taking the weighted Lifshitz traceover these pole residues and following the previously mentioned steps of the calculation In this section and the relevant appendixes, we denote the spacetime momenta using capital letters ( P , K , Q ). – 32 –ields the following result for the anomalous contribution to the two point Ward identity: − δ αα (cid:48) δ bβ (cid:104) W (2) W ) (cid:105) α (cid:48) b ( Q, P ) = − D µν (cid:104) T µν ( − P ) T αβ ( P ) (cid:105) (ren) = − i √ καn α n β ¯ P γ ¯ P γ ¯ P δ ¯ P δ π + i (3 − α ) n α ¯ P β ˆ P √ κπ + i √ καn β ¯ P α ¯ P γ ¯ P γ ˆ P π + i ( − α )¯ δ αβ ˆ P √ κπ . (5.16)Note that although the expression on the LHS of (5.16) seems to depend on two differentmomenta P and Q , due to the delta function δ ( P + Q ) in the definition (5.13) these momentaare not independent, and the RHS of (5.16) involves only one independent momentum. Thesame comment holds for other similar identities in this section. When comparing this resultto the first order variation (in the flat space limit) of the cohomologically trivial terms (orcoboundaries in cohomological terminology) in the two derivatives sector (E.9) and in thefour derivatives sector (E.31) with respect to the vielbeins, one finds the following results:In the two derivatives sector we have: i δ αα (cid:48) δ bβ (cid:104) W (2) W ) (cid:105) α (cid:48) b ( Q, P ) (cid:12)(cid:12)(cid:12)(cid:12) n D =2 = 2 α − √ κπ [ δ F ] αβ ( P ) , (5.17)where [ δX ] αβ ( P ) represents the Fourier transform of the first order variation of the ex-pression X with respect to the vielbeins in flat space, as defined in (E.1), and α is theregularization parameter defined in subsection 3.2.In the four derivatives sector we get: i δ αα (cid:48) δ bβ (cid:104) W (2) W ) (cid:105) α (cid:48) b ( Q, P ) (cid:12)(cid:12)(cid:12)(cid:12) n D =4 = − iC [ δ F ] αβ ( P ) − √ κα π [ δ F ] αβ ( P ) + 2 iC [ δ F ] αβ ( P )+ √ κα π [ δ F ] αβ ( P ) − iC [ δ F ] αβ ( P ) , (5.18)where C is a free parameter, whose value cannot be extracted from the two point corre-lation function. This is due to the fact that, in the basis of first order trivial terms weare using here, the first order variations of the trivial term densities in the four derivativessector are linearly dependent in the flat space limit (that is, there is a linear dependencebetween the first order variations of F , F and F with respect to the vielbeins in flatspace). When looking at higher point correlation functions, this dependence is removedand the coefficient C can be extracted, as we indeed show in the next subsection studyingthe three point function.We have also confirmed the pole residue of the two point function satisfies the conser- The results for the anomalous Ward identities in this section are in d t = 1 time dimension. In this case,we use ˆ P and ˆ K to denote the time components of the momenta P and K respectively (see appendix A.3). Only trivial terms are first order in the background fields. – 33 –ation Ward identity: (cid:104) I (2) D (cid:105) ρab ( Q, P ) = 0 . (5.19)This is expected from the argument made in subsection 3.2 that the conservation Wardidentity (2.45) holds separately on the pole part and on the regular finite part of thecorrelation functions. Using the previously mentioned calculation steps, we obtained the following result for theanomalous contribution to the three point Ward identity in the two derivative sector − δ αα (cid:48) δ γγ (cid:48) δ eβ δ fδ (cid:104) W (3) W (cid:105) α (cid:48) γ (cid:48) ef ( Q, P, K ) (cid:12)(cid:12)(cid:12)(cid:12) n D =2 =( − α ) ¯ K β ¯ K δ n α n γ √ κπ + (3 − α ) ¯ K δ ˆ Kn α n β n γ √ κπ + ( − α ) ¯ K β ˆ Kn γ ¯ δ αδ √ κπ + (3 − α ) ˆ K ¯ δ αδ ¯ δ βγ √ κπ + (3 − α ) ¯ K β ˆ Kn α ¯ δ γδ √ κπ + (3 − α ) ˆ Kn α n γ n δ ¯ P β √ κπ + ( − α ) ˆ Kn γ ¯ δ αδ ¯ P β √ κπ + (3 − α ) ˆ Kn α ¯ δ γδ ¯ P β √ κπ + ˆ Kn α ¯ δ βδ ¯ P γ √ κπ − ¯ K β n α n γ ¯ P δ √ κπ + (3 − α ) ˆ Kn γ ¯ δ αβ ¯ P δ √ κπ + α ˆ Kn α ¯ δ βγ ¯ P δ √ κπ − ¯ K µ n α n γ ¯ δ βδ ¯ P µ √ κπ + (3 − α ) ¯ K δ n α n β n γ ˆ P √ κπ + (3 − α ) ¯ K δ n γ ¯ δ αβ ˆ P √ κπ + α ¯ K β n γ ¯ δ αδ ˆ P √ κπ + ( − α ) ¯ K δ n α ¯ δ βγ ˆ P √ κπ + (3 − α ) ˆ K ¯ δ αδ ¯ δ βγ ˆ P √ κπ + ¯ K α n γ ¯ δ βδ ˆ P √ κπ (5.20)+ (3 − α ) ¯ K β n α ¯ δ γδ ˆ P √ κπ + ( − α ) ˆ Kn α n β ¯ δ γδ ˆ P √ κπ + ( − α ) ˆ K ¯ δ αβ ¯ δ γδ ˆ P √ κπ + (3 − α ) n α n γ n δ ¯ P β ˆ P √ κπ + ( − α ) n γ ¯ δ αδ ¯ P β ˆ P √ κπ + (3 − α ) n γ ¯ δ αβ ¯ P δ ˆ P √ κπ + ( − α ) n α ¯ δ βγ ¯ P δ ˆ P √ κπ + ( − α ) n γ n δ ¯ δ αβ ˆ P √ κπ + (3 − α )¯ δ αδ ¯ δ βγ ˆ P √ κπ + ( − α ) ¯ K δ ˆ Kn α ¯ δ βγ √ κπ + ( − α ) ˆ K n α n β ¯ δ γδ √ κπ + ( − α ) ¯ K δ n α n γ ¯ P β √ κπ + ( − α ) n α n γ ¯ P β ¯ P δ √ κπ + ( − α ) ˆ Kn γ n δ ¯ δ αβ ˆ P √ κπ − ˆ K ¯ δ αγ ¯ δ βδ ˆ P √ κπ . Writing this expression as a linear combination of the second order variations with respectto the vielbeins of the anomaly and trivial term densities in the two derivative sector (given We remind the reader that for the expressions in this subsection, the indexes α (cid:48) , e (and therefore α, β )correspond to insertions at the spacetime point y , or the momentum P after Fourier transform. Similarly,the indexes γ (cid:48) , f (and therefore γ, δ ) correspond to insertions at z , or the momentum K . See the definitionsin (2.50), (5.15) and (A.22). – 34 –n (E.11)–(E.12)), we get: − (cid:104) W (3) W (cid:105) αγef ( Q, P, K ) (cid:12)(cid:12)(cid:12)(cid:12) n D =2 = 132 √ κπ (cid:104) δ A (2 , , (cid:105) αγef ( P, K )+ 2 α − √ κπ (cid:2) δ F (cid:3) αγef ( P, K ) , (5.21)where [ δ X ] αγef ( P, K ) represents the Fourier transform of the second order variation of theexpression X with respect to the vielbeins in flat space, as defined in (E.2). Note that,as expected, the coefficient of the anomaly A (2 , , is independent of the regularizationparameter α and agrees with the result found in [11], while the coefficient of the trivialterm F depends on α , and vanishes when α = .The full result for the anomalous contribution to the three point Ward identity (2.50)in the four derivative sector is given in (E.33) in the appendix. Comparing this result tothe second order variations of the anomaly and trivial term densities in this sector we getthe following linear combination: − (cid:104) W (3) W (cid:105) αγef ( Q, P, K ) (cid:12)(cid:12)(cid:12)(cid:12) n D =4 = − √ κα π (cid:2) δ F (cid:3) αγef ( P, K ) + √ κα π (cid:2) δ F (cid:3) αγef ( P, K )+ √ κα π (cid:2) δ F (cid:3) αγef ( P, K ) + α √ κ π (cid:2) δ F (cid:3) αγef ( P, K )+ (cid:18) α √ κ π − C (cid:19) (cid:2) δ F (cid:3) αγef ( P, K ) + (cid:18) α √ κ π − C (cid:19) (cid:2) δ F (cid:3) αγef ( P, K )+ α √ κ π (cid:2) δ F (cid:3) αγef ( P, K ) + C (cid:2) δ F (cid:3) αγef ( P, K ) , (5.22)where the trivial terms (the F i -s) are defined in (E.31) and the anomalies are defined in(2.34) or (E.32). C is again a free parameter that cannot be extracted from the threepoint correlation function, similar to the appearance of the free parameter C in the twopoint level (5.18). Note that the three point function fully determines the coefficients of F , F and F , and therefore the value of the free parameter C from equation (5.18).Alternatively, the result in (5.22) can be written in terms of the second order variations ofthe scalars ( φ ’s) defined in (E.13): − (cid:104) W (3) W (cid:105) αγef ( Q, P, K ) (cid:12)(cid:12)(cid:12)(cid:12) n D =4 = α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )24 π + α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )24 π + α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )60 π + α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )15 π + α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )8 π + α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )24 π (5.23) − α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )12 π − α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )12 π + 7 α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )60 π − α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )30 π − α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )30 π − α √ κ (cid:2) δ φ (cid:3) αγef ( P, K )15 π . – 35 –t is apparent from these results that the coefficients of the 3 possible anomalies in thefour derivative sector that contribute to the three point correlation function vanish. Thisis consistent with previous results (see [11, 27]), that considered only the case where theFrobenius condition is satisfied (and therefore only the anomaly A , , = (cid:16) ˆ R + ¯ ∇ α a α (cid:17) ).All non-vanishing terms are proportional to α , and therefore represent only trivial termsthat can be removed by adding local counterterms to the effective action. It is also im-portant to note that these trivial terms contain contributions from φ , φ , φ , φ , φ and φ . These are terms that vanish in the Frobenius case (for which K A = 0, see [7]for details), and therefore their coefficients cannot be extracted from a curved spacetimecoupling that assumes the existence of a foliation structure. We conclude that for α (cid:54) = 0,violating the Frobenius condition is essential for describing the obtained trivial terms incurved spacetime, and for constructing the appropriate counterterms to cancel them.Finally, we have again verified that the pole residues of the various two and three pointfunctions satisfy the conservation Ward identity: (cid:104) I (3) D (cid:105) ραafe ( Q, P, K ) = 0 , (5.24)which is consistent with the argument made in subsection 3.2 that the conservation Wardidentity (2.46) holds separately on the pole part and on the regular finite part of thecorrelation functions (so that TPD invariance is not anomalous). In this work we developed a general scheme for field theory calculations of Lifshitz scaleanomalies. We analyzed the general structure of correlation functions of the stress-energytensor in Lifshitz field theories and constructed the corresponding anomalous Ward iden-tities. We presented a subtle ambiguity in the definition of the anomaly coefficients andclarified it. Our framework for calculating the anomaly coefficients was based on a splitdimensional regularization where space (momentum) and time (frequency) integrals areregulated separately, and pole residue calculations that allowed to extract the anomalycoefficients without a full calculation of the correlation functions.In order to implement the calculational scheme we had to analyze the coupling of a non-relativistic, Lifshitz invariant field theory in d t time dimensions and d s space dimensions toa curved spacetime manifold. This generalized the curved spacetime structure introducedin [6, 7] to the case of multiple time directions.We considered as a particular example the z = 2 free scalar field theory in 2 + 1spacetime dimensions. We showed that the only non-zero anomaly coefficient (of thoseappearing in the three point functions) is in the two derivatives sector, which agrees withthe heat kernel calculation in [11]. In order to account for some of the trivial terms arisingfrom this calculation, we found it necessary to give up the Frobenius condition (and thecorresponding foliation structure) of the curved spacetime description of the theory. Thisis because these terms are not in the span of possible anomalous contributions one obtainswhen assuming the Frobenius condition. – 36 –here are many directions for further studies that follow from our analysis. It would beinteresting to use the general scheme developed here to calculate the anomaly coefficientsof other Lifshitz field theories. One can also generalize the discussion and consider non-relativistic field theories that exhibit non-relativistic boost invariance. In such cases onehas in addition to the B-type scale anomalies also A-type ones [7] (see also [33]). In therelativistic CFT case, scale anomaly coefficients multiply universal terms in entanglemententropies. It would be of interest to analyze the entanglement entropy structure in Lifshitzfield theories and the role of the scale anomaly coefficients. In the CFT case, A-typeanomaly charges exhibit RG properties, i.e. a decrease from the UV to the IR. The non-relativistic versions of these are still lacking. Finally, it would be interesting to ask whetherthere are experimental observables of the anomaly charges in non-relativistic systems suchas low energy condensed matter ones. One potential path to consider is the hydrodymanicsof such systems [34, 35] and the role of scale anomalies in such descriptions. Acknowledgments
We would like to thank Itamar Hason, Carlos Hoyos, Zohar Komargodski, Adam Schwim-mer and Stefan Theisen for valuable discussions and comments. Many of the calculationsin this paper were performed using xAct [36] and xTras [37], tensor computer algebra pack-ages for
Mathematica . This work is supported in part by the I-CORE program of Planningand Budgeting Committee (grant number 1937/12), the US-Israel Binational Science Foun-dation, GIF and the ISF Center of Excellence. A.R.M gratefully acknowledges the supportof the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. I.Ais thankful for the support of the Alexander Zaks fellowship program.
A Notations and Conventions
In this appendix we describe the notations and conventions used in this paper.
A.1 Notations and Conventions in Curved Spacetime
Throughout this paper we use Greek letters ( µ, ν, ρ, . . . ) to denote spacetime indexes, bothin the relativistic and non-relativistic cases. For the relativistic case, we use a spacetimemetric g µν with signature diag(1 , − , − , . . . ) and denote the flat space metric by η µν . Forthe non-relativistic case, we use a signature of diag( − , − , . . . , , , . . . ) (with negativesigns for the time dimensions and positive for the space dimensions), and denote the flatspacetime metric by δ µν .In both cases we use the standard torsionless Levi-Civita connection associated withthe spacetime metric g µν . That is, the covariant derivative of a vector A µ is given by: ∇ ν A µ ≡ ∂ ν A µ + Γ µνρ A ρ , (A.1)where the Christoffel symbols are given by:Γ µνρ = 12 g µκ [ − ∂ κ g νρ + ∂ ν g κρ + ∂ ρ g κν ] . (A.2)– 37 –e use the following convention for the Riemann tensor: R λµκν = ∂ κ Γ λµν − ∂ ν Γ λµκ + Γ λκη Γ ηµν − Γ λνη Γ ηµκ , (A.3)while the Ricci tensor and scalar are given by: R µν = R γµγν , R = g µν R µν . (A.4)When using the vielbein formalism, the vielbeins e aµ are defined in the relativistic casesuch that: g µν = η ab e aµ e bν , (A.5)and in the non-relativistic case: g µν = δ ab e aµ e bν . (A.6)We use the following notations for the determinants of the metric and the vielbeins: g ≡ det ( g µν ) , e ≡ det ( e aµ ) . (A.7)We use the following formulas for variations of the metric: δg µν = − g µα g νβ δg αβ , δg µν = − g µα g νβ δg αβ , (A.8) δg αβ ( x ) δg µν ( x ) = − (cid:16) g αµ g βν + g αν g βµ (cid:17) δ ( x − x ) , (A.9) δg σρ δe cα = − g σλ g τρ δg λτ δe cα , (A.10) δg ρν δe cα = η ac ( δ αρ e aν + δ αν e aρ ) , (A.11) δ (cid:112) | g | = − (cid:112) | g | g aβ δg αβ , δ (cid:112) | g | = 12 (cid:112) | g | g αβ δg αβ . (A.12)Finally, for reference we give here the expressions for the Weyl tensor squared and theEuler density of the background manifold in the (3 + 1)-dimensional case: W = R αβγδ R αβγδ − R αβ R αβ + 13 R , (A.13) E = R αβγδ R αβγδ − R αβ R αβ + R . (A.14) A.2 Notations and Conventions for the Relativistic Scalar Case
As mentioned in appendix A.1, we use Greek letters ( µ, ν, ρ, . . . ) to denote spacetimeindexes. In the relativistic case, we use a flat spacetime metric of the form: η µν =diag(1 , − , − , . . . ). We use the following conventions for the Fourier transforms of twoand three point correlation functions in the relativistic case: F T (cid:2) I (2) ( x , x ) (cid:3) ≡ (cid:90) d d x d d x I (2) ( x , x ) e − i ( − k · x − q · x ) , (A.15)– 38 – T (cid:2) I (3) ( x , x , x ) (cid:3) ≡ (cid:90) d d x d d x d d x I (3) ( x , x , x ) e − i ( − k · x − q · x − p · x ) , (A.16)where the lower-case letters p, k, q denote spacetime momenta. We also use the followingnotations for the Fourier transformed two and three point correlation functions of thestress-energy tensor and the variations of the action in flat space:(2 π ) d δ ( − k − q ) (cid:104) T µν ( − q ) T ρσ ( q ) (cid:105) ≡ F T [ (cid:104) T µν ( x ) T ρσ ( x ) (cid:105) ] , (A.17)(2 π ) d δ ( − k − p − q ) (cid:68) T µν ( − p − q ) T ρσ ( q ) T αβ ( p ) (cid:69) ≡ F T (cid:104)(cid:68) T µν ( x ) T ρσ ( x ) T αβ ( x ) (cid:69)(cid:105) , (A.18)(2 π ) d δ ( − k − p − q ) (cid:28) δ Sδg µν δg ρσ ( k, q ) δSδg αβ ( p ) (cid:29) ≡ F T (cid:20)(cid:28) δ Sδg µν ( x ) δg ρσ ( x ) δSδg αβ ( x ) (cid:29)(cid:21) . (A.19)Finally, we use the following notation for the Fourier transformed second variation ofthe expression X with respect to the background metric, evaluated in flat spacetime:(2 π ) d δ ( − k − p − q ) (cid:2) δ X ( q, p ) (cid:3) ρσαβ ≡ F T (cid:34) δ X ( x ) δg ρσ ( x ) δg αβ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) g µν = η µν (cid:35) . (A.20) A.3 Notations and Conventions for the Lifshitz z = 2 Scalar Case
As explained in subsection 3.2, in the non-relativistic case we define the theory on a man-ifold M = M t × M s , where M t is a d t -dimensional time manifold and M s is a d s -dimensional space manifold, such that it is invariant both under rotations in the timemanifold and in the space manifold separately.Spacetime coordinates are denoted by x µ = ( x ˆ µ , x ¯ µ ) where ˆ µ = 1 , . . . , d t are timeindexes, ¯ µ = 1 , . . . , d s are space indexes and µ = 1 , . . . , d t + d s are spacetime indexes. Wedefine a flat metric ˆ δ ˆ µ ˆ ν = diag( − , . . . , −
1) on M t , and ¯ δ ¯ µ ¯ ν = diag(1 , . . . ,
1) on M s . Wealso define the time projector on M as ˆ δ µν = diag(ˆ δ ˆ µ ˆ ν ,
0) and similarly the space projectoras ¯ δ µν = diag(0 , ¯ δ ¯ µ ¯ ν ), so that δ µν = ˆ δ µν + ¯ δ µν . Given a vector v µ on M , we denote its timeprojection by ˆ v µ ≡ ˆ δ µν v ν , and its space projection by ¯ v µ ≡ ¯ δ µν v ν . In the case of d t = 1 timedimension, we use ˆ v (with no index) to denote the time component of the vector v µ , i.e.ˆ v µ = − ˆ vn µ where ˆ v ≡ v µ n µ and n µ = (1 , , , . . . ).We use capital letters P, K, Q to denote spacetime momenta. The notations ¯ P and ˆ P then refer to the spatial and temporal projections of the momentum P , respectively. Weuse the following conventions for the Fourier transforms of two and three point correlationfunctions in the non-relativistic case: F T (cid:2) I (2) ( x = (ˆ x, ¯ x ) , y = (ˆ y, ¯ y )) (cid:3) ≡ (cid:90) d d t + d s x d d t + d s y I (2) ( x, y ) e i ( Q · x + P · y ) , (A.21) F T (cid:2) I (3) ( x, y, z ) (cid:3) ≡ (cid:90) d d t + d s x d d t + d s y d d t + d s z I (3) ( x, y, z ) e i ( Q · x + P · y + K · z ) . (A.22)– 39 –e also use the following notations for the Fourier transformed two and three pointcorrelation functions of the stress-energy tensor and the variations of the action with respectto the vielbeins in flat space:(2 π ) d t + d s δ ( − P − Q ) (cid:104) T µν ( P ) T ρσ ( − P ) (cid:105) ≡ F T (cid:104)(cid:68) T µν ( x ) T αβ ( y ) (cid:69)(cid:105) , (A.23)(2 π ) d t + d s δ ( − P − K − Q ) (cid:68) T µν ( − P − K ) T αβ ( P ) T γδ ( K ) (cid:69) ≡ F T (cid:104)(cid:68) T µν ( x ) T αβ ( y ) T γδ ( z ) (cid:69)(cid:105) , (A.24)(2 π ) d t + d s δ ( − P − K − Q ) (cid:28) δ Sδe bα δe cγ ( P, K ) δSδe aµ ( Q ) (cid:29) ≡ F T (cid:20)(cid:28) δ Sδe bα ( y ) δe cγ ( z ) δSδe aµ ( x ) (cid:29)(cid:21) . (A.25)Finally, our notations for the various Ward identities of the flat space correlationfunctions (2.45)–(2.50) are as follows: We use I to denote the Ward identities (2.45)–(2.48)derived using a change of the variables in the path integral, whereas W denotes Wardidentities (2.49)–(2.50) derived from variations of the curved spacetime Ward identities.The subscript refers to the relevant symmetry: D corresponds to TPD symmetry, whereas W corresponds to anisotropic Weyl symmetry. The superscript ( n ) refers to the number ofpoints in the correlation function. The notation for the Fourier transforms of these Wardidentities is given in (5.12)–(5.15). B Non-Relativistic Curved Spacetime with Multiple Time Directions
In this appendix we discuss the coupling of a non-relativistic, Lifshitz invariant field theoryin d t time dimensions and d s space dimensions to a curved spacetime manifold. We firstgeneralize the curved spacetime structure introduced in [6, 7] to the case of multiple timedirections. We then discuss the local symmetries of the theory over curved spacetime.Finally we construct the curved spacetime action that corresponds to the free z = 2 Lifshitzscalar. B.1 Curved Spacetime Definitions
Consider a non-relativistic field theory defined over a spacetime manifold with d t timedimensions and d s space dimensions, such that it is invariant both under time rotationsand space rotations. In order to define the theory over a curved spacetime manifold, wegeneralize the structure introduced in [6, 7]. We require the background manifold to beequipped with a metric g µν , or alternatively vielbeins e aµ , as well as a distribution ofdimension d s , corresponding to the space directions at each point of the manifold.This distribution can be represented by the cotangent subbundle of 1-forms that an-nihilate space tangent vectors. Suppose this subbundle is spanned by a basis of d t linearlyindependent 1-forms t ( i ) µ ( i = 1 , . . . , d t ) that correspond to the d t time directions, so thata vector V α is space tangent if and only if t ( i ) µ V µ = 0 for all 1 ≤ i ≤ d t . Then physicalquantities, such as the curved spacetime action of the theory, will depend on the subbundle– 40 –panned by { t ( i ) µ } , but not on the choice of basis. We therefore expect them to be invariantunder transformations of the form: t ( i ) µ → L ij ( x ) t ( j ) µ , where L ij is an invertible matrix thatdepends on the spacetime coordinate x . Alternatively, we can choose an orthonormal basisof 1-forms n ( i ) µ , satisfying: g µν n ( i ) µ n ( j ) ν = − δ ij , (B.1)and require invariance under local rotations of the time directions – transformations of theform n ( i ) µ → Λ ij ( x ) n ( j ) µ , where Λ ij is an orthogonal matrix that depends on the coordinate x . Using the set of 1-forms n ( i ) µ we can make several definitions on the background man-ifold. A tensor ¯ T αβ... is space tangent if it satisfies: n α ( i ) ¯ T αβγ = n β ( i ) ¯ T αβγ = . . . = 0 , (B.2)for any 1 ≤ i ≤ d t . Any tensor can be rendered space tangent by projecting it on the spacedirections using the space projector (or spatial metric), defined as: P µν = g µν + n ( i ) µ n ( i ) ν . (B.3)The covariant derivative of the 1-form n ( i ) µ can be decomposed as follows: ∇ α n ( i ) β = ( K S ) ( i ) αβ + ( K A ) ( i ) αβ − a ( ij ) β n ( j ) α − b ( ij ) α n ( j ) β + c ( ijk ) n ( j ) α n ( k ) β , (B.4)where ( K S ) ( i ) αβ , ( K A ) ( i ) αβ , a ( ij ) α and b ( ij ) α are space tangent tensors. ( K S ) ( i ) αβ is symmetric, andgiven by: ( K S ) ( i ) αβ = P α (cid:48) α P β (cid:48) β ∇ ( α (cid:48) n ( i ) β (cid:48) ) = 12 ¯ L n ( i ) P αβ , (B.5)where the space projected Lie derivative ¯ L n ( i ) of a space tangent tensor ¯ T αβ... is defined asfollows: ¯ L n ( i ) ¯ T αβ... ≡ P α (cid:48) α P β (cid:48) β . . . L n ( i ) ¯ T α (cid:48) β (cid:48) ... . (B.6)We denote its trace by K ( i ) S ≡ ( K S ) ( i ) µµ . ( K A ) ( i ) αβ is antisymmetric and given by:( K A ) ( i ) αβ = P α (cid:48) α P β (cid:48) β ∇ [ α (cid:48) n ( i ) β (cid:48) ] . (B.7) a ( ij ) β is given by: a ( ij ) β = P β (cid:48) β n α ( j ) ∇ α n ( i ) β (cid:48) , (B.8)and we define the generalized acceleration vector a β as trace over the time indexes: a β ≡ a ( ii ) β . b ( ij ) α is given by: b ( ij ) α = P α (cid:48) α n β ( j ) ∇ α (cid:48) n ( i ) β , (B.9)and from (B.1) it is easy to show that b ( ij ) α is antisymmetric in its time indexes, i.e. b ( ji ) α = A summing convention is assumed for repeated time indexes ( i, j, . . . ). – 41 – b ( ij ) α . One can also obtain the following expression for the space projected Lie derivativeof the 1-form n ( j ) µ in the direction of n µ ( i ) :¯ L n ( i ) n ( j ) µ = a ( ji ) µ − b ( ji ) µ , (B.10)so that the generalized acceleration vector is also given by: a µ ≡ a ( ii ) µ = ¯ L n ( i ) n ( i ) µ . (B.11)Finally, c ( ijk ) is a scalar given by: c ( ijk ) = n α ( j ) n β ( k ) ∇ α n ( i ) β , (B.12)that satisfies c ( kji ) = − c ( ijk ) (again from (B.1)).Next we define the space tangent covariant derivative of a space tangent tensor ¯ T αβ... as follows: ¯ ∇ µ ¯ T αβ... ≡ P µ (cid:48) µ P α (cid:48) α P β (cid:48) β . . . ∇ µ (cid:48) ¯ T α (cid:48) β (cid:48) ... . (B.13)Note that the spatial metric P αβ is covariantly constant under this derivative:¯ ∇ µ P αβ = 0 . (B.14)Operating with the commutation of two space tangent derivatives on a space tangent tensor¯ T αβ... , one obtains the following expression: (cid:2) ¯ ∇ µ , ¯ ∇ ν (cid:3) ¯ T αβ... = (cid:101) R αρµν ¯ T ρβ... + (cid:101) R βρµν ¯ T αρ... + . . . + 2( K A ) ( i ) µν ¯ L n ( i ) ¯ T αβ... , (B.15)where (cid:101) R αρµν is a space tangent tensor defined by: (cid:101) R αρµν ≡ P α (cid:48) α P ρ (cid:48) ρ P µ (cid:48) µ P ν (cid:48) ν R α (cid:48) ρ (cid:48) µ (cid:48) ν (cid:48) − K A ) ( i ) µν K ( i ) αρ − K ( i ) µα K ( i ) νρ + K ( i ) να K ( i ) µρ , (B.16) K ( i ) αβ ≡ ( K S ) ( i ) αβ + ( K A ) ( i ) αβ is the total space tangent component of ∇ α n ( i ) β and R αρµν is thestandard Riemann curvature associated with the covariant derivative ∇ µ (see appendix Afor our conventions). Similarly to the one time direction case (see [7]), the tensor (cid:101) R αρµν does not have all of the standard symmetries of the Riemann tensor. It is therefore usefulto define a modified Riemann tensor: (cid:98) R αρµν ≡ (cid:101) R αρµν + 2( K A ) ( i ) µν ( K S ) ( i ) αρ + ( K A ) ( i ) µα ( K S ) ( i ) νρ + ( K S ) ( i ) µα ( K A ) ( i ) νρ − ( K A ) ( i ) να ( K S ) ( i ) µρ − ( K S ) ( i ) να ( K A ) ( i ) µρ , (B.17)which satisfies the usual Riemann tensor symmetries except for the second Bianchi identity.We then define the equivalents of the Ricci tensor and scalar for this modified Riemann– 42 –ensor (cid:98) R αρµν as follows: (cid:98) R ρν ≡ (cid:98) R µρµν = P αµ (cid:98) R αρµν , (cid:98) R ≡ (cid:98) R νν = P ρν (cid:98) R ρν . (B.18)Note that from the above definitions, one gets the following identity for the divergence ofa space tangent vector ¯ V µ : ∇ µ ¯ V µ = ( ¯ ∇ µ + a µ ) ¯ V µ . (B.19) B.2 Symmetries Over Curved Spacetime
Next we turn to discuss the symmetries of the curved spacetime field theory. Like in theone time direction case, the symmetries of the flat space Lifshitz theory translate to localsymmetries over curved spacetime:First, as mentioned in subsection 2.2.1 for the one time direction case, we require time-direction preserving diffeomorphism (TPD) invariance that corresponds to space rotationsymmetry in flat space. In this case, these are diffeomorphisms with a parameter ξ thatsatisfies: L ξ t ( i ) α = M ij ( x ) t ( j ) α where M ij is some spacetime dependent invertible d t × d t matrix. Similarly to the one time direction case (see [6, 7]), we can extend these to the fulldiffeomorphism group by having the 1-forms t ( i ) µ transform appropriately: δ Dξ g µν = ∇ µ ξ ν + ∇ ν ξ µ , δ Dξ t ( i ) α = L ξ t ( i ) α = ξ β ∇ β t ( i ) α + ∇ α ξ β t ( i ) β . (B.20)Second, as previously mentioned, we require invariance under local time rotations ofthe form n ( i ) µ → Λ ij ( x ) n ( j ) µ , where Λ ij is a spacetime dependent orthogonal matrix. Ininfinitesimal form, these transformations are given by: δ Tω n ( i ) α = ω ij n ( j ) α , δ Tω g µν = δ Tω P µν = 0 , (B.21)where ω ij is a transformation parameter that satisfies ω ji = − ω ij . From this transformationand the definition of the derivatives ¯ L n ( i ) , ¯ ∇ µ we obtain the following for a space tangenttensor ¯ T αβ... : δ Tω ¯ L n ( i ) ¯ T αβ... = ¯ L n ( i ) δ Tω ¯ T αβ... + ω ij ¯ L n ( j ) ¯ T αβ... ,δ Tω ¯ ∇ µ ¯ T αβ... = ¯ ∇ µ δ Tω ¯ T αβ... . (B.22)Using these formulas and the various definitions from appendix B.1, the following timerotation transformation rules can be derived: δ Tω ( K S ) ( i ) αβ = ω ij ( K S ) ( j ) αβ ,δ Tω K ( i ) S = ω ij K ( j ) S ,δ Tω ( K A ) ( i ) αβ = ω ij ( K A ) ( j ) αβ , (B.23) δ Tω a ( ij ) µ = ω ik a ( kj ) µ + ω jk a ( ik ) µ ,δ Tω a µ = 0 , – 43 – Tω (cid:98) R αρµν = δ Tω (cid:98) R ρν = δ Tω (cid:98) R = 0 . Finally we require anisotropic Weyl invariance, which is the local version of Lifshitzscale invariance in flat space (5.5). In the case of multiple time directions, the infinitesimalanisotropic Weyl transformation is given by: δ Wσ P µν = 2 σP µν , δ Wσ n µ ( i ) = − zσn µ ( i ) , δ Wσ n ( i ) µ = zσn ( i ) µ . (B.24)From this transformation and the definition of the derivatives ¯ L n ( i ) , ¯ ∇ µ we get the followingformulas for a space tangent tensor ¯ T αβ... : δ Wσ ¯ L n ( i ) ¯ T αβγ... = − zσ ¯ L n ( i ) ¯ T αβγ... + ¯ L n ( i ) δ Wσ ¯ T αβγ... ,δ Wσ ( ¯ ∇ µ ¯ T αβγ... ) = ¯ ∇ µ ( δ Wσ ¯ T αβγ... ) − I [ ¯ T ] ¯ ∇ µ σ ¯ T αβ... − ( ¯ ∇ α σ ) ¯ T µβγ... + ¯ ∇ ρ σP µα ¯ T ρβ... − . . . , (B.25)where I [ ¯ T ] is the rank of the tensor ¯ T αβ... . Using these formulas and the various definitionsfrom appendix B.1, the following anisotropic Weyl transformation rules can be derived: δ Wσ ( K S ) ( i ) µν = (2 − z ) σ ( K S ) ( i ) µν + P µν ¯ L n ( i ) σ,δ Wσ K ( i ) S = − zσK ( i ) S + d s ¯ L n ( i ) σ,δ Wσ a µ = zd t ¯ ∇ µ σ,δ Wσ ( K A ) ( i ) µν = zσ ( K A ) ( i ) µν ,δ Wσ (cid:98) R αρµν = 2 σ (cid:98) R αρµν + P αν ¯ ∇ ( ρ ¯ ∇ µ ) σ − P αµ ¯ ∇ ( ρ ¯ ∇ ν ) σ + P ρµ ¯ ∇ ( α ¯ ∇ ν ) σ − P ρν ¯ ∇ ( α ¯ ∇ µ ) σ,δ Wσ (cid:98) R αµ = (2 − d s ) ¯ ∇ ( α ¯ ∇ µ ) σ − P αµ ¯ ∇ σ,δ Wσ (cid:98) R = − σ (cid:98) R − d s −
1) ¯ ∇ σ. (B.26) B.3 Action of the Free z = 2 Scalar
Consider the free Lifshitz scalar in d s + d t dimensions with a dynamical critical exponentof z = 2. Its flat space action is given in (5.4). In order to couple it to a curved spacetimemanifold, we have to define its curved spacetime action S ( g µν , t ( i ) α , φ ) such that it is invariantunder TPD, local time rotation and anisotropic Weyl transformations, and it reduces tothe action (5.4) in the flat space limit. We also require it to be regular in the physical limit d t →
1, for the sake of using the split dimensional regularization procedure as described insubsection 3.2.Suppose that φ has a scaling dimension s under anisotropic Weyl transformations, thatis: δ Wσ φ = s σφ. (B.27)For the temporal part of the action, using the transformations given in appendix B.2 wecan find a linear combination of L n ( i ) φ and K ( i ) S φ which is covariant under both local timerotations and anisotropic Weyl transformations (it transforms with no contribution from– 44 –erivatives of the parameters ω ij and σ ): (cid:18) L n ( i ) − sd s K ( i ) S (cid:19) φ. (B.28)For the spatial part of the action, we can find a linear combination of ¯ ∇ φ , a µ ¯ ∇ µ φ , a φ and ¯ ∇ µ a µ φ which is anisotropic-Weyl-covariant, given by: (cid:20) ¯ ∇ + 2 − d s − szd t a µ ¯ ∇ µ − s (2 − d s − s ) z d t a − szd t ¯ ∇ µ a µ (cid:21) φ, (B.29)or alternatively, we can use ¯ ∇ φ , a µ ¯ ∇ µ φ , a φ and (cid:98) Rφ to obtain the following anisotropic-Weyl-covariant linear combination: (cid:20) ¯ ∇ + 2 − d s − szd t a µ ¯ ∇ µ − s (2 − d s − s )2 z d t a − s − d s ) (cid:98) R (cid:21) φ. (B.30)Finally, for the action to be anisotropic-Weyl-invariant with z = 2, the dimension ofthe scalar field is required to satisfy:2( s −
2) + 2 d t + d s = 0 ⇒ s = 2 − d t − d s . (B.31)Combining these expressions, we arrive at two possible options for the curved spacetimeaction of the free z = 2 scalar (corresponding to the two options for the spatial part). Thefirst option is given by: S = (cid:90) d d t + d s x (cid:112) | g | (cid:26) (cid:104) L n ( i ) φ + ξ K ( i ) S φ (cid:105) − κ (cid:2) ¯ ∇ φ + ξ a µ ¯ ∇ µ φ + ξ a φ + ξ ¯ ∇ µ a µ φ (cid:3) (cid:111) , (B.32)where: ξ ≡ d s (cid:18) d lif − (cid:19) , ξ ≡ d t − d t ,ξ ≡ d t (cid:18) d lif − (cid:19) (cid:18) d t − d s (cid:19) , ξ ≡ d t (cid:18) d lif − (cid:19) , (B.33)and d lif ≡ d t + d s . The second option is given by: S = (cid:90) d d t + d s x (cid:112) | g | (cid:26) (cid:104) L n ( i ) φ + ξ (cid:48) K ( i ) S φ (cid:105) − κ (cid:104) ¯ ∇ φ + ξ (cid:48) a µ ¯ ∇ µ φ + ξ (cid:48) a φ + ξ (cid:48) (cid:98) Rφ (cid:105) (cid:27) , (B.34) Of course, any linear combination of these two options could also be used. – 45 –here: ξ (cid:48) ≡ d s (cid:18) d lif − (cid:19) , ξ (cid:48) ≡ d t − d t ,ξ (cid:48) ≡ d t − d t (cid:18) d lif − (cid:19) , ξ (cid:48) ≡ − d s − (cid:18) d lif − (cid:19) . (B.35)These actions are indeed invariant under TPDs, local time rotations and anisotropic Weyltransformations. They are also regular in the d t → ε lif and therefore does not contribute to the ε lif pole residues of the flat space correlationfunctions (see subsection 3.2 for details). C Relativistic Scalar Field – Feynman Rules, Vertexes and Integrals
In this appendix we give some details for the calculations of the two and three pointcorrelation functions which are required for computing the conformal anomaly coefficientsof the relativistic scalar field in two and four spacetime dimensions, as explained in section 4.
C.1 Feynman Rules and Diagrams
The propagator of the relativistic scalar field is given by: (cid:104) φφ (cid:105) ( q ) = iq − m + i(cid:15) , (C.1)where, as explained in subsection 3.1, m is an IR mass regulator later taken to be zero.The (Fourier transformed) two and three point Feynman correlation functions of the stress-energy tensor are given by the expressions: (cid:104) T µν ( p ) T ρσ ( − p ) (cid:105) = ( i ) (cid:90) d d q (2 π ) d V T µν ( q, p − q )[ q − m + i(cid:15) ] V T ρσ ( q, p − q )[( p − q ) − m + i(cid:15) ] , (C.2)and (cid:68) T µν ( k = − p − q ) T ρσ ( q ) T αβ ( p ) (cid:69) F =( i ) (cid:90) d d l (2 π ) d V T µν ( − l, l − p − q )[ l − m + i(cid:15) ] V T ρσ ( l, q − l ) (cid:104) ( q − l ) − m + i(cid:15) (cid:105) × V T αβ ( l − q, − l + p + q ) (cid:104) ( p + q − l ) − m + i(cid:15) (cid:105) . (C.3)These expressions correspond to Feynman diagrams of the form given in figure 1 for thetwo point function of the stress-energy tensor, and figure 2 for the three point function of– 46 –he stress-energy tensor. The vertexes V µνT are given by: V T µν ( q, p ) = − (cid:18) d −
22 (1 − d ) (cid:19) A T µν ( q, p ) − (cid:18) d −
22 (1 − d ) (cid:19) C T µν ( q, p )+ (cid:18)
12 + d −
22 (1 − d ) (cid:19) B T µν ( q, p ) + (cid:18) d −
22 (1 − d ) (cid:19) D T µν ( q, p ) , (C.4)where A T , B T , C T and D T are defined by: A T µν ( q, p ) = ( i ) ( q µ p ν + q ν p µ ) , (C.5) B T µν ( q, p ) = 2( i ) η µν ( q · p ) , (C.6) C T µν ( q, p ) = ( i ) ( q µ q ν + p µ p ν ) , (C.7) D T µν ( q, p ) = ( i ) η µν (cid:0) q + p (cid:1) . (C.8)Note that terms which carry coefficients of order O ( ε ) do not contribute to the pole residuesof the correlation functions, and can therefore be ignored for the purpose of our calculations.For example, the terms C T and D T in equation (C.4) can be neglected when calculating thepole residues around two spacetime dimensions. However, these terms cannot be neglectedin four dimensions since their coefficients are no longer proportional to ε .The expression for the correlation function (A.19) (which corresponds to a Feynmandiagram of the form given in figure 3) is the following: (cid:28) δ Sδg µν δg ρσ ( k, q ) δSδg αβ ( p ) (cid:29) = ( i ) (cid:90) d d l (2 π ) d V µνρσ ( p + l, − l )[ l − m + i(cid:15) ] V T αβ ( − l, p + l ) (cid:104) ( p + l ) − m + i(cid:15) (cid:105) , (C.9)where the vertex V µνρσ is defined by: V µνρσ ( p, q ) ≡ V µνρσ ( p, q ) + V µνρσ ( p, q ) , (C.10)where: V µνρσ ( p, q ) ≡− η µσ η νρ p α q α − η µρ η νσ p α q α + η µν η ρσ p α q α − η ρσ p ν q µ + η νσ p ρ q µ + η νρ p σ q µ − η ρσ p µ q ν + η µσ p ρ q ν + η µρ p σ q ν + η νσ p µ q ρ + η µσ p ν q ρ − η µν p σ q ρ + η νρ p µ q σ + η µρ p ν q σ − η µν p ρ q σ , (C.11) V µνρσ ( p, q ) ≡− η µσ η νρ p α p α − d ) + dη µσ η νρ p α p α − d ) − η µρ η νσ p α p α − d ) + dη µρ η νσ p α p α − d ) − η µν η ρσ p α p α − d ) + dη µν η ρσ p α p α − d ) − η ρσ p µ p ν − d ) + 3 dη ρσ p µ p ν − d ) + η νσ p µ p ρ − d ) − dη νσ p µ p ρ − d ) + η µσ p ν p ρ − d ) − dη µσ p ν p ρ − d ) + η νρ p µ p σ − d ) − dη νρ p µ p σ − d )– 47 – η µρ p ν p σ − d ) − dη µρ p ν p σ − d ) − η µν p ρ p σ − d ) + 3 dη µν p ρ p σ − d ) − η µσ η νρ p α q α − d )+ dη µσ η νρ p α q α − d ) − η µρ η νσ p α q α − d ) + dη µρ η νσ p α q α − d ) − η µσ η νρ q α q α − d )+ dη µσ η νρ q α q α − d ) − η µρ η νσ q α q α − d ) + dη µρ η νσ q α q α − d ) − η µν η ρσ q α q α − d )+ dη µν η ρσ q α q α − d ) − η ρσ p ν q µ − d ) + dη ρσ p ν q µ − d ) + η νσ p ρ q µ − d ) − dη νσ p ρ q µ − d ) (C.12)+ η νρ p σ q µ − d ) − dη νρ p σ q µ − d ) − η ρσ p µ q ν − d ) + dη ρσ p µ q ν − d ) + η µσ p ρ q ν − d ) − dη µσ p ρ q ν − d ) + η µρ p σ q ν − d ) − dη µρ p σ q ν − d ) − η ρσ q µ q ν − d ) + 3 dη ρσ q µ q ν − d )+ η νσ p µ q ρ − d ) − dη νσ p µ q ρ − d ) + η µσ p ν q ρ − d ) − dη µσ p ν q ρ − d ) − η µν p σ q ρ − d )+ dη µν p σ q ρ − d ) + η νσ q µ q ρ − d ) − dη νσ q µ q ρ − d ) + η µσ q ν q ρ − d ) − dη µσ q ν q ρ − d )+ η νρ p µ q σ − d ) − dη νρ p µ q σ − d ) + η µρ p ν q σ − d ) − dη µρ p ν q σ − d ) − η µν p ρ q σ − d )+ dη µν p ρ q σ − d ) + η νρ q µ q σ − d ) − dη νρ q µ q σ − d ) + η µρ q ν q σ − d ) − dη µρ q ν q σ − d ) − η µν q ρ q σ − d ) + 3 dη µν q ρ q σ − d ) . C.2 Massless Integrals
The full evaluation of the two point correlation function of the stress-energy tensor as givenin equation (4.5) requires the use of the following dimensionally-regulated integrals (takenfrom [38], and transformed into the Lorentzian signature conventions): I ≡ (cid:90) d d q (2 π ) d q ( q − p ) = ( − d/ − i (4 π ) d/ Γ ( d/ −
1) Γ ( d/ −
1) Γ (2 − d/ d − (cid:0) p (cid:1) d/ − , (C.13) (cid:90) d d q (2 π ) d q µ q ( q − p ) = I p µ , (C.14) (cid:90) d d q (2 π ) d q µ q ν q ( q − p ) = I η µν + I p µ p ν , (C.15) (cid:90) d d q (2 π ) d q µ q ν q γ q ( q − p ) = I p µ p ν p γ + I E µνγ , (C.16) (cid:90) d d q (2 π ) d q µ q ν q γ q σ q ( q − p ) = I p µ p ν p γ p σ + I G µνγσ + I H µνγσ , (C.17)– 48 –here: E µνγ ≡ η µν p γ + η µγ p ν + η νγ p µ , (C.18) G µνγσ ≡ η µν p γ p σ + η µγ p ν p σ + η µσ p γ p ν + η νγ p µ p σ + η νσ p µ p γ + η γσ p µ p ν , (C.19) H µνγσ = η µν η γσ + η µσ η νγ + η µγ η νσ , (C.20)and: I = 12 I , (C.21) I = − p d − I , (C.22) I = d d − I , (C.23) I = d + 28 ( d − I , (C.24) I = − p d − I , (C.25) I = ( d + 2) ( d + 4)16 ( d − I , (C.26) I = − ( d + 2)16 ( d − p I , (C.27) I = 116 ( d − (cid:0) p (cid:1) I . (C.28) D Lifshitz z = 2 Scalar Field – Feynman Rules, Vertexes and Integrals
In this appendix we give some details for the calculation of the two and three point cor-relation functions which are required for computing the Lifshitz anomaly coefficients of a z = 2 free scalar field in 2 + 1 dimensions, as explained in section 5. These include theexpressions for the Feynman diagrams, the Feynman rules for the propagator and all thevertexes needed.The propagator of the Lifshitz z = 2 scalar is given by: (cid:104) φφ (cid:105) ( Q ) = − ii(cid:15) + m + κ ( ¯ Q α ¯ Q α ) + ( ˆ Q α ˆ Q α ) . (D.1)We denote the external momentum of the two point Feynman diagram by P µ = (cid:16) ˆ P µ , ¯ P µ (cid:17) ,and the “running” loop momentum by Q µ = (cid:16) ˆ Q µ , ¯ Q µ (cid:17) . The expression for the two point– 49 –unction Feynman diagram, as illustrated in figure 1, is given by: (cid:68) T µν ( P ) T αβ ( − P ) (cid:69) = − (cid:90) d d t ˆ Q (2 π ) d t (cid:90) d d s ¯ Q (2 π ) d s V µν ( Q, P − Q )[ i(cid:15) + m + κ ( ¯ Q α ¯ Q α ) + ( ˆ Q α ˆ Q α )] · V αβ ( Q, P − Q )[ i(cid:15) + m + κ (( ¯ Q − ¯ P ) α ( ¯ Q − ¯ P ) α ) + (( ˆ Q − ˆ P ) α ( ˆ Q − ˆ P ) α )] , (D.2)where (as in the relativistic calculation) m is an IR regulator later taken to be zero. Figure 1 . The Feynman diagram corresponding to the two point correlation function of the stress-energy tensor. Zigzag lines represent external momenta associated with the stress-energy tensorinsertions. Dashed lines represent the propagators of the scalar φ “running” in the loop. Theexpression corresponding to the vertexes is given in equation (D.3). Figure 2 . The Feynman diagram corresponding to the three point correlation function of thestress-energy tensor. Zigzag lines represent external momenta associated with the stress-energytensor insertions. Dashed lines represent the propagators of the scalar φ “running” in the loop.The expression corresponding to the vertexes is given in equation (D.3). Figure 3 . The Feynman diagram corresponding to the correlation function (D.5). Zigzag linesrepresent the external momenta. Dashed lines represent the propagators of the scalar φ “running”in the loop. The expression for the right vertex is given in equation (D.3). The expression for theleft vertex is constructed from several terms which are detailed in this appendix. It is straightforward to calculate the expression corresponding to the vertex V µν from– 50 –he stress-energy tensor given in (5.11). The result is: V αβ ( P, K ) = − κ ¯ K β ¯ K γ ¯ K γ ¯ P α + ( d s − d t ) κ ¯ K γ ¯ K γ ˆ K β ¯ P α d t − κ ¯ K α ¯ K γ ¯ K γ ¯ P β − ˆ K α ¯ P β + ( d s + 2 d t ) κ ¯ K α ˆ K β ¯ P γ ¯ P γ d t − κ ¯ K β ¯ P α ¯ P γ ¯ P γ − κ ˆ K β ¯ P α ¯ P γ ¯ P γ − κ ¯ K α ¯ P β ¯ P γ ¯ P γ + κ ¯ K γ ¯ K γ ¯ K δ ¯ δ αβ ¯ P δ − ( d s − d t ) κ ¯ K γ ¯ K γ ¯ K δ ˆ δ αβ ¯ P δ d t + κ ¯ K γ ¯ K γ ¯ δ αβ ¯ P δ ¯ P δ − d s κ ¯ K γ ¯ K γ ˆ δ αβ ¯ P δ ¯ P δ d t + κ ¯ K γ ¯ δ αβ ¯ P γ ¯ P δ ¯ P δ − ( d s − d t ) κ ¯ K γ ˆ δ αβ ¯ P γ ¯ P δ ¯ P δ d t − ¯ K β ˆ P α − ˆ K β ˆ P α − κ ¯ K α ¯ K γ ¯ K γ ˆ P β + ˆ K γ ˆ δ αβ ˆ P γ + ˆ K γ ¯ δ αβ ˆ P γ − ˆ K α ˆ P β + ( d s + 2 d t ) κ ¯ K γ ¯ K γ ¯ P α ˆ P β d t + ( d s − d t ) κ ¯ K α ¯ P γ ¯ P γ ˆ P β d t . (D.3)The expression (D.3) corresponds to the vertexes in figures 1 and 2, and to the right vertexin figure 3.The expression for the Feynman diagram corresponding to the three point function ofthe stress-energy tensor, as illustrated in figure 2, is given by: (cid:68) T µν ( L = − P − K ) T αβ ( P ) T ρσ ( K ) (cid:69) = i (cid:90) d d t ˆ Q (2 π ) d t (cid:90) d d s ¯ Q (2 π ) d s V αβ ( Q, P − Q )[ i(cid:15) + m + κ ( ¯ Q α ¯ Q α ) + ( ˆ Q α ˆ Q α )] · V ρσ ( − P + Q, K + P − Q )[ i(cid:15) + m + κ (( ¯ Q − ¯ P ) α ( ¯ Q − ¯ P ) α ) + (( ˆ Q − ˆ P ) α ( ˆ Q − ˆ P ) α )] · V µν ( − K − P + Q, − Q )[ i(cid:15) + m + κ (( ¯ Q − ¯ P − ¯ K ) α ( ¯ Q − ¯ P − ¯ K ) α ) + (( ˆ Q − ˆ P − ˆ K ) α ( ˆ Q − ˆ P − ˆ K ) α )] . (D.4)The expression for the Feynman diagram corresponding to the correlation function (A.25),as illustrated in figure 3, is given by: (cid:28) δ Sδe bα δe cγ ( P, K ) δSδe aµ ( − P − K ) (cid:29) = − (cid:90) d d t ˆ Q (2 π ) d t (cid:90) d d s ¯ Q (2 π ) d s (cid:2) δ S (cid:3) αγbc ( P, K, P + K − Q, Q )[ i(cid:15) + m + κ ( ¯ Q α ¯ Q α ) + ( ˆ Q α ˆ Q α )] · V µa ( P + K − Q, Q )[ i(cid:15) + m + κ (( ¯ Q − ¯ P − ¯ K ) α ( ¯ Q − ¯ P − ¯ K ) α ) + (( ˆ Q − ˆ P − ˆ K ) α ( ˆ Q − ˆ P − ˆ K ) α )] , (D.5)where the expression (cid:2) δ S (cid:3) αβγδ ( P, K, Q, L ) ≡ δ αα (cid:48) δ γγ (cid:48) δ bβ δ cδ (cid:2) δ S (cid:3) α (cid:48) γ (cid:48) bc ( P, K, P + K − Q, Q )corresponds to the insertion of the second variation of the action δ Sδe bα δe cγ in flat space. P and K in this expression refer to the external momenta, whereas the two momenta Q and L correspond to the two scalar propagator lines attached to the vertex, as illustrated infigure 3. – 51 –he expression for the vertex (cid:2) δ S (cid:3) αβγδ ( P, K, Q, L ) is quite long, and the remainderof this appendix is dedicated to the details of its calculation. For convenience, we split itinto several parts as follows. Ignoring the terms of order O ( ε lif ), the action (5.2) takes theform: S = (cid:90) d d t + d s x ( e ) (cid:32) (cid:16) L n ( i ) φ (cid:17) − κ (cid:18) ¯ ∇ φ + (cid:18) d t − d t (cid:19) a µ ¯ ∇ µ φ (cid:19) (cid:33) . (D.6)Let us define: S I ≡ (cid:90) d d t + d s x ( e ) (cid:18) (cid:16) L n ( i ) φ (cid:17) (cid:19) , (D.7)and S II ≡ − κ (cid:90) d d t + d s x ( e ) (cid:18) ¯ ∇ φ + (cid:18) d t − d t (cid:19) a µ ¯ ∇ µ φ (cid:19) , (D.8)so that: S = S I + S II . (D.9)The contribution of the first part of the action (D.7) to the vertex (cid:2) δ S (cid:3) αβγδ ( P, K, Q, L )can be easily derived: (cid:2) δ S I (cid:3) αβγδ ( P, K, Q, L ) = (cid:0) − δ γδ ˆ L α Q β + δ αδ ˆ L γ Q β + L δ ˆ δ αγ Q β + δ γβ ˆ L α Q δ − δ αβ ˆ L γ Q δ + L β ˆ δ αγ Q δ − δ γδ L β ˆ Q α + δ γβ L δ ˆ Q α + δ αδ L β ˆ Q γ − δ αβ L δ ˆ Q γ − ( δ αδ δ γβ − δ αβ δ γδ ) ˆ L µ ˆ Q µ − ( δ αδ δ γβ − δ αβ δ γδ ) ˆ L ν ˆ Q ν (cid:1) . (D.10)For the contribution of the second part of the action (D.8), it is convenient to use thefollowing identity (which follows from (B.19)):¯ ∇ φ = ∇ µ ¯ ∇ µ φ − a ν ¯ ∇ ν φ. (D.11) S II can then be written as follows: S II = − κ (cid:90) d d t + d s x ( e ) I ≡ − κ (cid:90) d d t + d s x ( e ) (cid:18) I − d t I (cid:19) , (D.12)where I , I and I are defined by: I ≡ ∇ µ ¯ ∇ µ φ = 1 e ∂ µ (cid:16) eP ab e µa e ν b ∂ ν φ (cid:17) , (D.13) I ≡ a ν ¯ ∇ ν φ, (D.14)and I ≡ I − d t I . (D.15)– 52 –he second variation of the expression S II with respect to the vielbeins, evaluated in flatspace, can be decomposed as follows: δ S II δe bα ( y ) δe cγ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) flat space = − κ (cid:90) d d t + d s x (cid:20) δ e ( x ) δe bα ( y ) δe cγ ( z ) (cid:0) ¯ ∂ µ ¯ ∂ µ φ ( x ) (cid:1) +2 δe ( x ) δe bα ( y ) (cid:0) ¯ ∂ µ ¯ ∂ µ φ ( x ) (cid:1) δI ( x ) δe cγ ( z )+2 δe ( x ) δe cγ ( z ) (cid:0) ¯ ∂ µ ¯ ∂ µ φ ( x ) (cid:1) δI ( x ) δe bα ( y ) + 2 δI ( x ) δe bα ( y ) δI ( x ) δe cγ ( z )+2 (cid:0) ¯ ∂ µ ¯ ∂ µ φ ( x ) (cid:1) δ I ( x ) δe bα ( y ) δe cγ ( z ) (cid:21) , (D.16)where all variations with respect to the vielbeins in this expression are evaluated at flatspace. Each instance of the first order variations of I that appear in the second, third andfourth terms in (D.16) contributes an expression of the following form to the vertex:[ δ ( I )] αβ ( P, Q ) ≡ − ¯ P α Q β − P β ¯ Q α + ˆ P β ¯ Q α d t + 2 Q β ¯ Q α + δ αβ ¯ P γ ¯ Q γ − ˆ δ αβ ¯ P γ ¯ Q γ d t , (D.17)where the momentum P here refers to one of the external momenta associated with thevertex, and Q to the momentum of one of the scalar propagator lines attached to it. Thecontribution of the expression (cid:82) d d t + d s x ¯ ∂ µ ¯ ∂ µ φ ( x ) δ I ( x ) δe bα ( y ) δe cγ ( z ) (contained in the last termof (D.16)) to the vertex is given by: (cid:104) δ ˜ I (cid:105) αβγδ ( P, K, Q, L ) ≡ (cid:18) − (cid:16) − − δ γδ L β ¯ L α + δ γβ L δ ¯ L α ) − δ αδ L β ¯ L γ − δ αβ L δ ¯ L γ ) − ( − δ αδ δ γβ + δ αβ δ γδ ) ¯ L ρ ¯ L ρ − ( δ αδ δ γβ + δ αβ δ γδ ) ¯ L ρ ¯ L ρ + δ γδ ( − L β ¯ L α + δ αβ ¯ L ρ ¯ L ρ )+ δ αβ ( − L δ ¯ L γ + δ γδ ¯ L ρ ¯ L ρ ) − L β L δ ¯ δ αγ − δ γδ P ρ ( − δ βρ ¯ L α + δ αβ ¯ L ρ − L β ¯ δ ρα ) − δ αβ K ρ ( − δ δρ ¯ L γ + δ γδ ¯ L ρ − L δ ¯ δ ργ ) + ( K ρ + P ρ ) (cid:0) ( δ δρ δ γβ − δ βρ δ γδ ) ¯ L α + ( − δ δρ δ αβ + δ βρ δ αδ ) ¯ L γ + ( − δ αδ δ γβ + δ αβ δ γδ ) ¯ L ρ + ( δ δρ L β + δ βρ L δ )¯ δ γα + ( − δ γδ L β + δ γβ L δ )¯ δ ρα + ( δ αδ L β − δ αβ L δ )¯ δ ργ (cid:1)(cid:17) ¯ Q µ ¯ Q µ − ¯ L µ ¯ L µ (cid:16) − δ αγ Q β Q δ − − δ γδ Q β ¯ Q α + δ γβ Q δ ¯ Q α ) (D.18) − δ αδ Q β ¯ Q γ − δ αβ Q δ ¯ Q γ ) − ( − δ αδ δ γβ + δ αβ δ γδ ) ¯ Q ν ¯ Q ν − ( δ αδ δ γβ + δ αβ δ γδ ) ¯ Q ν ¯ Q ν − δ γδ P ν ( − ¯ δ ν α Q β − δ βν ¯ Q α + δ αβ ¯ Q ν ) − δ αβ K ν ( − ¯ δ ν γ Q δ − δ δν ¯ Q γ + δ γδ ¯ Q ν ) + ( K ν + P ν ) (cid:0) ¯ δ γα ( δ δν Q β + δ βν Q δ ) + ¯ δ ν γ ( δ αδ Q β − δ αβ Q δ )+ ¯ δ ν α ( − δ γδ Q β + δ γβ Q δ ) + ( δ δν δ γβ − δ βν δ γδ ) ¯ Q α + ( − δ δν δ αβ + δ βν δ αδ ) ¯ Q γ + ( − δ αδ δ γβ + δ αβ δ γδ ) ¯ Q ν (cid:1) + δ γδ ( − Q β ¯ Q α + δ αβ ¯ Q ν ¯ Q ν )– 53 – δ αβ ( − Q δ ¯ Q γ + δ γδ ¯ Q ν ¯ Q ν ) (cid:17)(cid:19) . Finally, the contribution of the expression (cid:82) d d t + d s x ¯ ∂ µ ¯ ∂ µ φ ( x ) δ I ( x ) δe bα ( y ) δe cγ ( z ) (also containedin the last term of (D.16)) to the vertex is given by: (cid:104) δ ˜ I (cid:105) αβγδ ( P, K, Q, L ) ≡ (cid:18) − (cid:16) − ( δ ρα ¯ L β − L β ¯ δ ρα )( δ γρ ˆ K δ − K ρ ˆ δ γδ ) − ( δ ργ ¯ L δ − L δ ¯ δ ργ )( − P ρ ˆ δ αβ + δ αρ ˆ P β ) + i ¯ L ρ (cid:0) i ( − δ γρ K β ˆ δ αδ + δ γβ K ρ ˆ δ αδ − δ αρ P δ ˆ δ γβ + δ αδ P ρ ˆ δ γβ ) − i ( δ βρ δ γα ˆ K δ + δ αρ δ γβ ˆ K δ − δ βρ K α ˆ δ γδ − δ αρ K β ˆ δ γδ ) − i ( − δ δρ P γ ˆ δ αβ − δ γρ P δ ˆ δ αβ + δ γρ δ αδ ˆ P β + δ δρ δ γα ˆ P β ) (cid:1)(cid:17) ¯ Q µ ¯ Q µ − ¯ L µ ¯ L µ (cid:16) − ( δ γν ˆ K δ (D.19) − K ν ˆ δ γδ )( − ¯ δ να Q β + δ να ¯ Q β ) − ( − P ν ˆ δ αβ + δ αν ˆ P β )( − ¯ δ νγ Q δ + δ νγ ¯ Q δ )+ i (cid:0) i ( − δ γν K β ˆ δ αδ + δ γβ K ν ˆ δ αδ − δ αν P δ ˆ δ γβ + δ αδ P ν ˆ δ γβ ) − i ( δ βν δ γα ˆ K δ + δ αν δ γβ ˆ K δ − δ βν K α ˆ δ γδ − δ αν K β ˆ δ γδ ) − i ( − δ δν P γ ˆ δ αβ − δ γν P δ ˆ δ αβ + δ γν δ αδ ˆ P β + δ δν δ γα ˆ P β ) (cid:1) ¯ Q ν (cid:17)(cid:19) . Using the expressions (D.17)–(D.19) and the standard relations: δe ( x ) δe aµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) flat space = δ aµ δ ( x − y ) , (D.20) δ e ( x ) δe aµ ( y ) δe bρ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) flat space = ( δ µa δ ρb − δ µb δ ρa ) δ ( x − y ) δ ( x − z ) , (D.21)the total contribution of S II to the vertex (cid:2) δ S (cid:3) αβγδ ( P, K, Q, L ) can be assembled asfollows: (cid:2) δ S II (cid:3) αβγδ ( P, K, Q, L ) = κδ αδ δ βγ ¯ L µ ¯ L µ ¯ Q ν ¯ Q ν − κδ αβ δ γδ ¯ L µ ¯ L µ ¯ Q ν ¯ Q ν + κ (cid:16) ¯ Q ν ¯ Q ν [ δ ( I )] γδ ( K, L ) + ¯ L ν ¯ L ν [ δ ( I )] γδ ( K, L ) (cid:17) δ αβ + κ (cid:16) ¯ Q ν ¯ Q ν [ δ ( I )] αβ ( P, L ) + ¯ L ν ¯ L ν [ δ ( I )] αβ ( P, L ) (cid:17) δ γδ − κ (cid:16) [ δ ( I )] αβ ( P, Q ) [ δ ( I )] γδ ( K, L ) + [ δ ( I )] αβ ( P, L ) [ δ ( I )] γδ ( K, Q ) (cid:17) − κ (cid:104) δ ˜ I (cid:105) αβγδ ( P, K, Q, L ) + κd t (cid:104) δ ˜ I (cid:105) αβγδ ( P, K, Q, L ) . (D.22)– 54 – Variations of the Anomaly and Trivial Term Densities
In [7], the possible forms of Lifshitz scale anomalies (and the cohomologically trivial terms)were derived for the case of 2 + 1 dimensions with a dynamical critical exponent of z = 2,using a cohomological formulation of the Wess-Zumino consistency condition in curvedspacetime. In this appendix we detail the first and second order variations of these anomalyand trivial term densities with respect to the background vielbeins, evaluated in flatspacetime. These variations are used in conjunction with the anomalous Ward identitiesfor the flat space correlation functions given in (2.49)–(2.50) to extract the anomaly andtrivial term coefficients in section 5. The appendix is divided into two parts, for the twoand four derivatives sectors respectively.Throughout this appendix we use the following notation for the Fourier transformedvariation of some scalar X with respect to the vielbeins:(2 π ) d t + d s δ ( − P − Q )[ δX ] αβ ( P ) ≡ δ bβ δ αα (cid:48) F T (cid:20) δX ( x ) δe bα (cid:48) ( y ) (cid:21) , (E.1)(2 π ) d t + d s δ ( − P − K − Q ) (cid:2) δ X (cid:3) αβγδ ( P, K ) ≡ δ αα (cid:48) δ γγ (cid:48) δ bβ δ cδ F T (cid:20) δ X ( x ) δe bα (cid:48) ( y ) δe cγ (cid:48) ( z ) (cid:21) , (E.2)where the variations are evaluated in flat space, and the Fourier transforms use the con-ventions (A.21)–(A.22). Note that the expressions in this appendix are all in d t = 1 timedimension, and therefore the time components of the momenta are given by:ˆ P µ = − ˆ P n µ , ˆ K µ = − ˆ Kn µ , (E.3)where ˆ P ≡ P µ n µ and ˆ K ≡ K µ n µ , as explained in appendix A.3. E.1 The Two Derivatives Sector
Taken from [7], this sector contains the following scalar terms: φ = Tr (cid:0) K S (cid:1) , φ = K S , φ = L n K S . (E.4)Out of these terms, the only one which is first order in the background fields is φ , whosefirst order variation with respect to the vielbeins (evaluated in flat space and Fouriertransformed) is given by: [ δφ ] αβ = n α ¯ P β ˆ P − ¯ δ αβ ˆ P . (E.5)The second order variations of these terms with respect to the vielbeins (evaluated inflat space and Fourier transformed) are as follows: (cid:2) δ φ (cid:3) αβγδ ( P, K ) = ˆ Kn α ¯ δ βδ ¯ P γ − ¯ K β n α n γ ¯ P δ + ˆ Kn α ¯ δ βγ ¯ P δ − ¯ K µ n α n γ ¯ δ βδ ¯ P µ + ¯ K β n γ ¯ δ αδ ˆ P − ˆ K ¯ δ αδ ¯ δ βγ ˆ P + ¯ K α n γ ¯ δ βδ ˆ P − ˆ K ¯ δ αγ ¯ δ βδ ˆ P , (E.6) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − K δ n α n γ ¯ P β + 2 ˆ Kn α ¯ δ γδ ¯ P β + 2 ¯ K δ n γ ¯ δ αβ ˆ P − K ¯ δ αβ ¯ δ γδ ˆ P , (E.7) We use the results of the more general non-Frobenius case (See [7] for details and discussion). – 55 – δ φ (cid:3) αβγδ ( P, K ) = − ¯ K β ¯ K δ n α n γ + 2 ¯ K δ ˆ Kn α n β n γ − ¯ K β ˆ Kn γ ¯ δ αδ + ˆ K ¯ δ αδ ¯ δ βγ + 2 ¯ K β ˆ Kn α ¯ δ γδ − K n α n β ¯ δ γδ + ˆ Kn α n γ n δ ¯ P β − ˆ Kn γ ¯ δ αδ ¯ P β + ˆ Kn γ ¯ δ αβ ¯ P δ − ˆ Kn α ¯ δ βγ ¯ P δ − n α n γ ¯ P β ¯ P δ + ¯ K δ n α n β n γ ˆ P − ˆ Kn γ n δ ¯ δ αβ ˆ P − ¯ K β n γ ¯ δ αδ ˆ P − ¯ K δ n α ¯ δ βγ ˆ P + 2 ˆ K ¯ δ αδ ¯ δ βγ ˆ P + ¯ K β n α ¯ δ γδ ˆ P − ˆ Kn α n β ¯ δ γδ ˆ P + 2 n α n γ n δ ¯ P β ˆ P − n γ ¯ δ αδ ¯ P β ˆ P + 2 n γ ¯ δ αβ ¯ P δ ˆ P − n α ¯ δ βγ ¯ P δ ˆ P − n γ n δ ¯ δ αβ ˆ P + ¯ δ αδ ¯ δ βγ ˆ P − ¯ K δ ˆ Kn α ¯ δ βγ . (E.8)As found in [7], this sector contains a trivial term (coboundary) with the following density: F = − φ − φ , (E.9)as well as an anomaly term with the following density: A (2 , , = φ − φ . (E.10)The second variation of the trivial term F with respect to the vielbeins (evaluated in flatspace and Fourier transformed) is therefore given by the following expression: (cid:2) δ F (cid:3) αβγδ ( P, K ) = 2 ¯ K β ¯ K δ n α n γ − K δ ˆ Kn α n β n γ + 2 ¯ K β ˆ Kn γ ¯ δ αδ − K ¯ δ αδ ¯ δ βγ − K β ˆ Kn α ¯ δ γδ + 4 ˆ K n α n β ¯ δ γδ + 4 ¯ K δ n α n γ ¯ P β − Kn α n γ n δ ¯ P β + 2 ˆ Kn γ ¯ δ αδ ¯ P β − Kn α ¯ δ γδ ¯ P β − Kn γ ¯ δ αβ ¯ P δ + 2 ˆ Kn α ¯ δ βγ ¯ P δ + 2 n α n γ ¯ P β ¯ P δ − K δ n α n β n γ ˆ P − K δ n γ ¯ δ αβ ˆ P + 2 ˆ Kn γ n δ ¯ δ αβ ˆ P + 2 ¯ K β n γ ¯ δ αδ ˆ P + 2 ¯ K δ n α ¯ δ βγ ˆ P − K ¯ δ αδ ¯ δ βγ ˆ P − K β n α ¯ δ γδ ˆ P + 2 ˆ Kn α n β ¯ δ γδ ˆ P + 4 ˆ K ¯ δ αβ ¯ δ γδ ˆ P − n α n γ n δ ¯ P β ˆ P + 2 n γ ¯ δ αδ ¯ P β ˆ P − n γ ¯ δ αβ ¯ P δ ˆ P + 2 n α ¯ δ βγ ¯ P δ ˆ P + 4 n γ n δ ¯ δ αβ ˆ P − δ αδ ¯ δ βγ ˆ P + 2 ¯ K δ ˆ Kn α ¯ δ βγ , (E.11)whereas the second variation of the anomaly density A (2 , , is given by: (cid:104) δ A (2 , , (cid:105) αβγδ ( P, K ) = ¯ K δ n α n γ ¯ P β − ˆ Kn α ¯ δ γδ ¯ P β + ˆ Kn α ¯ δ βδ ¯ P γ − ¯ K µ n α n γ ¯ δ βδ ¯ P µ − ¯ K δ n γ ¯ δ αβ ˆ P + ¯ K β n γ ¯ δ αδ ˆ P − ˆ K ¯ δ αδ ¯ δ βγ ˆ P + ¯ K α n γ ¯ δ βδ ˆ P − ˆ K ¯ δ αγ ¯ δ βδ ˆ P + ˆ K ¯ δ αβ ¯ δ γδ ˆ P − ¯ K β n α n γ ¯ P δ + ˆ Kn α ¯ δ βγ ¯ P δ . (E.12)– 56 – .2 The Four Derivatives Sector The independent scalar terms in the four derivatives sector are given in [7]. Out of those,the ones which are at most second order in the background fields are the following: φ = ˆ R , φ = a α ¯ ∇ α ˆ R, φ = ˆ R ¯ ∇ α a α ,φ = ¯ ∇ ˆ R, φ = (cid:0) ¯ ∇ α a α (cid:1) , φ = ¯ ∇ ( α a β ) ¯ ∇ ( α a β ) , φ = a α ¯ ∇ α ¯ ∇ β a β ,φ = ¯ ∇ ¯ ∇ α a α , φ = ¯ (cid:15) µν ¯ ∇ µ K A ¯ ∇ ν K S , φ = K A ¯ ∇ α ¯ ∇ β ˜ K αβS ,φ = ¯ ∇ α K A ¯ ∇ β ˜ K αβS , φ = ¯ ∇ α ¯ ∇ β K A ˜ K αβS , φ = ¯ (cid:15) µν L n ¯ ∇ µ K A a ν ,φ = ¯ (cid:15) µν ¯ ∇ µ K A L n a ν , φ = K A L n K A , φ = ( L n K A ) . (E.13)Note that terms which are of higher order in the background fields will not show up inour analysis, since we consider here only the two and three point correlation functions, asexplained in section 5.Out of the scalar terms in (E.13), the only ones which are first order in the backgroundfields are φ and φ . Their first order variations with respect to the vielbeins (evaluatedin flat space and Fourier transformed) are given by:[ δφ ] αβ ( P ) = 2 ¯ P α ¯ P β ¯ P γ ¯ P γ − δ αβ ¯ P γ ¯ P γ ¯ P δ ¯ P δ , (E.14)[ δφ ] αβ ( P ) = − n α n β ¯ P γ ¯ P γ ¯ P δ ¯ P δ + n β ¯ P α ¯ P γ ¯ P γ ˆ P . (E.15)The second order variation of the terms in (E.13) with respect to the vielbeins (evaluatedin flat space and Fourier transformed) are as follows: (cid:2) δ φ (cid:3) αβγδ ( P, K ) = 8 ¯ K γ ¯ K δ ¯ P α ¯ P β − K µ ¯ K µ ¯ δ γδ ¯ P α ¯ P β + 8 ¯ K µ ¯ K µ ¯ δ αβ ¯ δ γδ ¯ P ν ¯ P ν − K γ ¯ K δ ¯ δ αβ ¯ P µ ¯ P µ , (E.16) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = 2 ˆ Kn δ ¯ P α ¯ P β ¯ P γ − K γ ¯ K δ ¯ K µ n α n β ¯ P µ − K µ n γ n δ ¯ P α ¯ P β ¯ P µ + 2 ¯ K µ ¯ K µ ¯ K ν n α n β ¯ δ γδ ¯ P ν + 2 ¯ K µ n γ n δ ¯ δ αβ ¯ P µ ¯ P ν ¯ P ν + 2 ¯ K α ¯ K γ ¯ K δ n β ˆ P − K α ¯ K µ ¯ K µ n β ¯ δ γδ ˆ P − Kn δ ¯ δ αβ ¯ P γ ¯ P µ ¯ P µ , (E.17) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = 2 ¯ K γ ˆ Kn δ ¯ P α ¯ P β − K µ ¯ K µ n γ n δ ¯ P α ¯ P β + 2 ¯ K µ ¯ K µ n γ n δ ¯ δ αβ ¯ P ν ¯ P ν + 2 ¯ K µ ¯ K µ n α n β ¯ δ γδ ¯ P ν ¯ P ν + 2 ¯ K γ ¯ K δ n β ¯ P α ˆ P − K µ ¯ K µ n β ¯ δ γδ ¯ P α ˆ P − K γ ¯ K δ n α n β ¯ P µ ¯ P µ − K γ ˆ Kn δ ¯ δ αβ ¯ P µ ¯ P µ , (E.18) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − K γ ˆ Kn α n β n δ ¯ P µ ¯ P µ + 2 ¯ K µ ¯ K µ n α n β n γ n δ ¯ P ν ¯ P ν − K µ ¯ K µ n β n γ n δ ¯ P α ˆ P + 2 ¯ K γ ˆ Kn β n δ ¯ P α ˆ P , (E.19) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − K µ ˆ Kn α n β n δ ¯ P γ ¯ P µ + 2 ¯ K µ ¯ K ν n α n β n γ n δ ¯ P µ ¯ P ν + 2 ¯ K µ ˆ Kn β n δ ¯ δ αγ ¯ P µ ˆ P − K α ¯ K µ n β n γ n δ ¯ P µ ˆ P , (E.20) Note that we changed the notations of (cid:101) ∇ and ˜ (cid:15) µν from [7] to ¯ ∇ and ¯ (cid:15) µν respectively, to better fit thenotations of this paper. – 57 – δ φ (cid:3) αβγδ ( P, K ) = − ¯ K γ ¯ K µ ˆ Kn α n β n δ ¯ P µ − ˆ Kn α n β n δ ¯ P γ ¯ P µ ¯ P µ + ¯ K µ n α n β n γ n δ ¯ P µ ¯ P ν ¯ P ν + ¯ K α ¯ K γ ˆ Kn β n δ ˆ P − ¯ K α ¯ K µ ¯ K µ n β n γ n δ ˆ P − ¯ K µ n β n γ n δ ¯ P α ¯ P µ ˆ P + ¯ K µ ¯ K µ ¯ K ν n α n β n γ n δ ¯ P ν + ˆ Kn β n δ ¯ P α ¯ P γ ˆ P , (E.21) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − K α ¯ K β ¯ K γ ¯ K δ + 4 ¯ K α ¯ K γ ¯ K δ ˆ Kn β + 2 ¯ K α ¯ K δ ¯ K µ ¯ K µ n β n γ − K β ¯ K δ ¯ K µ ¯ K µ ¯ δ αγ + 2 ¯ K δ ¯ K µ ¯ K µ ˆ Kn β ¯ δ αγ − K β ¯ K γ ¯ K µ ¯ K µ ¯ δ αδ + 2 ¯ K γ ¯ K µ ¯ K µ ˆ Kn β ¯ δ αδ − K µ ¯ K µ ¯ K ν ¯ K ν n β n γ ¯ δ αδ − K α ¯ K δ ¯ K µ ¯ K µ ¯ δ βγ + 2 ¯ K µ ¯ K µ ¯ K ν ¯ K ν ¯ δ αδ ¯ δ βγ + 8 ¯ K α ¯ K β ¯ K µ ¯ K µ ¯ δ γδ − K α ¯ K µ ¯ K µ ˆ Kn β ¯ δ γδ − K β ¯ K γ ¯ K δ ¯ P α + 2 ¯ K γ ¯ K δ ˆ Kn β ¯ P α + 2 ¯ K δ ¯ K µ ¯ K µ n β n γ ¯ P α − K δ ¯ K µ ¯ K µ ¯ δ βγ ¯ P α + 4 ¯ K β ¯ K µ ¯ K µ ¯ δ γδ ¯ P α − K µ ¯ K µ ˆ Kn β ¯ δ γδ ¯ P α − K α ¯ K γ ¯ K δ ¯ P β + 2 ¯ K γ ¯ K µ ¯ K µ n α n δ ¯ P β − K δ ¯ K µ ¯ K µ ¯ δ αγ ¯ P β − K γ ¯ K µ ¯ K µ ¯ δ αδ ¯ P β + 4 ¯ K α ¯ K µ ¯ K µ ¯ δ γδ ¯ P β − ¯ K β ¯ K µ ¯ K µ n α n δ ¯ P γ + 2 ¯ K δ ¯ K µ ¯ K µ ¯ δ αβ ¯ P γ − ¯ K β ¯ K µ ¯ K µ ¯ δ αδ ¯ P γ + ¯ K µ ¯ K µ ˆ Kn β ¯ δ αδ ¯ P γ + ¯ K α ¯ K µ ¯ K µ ¯ δ βδ ¯ P γ + 2 ¯ K µ ¯ K µ n α n δ ¯ P β ¯ P γ − K µ ¯ K µ ¯ δ αδ ¯ P β ¯ P γ − K δ ¯ P α ¯ P β ¯ P γ − ¯ K α ¯ K µ ¯ K µ n β n γ ¯ P δ + 2 ¯ K γ ¯ K µ ¯ K µ ¯ δ αβ ¯ P δ − ¯ K β ¯ K µ ¯ K µ ¯ δ αγ ¯ P δ + ¯ K µ ¯ K µ ˆ Kn β ¯ δ αγ ¯ P δ − ¯ K α ¯ K µ ¯ K µ ¯ δ βγ ¯ P δ − K µ ¯ K µ ¯ δ βγ ¯ P α ¯ P δ − K µ ¯ K µ ¯ δ αγ ¯ P β ¯ P δ − K γ ¯ P α ¯ P β ¯ P δ + 4 ¯ K µ ¯ K µ ¯ δ αβ ¯ P γ ¯ P δ − P α ¯ P β ¯ P γ ¯ P δ + 4 ¯ K α ¯ K δ ¯ K µ n β n γ ¯ P µ + 2 ¯ K γ ¯ K δ ¯ K µ ¯ δ αβ ¯ P µ − K β ¯ K δ ¯ K µ ¯ δ αγ ¯ P µ + 4 ¯ K δ ¯ K µ ˆ Kn β ¯ δ αγ ¯ P µ − K β ¯ K γ ¯ K µ ¯ δ αδ ¯ P µ + 4 ¯ K γ ¯ K µ ˆ Kn β ¯ δ αδ ¯ P µ − K α ¯ K δ ¯ K µ ¯ δ βγ ¯ P µ + 8 ¯ K α ¯ K β ¯ K µ ¯ δ γδ ¯ P µ − K α ¯ K µ ˆ Kn β ¯ δ γδ ¯ P µ + 4 ¯ K δ ¯ K µ n β n γ ¯ P α ¯ P µ − K δ ¯ K µ ¯ δ βγ ¯ P α ¯ P µ + 4 ¯ K β ¯ K µ ¯ δ γδ ¯ P α ¯ P µ − K µ ˆ Kn β ¯ δ γδ ¯ P α ¯ P µ + 4 ¯ K γ ¯ K µ n α n δ ¯ P β ¯ P µ − K δ ¯ K µ ¯ δ αγ ¯ P β ¯ P µ − K γ ¯ K µ ¯ δ αδ ¯ P β ¯ P µ + 4 ¯ K α ¯ K µ ¯ δ γδ ¯ P β ¯ P µ + 2 ¯ K µ ¯ δ γδ ¯ P α ¯ P β ¯ P µ − K β ¯ K µ n α n δ ¯ P γ ¯ P µ + 4 ¯ K δ ¯ K µ ¯ δ αβ ¯ P γ ¯ P µ − K β ¯ K µ ¯ δ αδ ¯ P γ ¯ P µ + 2 ¯ K µ ˆ Kn β ¯ δ αδ ¯ P γ ¯ P µ + 2 ¯ K α ¯ K µ ¯ δ βδ ¯ P γ ¯ P µ + 4 ¯ K µ n α n δ ¯ P β ¯ P γ ¯ P µ − K µ ¯ δ αδ ¯ P β ¯ P γ ¯ P µ − K α ¯ K µ n β n γ ¯ P δ ¯ P µ + 4 ¯ K γ ¯ K µ ¯ δ αβ ¯ P δ ¯ P µ − K β ¯ K µ ¯ δ αγ ¯ P δ ¯ P µ + 2 ¯ K µ ˆ Kn β ¯ δ αγ ¯ P δ ¯ P µ − K α ¯ K µ ¯ δ βγ ¯ P δ ¯ P µ − K µ ¯ δ βγ ¯ P α ¯ P δ ¯ P µ − K µ ¯ δ αγ ¯ P β ¯ P δ ¯ P µ + 8 ¯ K µ ¯ δ αβ ¯ P γ ¯ P δ ¯ P µ + 2 ¯ K α ¯ K δ n β n γ ¯ P µ ¯ P µ − K β ¯ K δ ¯ δ αγ ¯ P µ ¯ P µ + 2 ¯ K δ ˆ Kn β ¯ δ αγ ¯ P µ ¯ P µ − K β ¯ K γ ¯ δ αδ ¯ P µ ¯ P µ + 2 ¯ K γ ˆ Kn β ¯ δ αδ ¯ P µ ¯ P µ − K α ¯ K δ ¯ δ βγ ¯ P µ ¯ P µ + 4 ¯ K α ¯ K β ¯ δ γδ ¯ P µ ¯ P µ − K α ˆ Kn β ¯ δ γδ ¯ P µ ¯ P µ + 2 ¯ K δ n β n γ ¯ P α ¯ P µ ¯ P µ − K δ ¯ δ βγ ¯ P α ¯ P µ ¯ P µ + 2 ¯ K β ¯ δ γδ ¯ P α ¯ P µ ¯ P µ (E.22) − Kn β ¯ δ γδ ¯ P α ¯ P µ ¯ P µ + 2 ¯ K γ n α n δ ¯ P β ¯ P µ ¯ P µ − K δ ¯ δ αγ ¯ P β ¯ P µ ¯ P µ − K γ ¯ δ αδ ¯ P β ¯ P µ ¯ P µ + 2 ¯ K α ¯ δ γδ ¯ P β ¯ P µ ¯ P µ − ¯ K β n α n δ ¯ P γ ¯ P µ ¯ P µ + 4 ¯ K δ ¯ δ αβ ¯ P γ ¯ P µ ¯ P µ − ¯ K β ¯ δ αδ ¯ P γ ¯ P µ ¯ P µ + ˆ Kn β ¯ δ αδ ¯ P γ ¯ P µ ¯ P µ – 58 – ¯ K α ¯ δ βδ ¯ P γ ¯ P µ ¯ P µ + 2 n α n δ ¯ P β ¯ P γ ¯ P µ ¯ P µ − δ αδ ¯ P β ¯ P γ ¯ P µ ¯ P µ − ¯ K α n β n γ ¯ P δ ¯ P µ ¯ P µ + 4 ¯ K γ ¯ δ αβ ¯ P δ ¯ P µ ¯ P µ − ¯ K β ¯ δ αγ ¯ P δ ¯ P µ ¯ P µ + ˆ Kn β ¯ δ αγ ¯ P δ ¯ P µ ¯ P µ − ¯ K α ¯ δ βγ ¯ P δ ¯ P µ ¯ P µ − δ βγ ¯ P α ¯ P δ ¯ P µ ¯ P µ − δ αγ ¯ P β ¯ P δ ¯ P µ ¯ P µ + 8¯ δ αβ ¯ P γ ¯ P δ ¯ P µ ¯ P µ − K µ ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P ν − ¯ K µ ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P ν + 7 ¯ K µ ¯ K µ ¯ K ν ¯ δ αδ ¯ δ βγ ¯ P ν − ¯ K µ ¯ K µ ¯ K ν ¯ δ αγ ¯ δ βδ ¯ P ν − K µ ¯ K µ ¯ K ν ¯ δ αβ ¯ δ γδ ¯ P ν − K µ ¯ K ν n β n γ ¯ δ αδ ¯ P µ ¯ P ν − K µ ¯ K ν n α n δ ¯ δ βγ ¯ P µ ¯ P ν + 6 ¯ K µ ¯ K ν ¯ δ αδ ¯ δ βγ ¯ P µ ¯ P ν − K µ ¯ K ν ¯ δ αγ ¯ δ βδ ¯ P µ ¯ P ν − K µ ¯ K ν ¯ δ αβ ¯ δ γδ ¯ P µ ¯ P ν − K µ ¯ K µ n β n γ ¯ δ αδ ¯ P ν ¯ P ν − K µ ¯ K µ n α n δ ¯ δ βγ ¯ P ν ¯ P ν + 4 ¯ K µ ¯ K µ ¯ δ αδ ¯ δ βγ ¯ P ν ¯ P ν − ¯ K µ n β n γ ¯ δ αδ ¯ P µ ¯ P ν ¯ P ν − K µ n α n δ ¯ δ βγ ¯ P µ ¯ P ν ¯ P ν + 7 ¯ K µ ¯ δ αδ ¯ δ βγ ¯ P µ ¯ P ν ¯ P ν − ¯ K µ ¯ δ αγ ¯ δ βδ ¯ P µ ¯ P ν ¯ P ν − K µ ¯ δ αβ ¯ δ γδ ¯ P µ ¯ P ν ¯ P ν − n α n δ ¯ δ βγ ¯ P µ ¯ P µ ¯ P ν ¯ P ν + 2¯ δ αδ ¯ δ βγ ¯ P µ ¯ P µ ¯ P ν ¯ P ν − K γ ¯ K µ ¯ K µ n δ ¯ δ αβ ˆ P + ¯ K β ¯ K µ ¯ K µ n δ ¯ δ αγ ˆ P + ¯ K α ¯ K µ ¯ K µ n δ ¯ δ βγ ˆ P + 2 ¯ K µ ¯ K µ n δ ¯ δ βγ ¯ P α ˆ P + 2 ¯ K µ ¯ K µ n δ ¯ δ αγ ¯ P β ˆ P + 2 ¯ K γ n δ ¯ P α ¯ P β ˆ P − K µ ¯ K µ n δ ¯ δ αβ ¯ P γ ˆ P + 4 n δ ¯ P α ¯ P β ¯ P γ ˆ P − K γ ¯ K µ n δ ¯ δ αβ ¯ P µ ˆ P + 2 ¯ K β ¯ K µ n δ ¯ δ αγ ¯ P µ ˆ P + 2 ¯ K α ¯ K µ n δ ¯ δ βγ ¯ P µ ˆ P + 4 ¯ K µ n δ ¯ δ βγ ¯ P α ¯ P µ ˆ P + 4 ¯ K µ n δ ¯ δ αγ ¯ P β ¯ P µ ˆ P − K µ n δ ¯ δ αβ ¯ P γ ¯ P µ ˆ P − K γ n δ ¯ δ αβ ¯ P µ ¯ P µ ˆ P + ¯ K β n δ ¯ δ αγ ¯ P µ ¯ P µ ˆ P + ¯ K α n δ ¯ δ βγ ¯ P µ ¯ P µ ˆ P + 2 n δ ¯ δ βγ ¯ P α ¯ P µ ¯ P µ ˆ P + 2 n δ ¯ δ αγ ¯ P β ¯ P µ ¯ P µ ˆ P − n δ ¯ δ αβ ¯ P γ ¯ P µ ¯ P µ ˆ P , (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − K α ¯ K β ¯ K γ ˆ Kn δ − ¯ K β ¯ K γ ¯ K µ ¯ K µ n α n δ + 2 ¯ K α ¯ K γ ˆ K n β n δ + ¯ K γ ¯ K µ ¯ K µ ˆ Kn α n β n δ + 4 ¯ K α ¯ K β ¯ K µ ¯ K µ n γ n δ − K α ¯ K µ ¯ K µ ˆ Kn β n γ n δ − ¯ K µ ¯ K µ ¯ K ν ¯ K ν n α n β n γ n δ − ¯ K β ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αγ + ¯ K µ ¯ K µ ˆ K n β n δ ¯ δ αγ − ¯ K α ¯ K µ ¯ K µ ˆ Kn δ ¯ δ βγ + ¯ K µ ¯ K µ ¯ K ν ¯ K ν n α n δ ¯ δ βγ − ¯ K β ¯ K γ ˆ Kn δ ¯ P α + ¯ K γ ˆ K n β n δ ¯ P α + 2 ¯ K β ¯ K µ ¯ K µ n γ n δ ¯ P α − ¯ K µ ¯ K µ ˆ Kn β n γ n δ ¯ P α − ¯ K µ ¯ K µ ˆ Kn δ ¯ δ βγ ¯ P α − ¯ K α ¯ K γ ˆ Kn δ ¯ P β + 2 ¯ K α ¯ K µ ¯ K µ n γ n δ ¯ P β − ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αγ ¯ P β + ¯ K δ ¯ K µ ¯ K µ n α n β ¯ P γ − ¯ K β ¯ K µ ¯ K µ n α n δ ¯ P γ + ¯ K µ ¯ K µ ˆ Kn α n β n δ ¯ P γ + ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P γ + ¯ K γ ¯ K µ ¯ K µ n α n β ¯ P δ − ¯ K α ¯ K µ ¯ K µ n β n γ ¯ P δ − ¯ K µ ¯ K µ n β n γ ¯ P α ¯ P δ + 2 ¯ K µ ¯ K µ n α n β ¯ P γ ¯ P δ − K β ¯ K γ ¯ K µ n α n δ ¯ P µ + 2 ¯ K γ ¯ K µ ˆ Kn α n β n δ ¯ P µ + 4 ¯ K α ¯ K β ¯ K µ n γ n δ ¯ P µ − K α ¯ K µ ˆ Kn β n γ n δ ¯ P µ + ¯ K γ ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P µ − K β ¯ K µ ˆ Kn δ ¯ δ αγ ¯ P µ + 2 ¯ K µ ˆ K n β n δ ¯ δ αγ ¯ P µ − K α ¯ K µ ˆ Kn δ ¯ δ βγ ¯ P µ + 2 ¯ K β ¯ K µ n γ n δ ¯ P α ¯ P µ − K µ ˆ Kn δ ¯ δ βγ ¯ P α ¯ P µ + 2 ¯ K α ¯ K µ n γ n δ ¯ P β ¯ P µ − K µ ˆ Kn δ ¯ δ αγ ¯ P β ¯ P µ + 2 ¯ K δ ¯ K µ n α n β ¯ P γ ¯ P µ − K β ¯ K µ n α n δ ¯ P γ ¯ P µ + 2 ¯ K µ ˆ Kn α n β n δ ¯ P γ ¯ P µ – 59 – 2 ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P γ ¯ P µ + 2 ¯ K γ ¯ K µ n α n β ¯ P δ ¯ P µ − K α ¯ K µ n β n γ ¯ P δ ¯ P µ − K µ n β n γ ¯ P α ¯ P δ ¯ P µ + 4 ¯ K µ n α n β ¯ P γ ¯ P δ ¯ P µ − ¯ K β ¯ K γ n α n δ ¯ P µ ¯ P µ + ¯ K γ ˆ Kn α n β n δ ¯ P µ ¯ P µ + 2 ¯ K α ¯ K β n γ n δ ¯ P µ ¯ P µ − ¯ K α ˆ Kn β n γ n δ ¯ P µ ¯ P µ − ¯ K β ˆ Kn δ ¯ δ αγ ¯ P µ ¯ P µ + ˆ K n β n δ ¯ δ αγ ¯ P µ ¯ P µ − ¯ K α ˆ Kn δ ¯ δ βγ ¯ P µ ¯ P µ + ¯ K β n γ n δ ¯ P α ¯ P µ ¯ P µ − ˆ Kn δ ¯ δ βγ ¯ P α ¯ P µ ¯ P µ + ¯ K α n γ n δ ¯ P β ¯ P µ ¯ P µ (E.23) − ˆ Kn δ ¯ δ αγ ¯ P β ¯ P µ ¯ P µ + 2 ¯ K δ n α n β ¯ P γ ¯ P µ ¯ P µ − ¯ K β n α n δ ¯ P γ ¯ P µ ¯ P µ + ˆ Kn α n β n δ ¯ P γ ¯ P µ ¯ P µ + ˆ Kn δ ¯ δ αβ ¯ P γ ¯ P µ ¯ P µ + 2 ¯ K γ n α n β ¯ P δ ¯ P µ ¯ P µ − ¯ K α n β n γ ¯ P δ ¯ P µ ¯ P µ − n β n γ ¯ P α ¯ P δ ¯ P µ ¯ P µ + 4 n α n β ¯ P γ ¯ P δ ¯ P µ ¯ P µ − K µ ¯ K µ ¯ K ν n α n β n γ n δ ¯ P ν − K µ ¯ K µ ¯ K ν n γ n δ ¯ δ αβ ¯ P ν + ¯ K µ ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P ν + 3 ¯ K µ ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P ν − ¯ K µ ¯ K µ ¯ K ν n α n β ¯ δ γδ ¯ P ν − K µ ¯ K ν n α n β n γ n δ ¯ P µ ¯ P ν − K µ ¯ K ν n γ n δ ¯ δ αβ ¯ P µ ¯ P ν + 2 ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P µ ¯ P ν + 2 ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P µ ¯ P ν − K µ ¯ K ν n α n β ¯ δ γδ ¯ P µ ¯ P ν − K µ ¯ K µ n α n β n γ n δ ¯ P ν ¯ P ν + ¯ K µ ¯ K µ n β n γ ¯ δ αδ ¯ P ν ¯ P ν + ¯ K µ ¯ K µ n α n δ ¯ δ βγ ¯ P ν ¯ P ν − K µ n α n β n γ n δ ¯ P µ ¯ P ν ¯ P ν − ¯ K µ n γ n δ ¯ δ αβ ¯ P µ ¯ P ν ¯ P ν + 3 ¯ K µ n β n γ ¯ δ αδ ¯ P µ ¯ P ν ¯ P ν + ¯ K µ n α n δ ¯ δ βγ ¯ P µ ¯ P ν ¯ P ν − K µ n α n β ¯ δ γδ ¯ P µ ¯ P ν ¯ P ν − n α n β n γ n δ ¯ P µ ¯ P µ ¯ P ν ¯ P ν + n β n γ ¯ δ αδ ¯ P µ ¯ P µ ¯ P ν ¯ P ν + ¯ K α ¯ K µ ¯ K µ n β n γ n δ ˆ P − ¯ K δ ¯ K µ ¯ K µ n β ¯ δ αγ ˆ P − ¯ K γ ¯ K µ ¯ K µ n β ¯ δ αδ ˆ P + ¯ K α ¯ K µ ¯ K µ n β ¯ δ γδ ˆ P + ¯ K µ ¯ K µ n β n γ n δ ¯ P α ˆ P − ¯ K µ ¯ K µ n α n β n δ ¯ P γ ˆ P − ¯ K µ ¯ K µ n β ¯ δ αδ ¯ P γ ˆ P − ¯ K δ n β ¯ P α ¯ P γ ˆ P − ¯ K µ ¯ K µ n β ¯ δ αγ ¯ P δ ˆ P − ¯ K γ n β ¯ P α ¯ P δ ˆ P − n β ¯ P α ¯ P γ ¯ P δ ˆ P + 2 ¯ K α ¯ K µ n β n γ n δ ¯ P µ ˆ P − K δ ¯ K µ n β ¯ δ αγ ¯ P µ ˆ P − K γ ¯ K µ n β ¯ δ αδ ¯ P µ ˆ P + 2 ¯ K α ¯ K µ n β ¯ δ γδ ¯ P µ ˆ P + 2 ¯ K µ n β n γ n δ ¯ P α ¯ P µ ˆ P + ¯ K µ n β ¯ δ γδ ¯ P α ¯ P µ ˆ P − K µ n α n β n δ ¯ P γ ¯ P µ ˆ P − K µ n β ¯ δ αδ ¯ P γ ¯ P µ ˆ P − K µ n β ¯ δ αγ ¯ P δ ¯ P µ ˆ P − ¯ K γ n α n β n δ ¯ P µ ¯ P µ ˆ P + ¯ K α n β n γ n δ ¯ P µ ¯ P µ ˆ P − ¯ K δ n β ¯ δ αγ ¯ P µ ¯ P µ ˆ P − ¯ K γ n β ¯ δ αδ ¯ P µ ¯ P µ ˆ P + ¯ K α n β ¯ δ γδ ¯ P µ ¯ P µ ˆ P + n β n γ n δ ¯ P α ¯ P µ ¯ P µ ˆ P − n α n β n δ ¯ P γ ¯ P µ ¯ P µ ˆ P − n β ¯ δ αδ ¯ P γ ¯ P µ ¯ P µ ˆ P − n β ¯ δ αγ ¯ P δ ¯ P µ ¯ P µ ˆ P + ¯ K µ ¯ K µ n β n δ ¯ δ αγ ˆ P + ¯ K γ n β n δ ¯ P α ˆ P + 2 n β n δ ¯ P α ¯ P γ ˆ P + 2 ¯ K µ n β n δ ¯ δ αγ ¯ P µ ˆ P + n β n δ ¯ δ αγ ¯ P µ ¯ P µ ˆ P , (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ¯ K µ ¯ K µ n α n δ ¯ P β ¯ P γ + ¯ K δ ¯ K µ n β n γ ¯ P α ¯ P µ − ¯ K µ ˆ Kn β ¯ δ γδ ¯ P α ¯ P µ + ¯ K γ ¯ K µ n α n δ ¯ P β ¯ P µ − ¯ K α ¯ K δ n β n γ ¯ P µ ¯ P µ + ¯ K µ ¯ K µ n δ ¯ δ αβ ¯ P γ ˆ P − ¯ K γ ¯ K µ n δ ¯ δ αβ ¯ P µ ˆ P + ¯ K α ˆ Kn β ¯ δ γδ ¯ P µ ¯ P µ , (E.24)– 60 – δ φ (cid:3) αβγδ ( P, K ) = − ¯ K α ¯ K δ ˆ Kn β ¯ P γ − ¯ K α ¯ K γ ˆ Kn β ¯ P δ + ¯ K δ ¯ K µ ˆ Kn β ¯ δ αγ ¯ P µ + ¯ K γ ¯ K µ ˆ Kn β ¯ δ αδ ¯ P µ − ¯ K µ ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P ν − ¯ K µ n α n δ ¯ δ βγ ¯ P µ ¯ P ν ¯ P ν − ¯ K α n δ ¯ P β ¯ P γ ˆ P + ¯ K µ n δ ¯ δ βγ ¯ P α ¯ P µ ˆ P + ¯ K µ n δ ¯ δ αγ ¯ P β ¯ P µ ˆ P + ¯ K α ¯ K µ ¯ K µ n β n γ ¯ P δ + ¯ K β n α n δ ¯ P γ ¯ P µ ¯ P µ − ¯ K β n δ ¯ P α ¯ P γ ˆ P , (E.25) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ¯ K δ ˆ Kn β ¯ P α ¯ P γ − ¯ K µ ¯ K µ n α n δ ¯ P β ¯ P γ − ¯ K γ ˆ Kn β ¯ P α ¯ P δ + ¯ K µ ¯ K µ n β n γ ¯ P α ¯ P δ − ¯ K α ˆ Kn β ¯ P γ ¯ P δ + ¯ K δ ¯ K µ n β n γ ¯ P α ¯ P µ + ¯ K γ ¯ K µ n α n δ ¯ P β ¯ P µ + ¯ K β ¯ K µ n α n δ ¯ P γ ¯ P µ + ¯ K µ ˆ Kn β ¯ δ αδ ¯ P γ ¯ P µ + ¯ K α ¯ K µ n β n γ ¯ P δ ¯ P µ + ¯ K µ ˆ Kn β ¯ δ αγ ¯ P δ ¯ P µ − ¯ K α ¯ K δ n β n γ ¯ P µ ¯ P µ + ¯ K β ¯ K γ n α n δ ¯ P µ ¯ P µ + ¯ K δ ˆ Kn β ¯ δ αγ ¯ P µ ¯ P µ + ¯ K γ ˆ Kn β ¯ δ αδ ¯ P µ ¯ P µ − ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P µ ¯ P ν − ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P µ ¯ P ν − ¯ K µ ¯ K µ n β n γ ¯ δ αδ ¯ P ν ¯ P ν − ¯ K µ ¯ K µ n α n δ ¯ δ βγ ¯ P ν ¯ P ν − ¯ K β ¯ K γ n δ ¯ P α ˆ P + ¯ K µ ¯ K µ n δ ¯ δ βγ ¯ P α ˆ P − ¯ K α ¯ K γ n δ ¯ P β ˆ P + ¯ K µ ¯ K µ n δ ¯ δ αγ ¯ P β ˆ P − ¯ K α ¯ K β n δ ¯ P γ ˆ P + ¯ K β ¯ K µ n δ ¯ δ αγ ¯ P µ ˆ P + ¯ K α ¯ K µ n δ ¯ δ βγ ¯ P µ ˆ P , (E.26) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ¯ K β ¯ K µ ¯ K µ n α n δ ¯ P γ − ˆ Kn β ¯ P α ¯ P γ ¯ P δ + ˆ Kn β ¯ δ αδ ¯ P γ ¯ P µ ¯ P µ − ¯ K α n β n γ ¯ P δ ¯ P µ ¯ P µ + ˆ Kn β ¯ δ αγ ¯ P δ ¯ P µ ¯ P µ − ¯ K µ ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P ν − ¯ K µ n β n γ ¯ δ αδ ¯ P µ ¯ P ν ¯ P ν − ¯ K α ¯ K β ¯ K γ n δ ˆ P + ¯ K β ¯ K µ ¯ K µ n δ ¯ δ αγ ˆ P + ¯ K α ¯ K µ ¯ K µ n δ ¯ δ βγ ˆ P + ¯ K β ¯ K γ ¯ K µ n α n δ ¯ P µ + ¯ K µ n β n γ ¯ P α ¯ P δ ¯ P µ , (E.27) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ˆ K n β n δ ¯ P α ¯ P γ + ¯ K µ ˆ Kn β n γ n δ ¯ P α ¯ P µ + ˆ K n β n δ ¯ δ αγ ¯ P µ ¯ P µ − ¯ K µ ¯ K µ n α n β n δ ¯ P γ ˆ P + ¯ K γ ¯ K µ n α n β n δ ¯ P µ ˆ P − ¯ K α ¯ K γ n β n δ ˆ P + ¯ K µ ¯ K µ n β n δ ¯ δ αγ ˆ P − ¯ K α ˆ Kn β n γ n δ ¯ P µ ¯ P µ , (E.28) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ¯ K α ˆ K n β n δ ¯ P γ + ¯ K µ ˆ K n β n δ ¯ δ αγ ¯ P µ + ¯ K µ n β n δ ¯ δ αγ ¯ P µ ˆ P − ¯ K α n β n δ ¯ P γ ˆ P , (E.29) (cid:2) δ φ (cid:3) αβγδ ( P, K ) = − ¯ K α ˆ Kn β n δ ¯ P γ ˆ P + ¯ K µ ˆ Kn β n δ ¯ δ αγ ¯ P µ ˆ P , (E.30)As found in [7], this sector contains 12 independent trivial terms (coboundaries). The10 of them which are at most second order in the background fields have the followingdensities: F = − − φ − φ + . . . , F = − φ − φ − φ − φ + 8 φ − φ − φ + . . . , F = 2 φ − φ + 2 φ + 4 φ + 8 φ + 2 φ − φ + 8 φ + 8 φ + . . . , The expressions in equations (E.31)–(E.32) are taken from equations (D.12)–(D.13) in [7]. The ellipsis( . . . ) in these expressions stands for terms which are more than second order in the background fields, i.e.terms whose second order variations with respect to the vielbeins vanish in flat space. These terms willtherefore not contribute to the two and three point anomalous Ward identities. – 61 – = − φ + 4 φ − φ + 8 φ − φ − φ + . . . , F = 2 φ + 2 φ − φ + 8 φ + 10 φ + 4 φ − φ + 12 φ − φ − φ + . . . , (E.31) F = − φ − φ − φ − φ − φ + . . . , F = 2 φ + 2 φ + 2 φ + . . . , F = 2 φ + 4 φ + 2 φ + . . . , F = − φ − φ + . . . , F = 2 φ + 2 φ + 2 φ − φ − φ + . . . . This sector also contains 4 independent anomaly terms. The 3 of them which are at mostsecond order in the background fields have the following densities: A , , = φ + . . . = K A L n K A + . . . , A , , = φ + . . . = ˜ K αβS ¯ ∇ α ¯ ∇ β K A + . . . , A , , = φ + 2 φ + φ = (cid:16) ˆ R + ¯ ∇ α a α (cid:17) . (E.32)The second variations of these anomaly and trivial term densities can now be obtainedfrom linear combinations of the expressions (E.16)–(E.30).Finally, we give here the complete result for the anomalous contribution to the threepoint Ward identity (2.50) in the four derivative sector: − δ αα (cid:48) δ γγ (cid:48) δ fδ δ eβ (cid:104) W (3) W (cid:105) α (cid:48) γ (cid:48) ef ( Q, P, K ) (cid:12)(cid:12)(cid:12)(cid:12) n D =4 = − κ / α ¯ K α ¯ K β ¯ K γ ˆ Kn δ π − κ / α ¯ K β ¯ K γ ¯ K µ ¯ K µ n α n δ π + κ / α ¯ K α ¯ K γ ˆ K n β n δ π + κ / α ¯ K γ ¯ K µ ¯ K µ ˆ Kn α n β n δ π + κ / α ¯ K α ¯ K β ¯ K µ ¯ K µ n γ n δ π − κ / α ¯ K α ¯ K µ ¯ K µ ˆ Kn β n γ n δ π − κ / α ¯ K µ ¯ K µ ¯ K ν ¯ K ν n α n β n γ n δ π − κ / α ¯ K β ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αγ π + κ / α ¯ K µ ¯ K µ ˆ K n β n δ ¯ δ αγ π − κ / α ¯ K α ¯ K µ ¯ K µ ˆ Kn δ ¯ δ βγ π + κ / α ¯ K µ ¯ K µ ¯ K ν ¯ K ν n α n δ ¯ δ βγ π − κ / α ¯ K β ¯ K γ ˆ Kn δ ¯ P α π + κ / α ¯ K γ ˆ K n β n δ ¯ P α π + κ / α ¯ K β ¯ K µ ¯ K µ n γ n δ ¯ P α π − κ / α ¯ K µ ¯ K µ ˆ Kn β n γ n δ ¯ P α π − κ / α ¯ K µ ¯ K µ ˆ Kn δ ¯ δ βγ ¯ P α π − κ / α ¯ K α ¯ K γ ˆ Kn δ ¯ P β π + κ / α ¯ K α ¯ K µ ¯ K µ n γ n δ ¯ P β π − κ / α ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αγ ¯ P β π + κ / α ¯ K γ ˆ Kn δ ¯ P α ¯ P β π − κ / α ¯ K µ ¯ K µ n γ n δ ¯ P α ¯ P β π + κ / α ¯ K α ¯ K δ ˆ Kn β ¯ P γ π + κ / α ¯ K δ ¯ K µ ¯ K µ n α n β ¯ P γ π − κ / α ¯ K β ¯ K µ ¯ K µ n α n δ ¯ P γ π + 7 κ / α ¯ K α ˆ K n β n δ ¯ P γ π – 62 – κ / α ¯ K µ ¯ K µ ˆ Kn α n β n δ ¯ P γ π + κ / α ¯ K µ ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P γ π + κ / α ¯ K δ ˆ Kn β ¯ P α ¯ P γ π + κ / α ¯ K β ˆ Kn δ ¯ P α ¯ P γ π + κ / α ˆ K n β n δ ¯ P α ¯ P γ π + κ / α ¯ K α ˆ Kn δ ¯ P β ¯ P γ π + κ / α ¯ K µ ¯ K µ n α n δ ¯ P β ¯ P γ π + κ / α ˆ Kn δ ¯ P α ¯ P β ¯ P γ π + κ / α ¯ K α ¯ K γ ˆ Kn β ¯ P δ π + κ / α ¯ K γ ¯ K µ ¯ K µ n α n β ¯ P δ π − κ / α ¯ K α ¯ K µ ¯ K µ n β n γ ¯ P δ π + κ / α ¯ K γ ˆ Kn β ¯ P α ¯ P δ π − κ / α ¯ K µ ¯ K µ n β n γ ¯ P α ¯ P δ π + κ / α ¯ K α ˆ Kn β ¯ P γ ¯ P δ π + κ / α ¯ K µ ¯ K µ n α n β ¯ P γ ¯ P δ π − κ / α ¯ K γ ¯ K δ ¯ K µ n α n β ¯ P µ π − κ / α ¯ K β ¯ K γ ¯ K µ n α n δ ¯ P µ π + κ / α ¯ K γ ¯ K µ ˆ Kn α n β n δ ¯ P µ π + κ / α ¯ K α ¯ K β ¯ K µ n γ n δ ¯ P µ π − κ / α ¯ K α ¯ K µ ˆ Kn β n γ n δ ¯ P µ π + κ / α ¯ K γ ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P µ π − κ / α ¯ K δ ¯ K µ ˆ Kn β ¯ δ αγ ¯ P µ π − κ / α ¯ K β ¯ K µ ˆ Kn δ ¯ δ αγ ¯ P µ π + κ / α ¯ K µ ˆ K n β n δ ¯ δ αγ ¯ P µ π − κ / α ¯ K γ ¯ K µ ˆ Kn β ¯ δ αδ ¯ P µ π − κ / α ¯ K α ¯ K µ ˆ Kn δ ¯ δ βγ ¯ P µ π − κ / α ¯ K δ ¯ K µ n β n γ ¯ P α ¯ P µ π + κ / α ¯ K β ¯ K µ n γ n δ ¯ P α ¯ P µ π − κ / α ¯ K µ ˆ Kn β n γ n δ ¯ P α ¯ P µ π − κ / α ¯ K µ ˆ Kn δ ¯ δ βγ ¯ P α ¯ P µ π + κ / α ¯ K µ ˆ Kn β ¯ δ γδ ¯ P α ¯ P µ π − κ / α ¯ K γ ¯ K µ n α n δ ¯ P β ¯ P µ π + κ / α ¯ K α ¯ K µ n γ n δ ¯ P β ¯ P µ π − κ / α ¯ K µ ˆ Kn δ ¯ δ αγ ¯ P β ¯ P µ π − κ / α ¯ K µ n γ n δ ¯ P α ¯ P β ¯ P µ π + κ / α ¯ K δ ¯ K µ n α n β ¯ P γ ¯ P µ π − κ / α ¯ K β ¯ K µ n α n δ ¯ P γ ¯ P µ π − κ / α ¯ K µ ˆ Kn α n β n δ ¯ P γ ¯ P µ π + κ / α ¯ K µ ˆ Kn δ ¯ δ αβ ¯ P γ ¯ P µ π − κ / α ¯ K µ ˆ Kn β ¯ δ αδ ¯ P γ ¯ P µ π + κ / α ¯ K γ ¯ K µ n α n β ¯ P δ ¯ P µ π − κ / α ¯ K α ¯ K µ n β n γ ¯ P δ ¯ P µ π − κ / α ¯ K µ ˆ Kn β ¯ δ αγ ¯ P δ ¯ P µ π − κ / α ¯ K µ n β n γ ¯ P α ¯ P δ ¯ P µ π + κ / α ¯ K µ n α n β ¯ P γ ¯ P δ ¯ P µ π − κ / α ¯ K γ ¯ K δ n α n β ¯ P µ ¯ P µ π + κ / α ¯ K α ¯ K δ n β n γ ¯ P µ ¯ P µ π − κ / α ¯ K β ¯ K γ n α n δ ¯ P µ ¯ P µ π + κ / α ¯ K γ ˆ Kn α n β n δ ¯ P µ ¯ P µ π + κ / α ¯ K α ¯ K β n γ n δ ¯ P µ ¯ P µ π − κ / α ¯ K α ˆ Kn β n γ n δ ¯ P µ ¯ P µ π − κ / α ¯ K γ ˆ Kn δ ¯ δ αβ ¯ P µ ¯ P µ π − κ / α ¯ K δ ˆ Kn β ¯ δ αγ ¯ P µ ¯ P µ π − κ / α ¯ K β ˆ Kn δ ¯ δ αγ ¯ P µ ¯ P µ π + κ / α ˆ K n β n δ ¯ δ αγ ¯ P µ ¯ P µ π − κ / α ¯ K γ ˆ Kn β ¯ δ αδ ¯ P µ ¯ P µ π – 63 – κ / α ¯ K α ˆ Kn δ ¯ δ βγ ¯ P µ ¯ P µ π − κ / α ¯ K α ˆ Kn β ¯ δ γδ ¯ P µ ¯ P µ π + κ / α ¯ K β n γ n δ ¯ P α ¯ P µ ¯ P µ π − κ / α ˆ Kn δ ¯ δ βγ ¯ P α ¯ P µ ¯ P µ π + κ / α ¯ K α n γ n δ ¯ P β ¯ P µ ¯ P µ π − κ / α ˆ Kn δ ¯ δ αγ ¯ P β ¯ P µ ¯ P µ π + κ / α ¯ K δ n α n β ¯ P γ ¯ P µ ¯ P µ π − κ / α ¯ K β n α n δ ¯ P γ ¯ P µ ¯ P µ π − κ / α ˆ Kn α n β n δ ¯ P γ ¯ P µ ¯ P µ π − κ / α ˆ Kn δ ¯ δ αβ ¯ P γ ¯ P µ ¯ P µ π + κ / α ¯ K γ n α n β ¯ P δ ¯ P µ ¯ P µ π − κ / α ¯ K α n β n γ ¯ P δ ¯ P µ ¯ P µ π − κ / αn β n γ ¯ P α ¯ P δ ¯ P µ ¯ P µ π + κ / αn α n β ¯ P γ ¯ P δ ¯ P µ ¯ P µ π − κ / α ¯ K µ ¯ K µ ¯ K ν n α n β n γ n δ ¯ P ν π − κ / α ¯ K µ ¯ K µ ¯ K ν n γ n δ ¯ δ αβ ¯ P ν π + κ / α ¯ K µ ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P ν π + κ / α ¯ K µ ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P ν π + κ / α ¯ K µ ¯ K µ ¯ K ν n α n β ¯ δ γδ ¯ P ν π − κ / α ¯ K µ ¯ K ν n α n β n γ n δ ¯ P µ ¯ P ν π − κ / α ¯ K µ ¯ K ν n γ n δ ¯ δ αβ ¯ P µ ¯ P ν π + κ / α ¯ K µ ¯ K ν n β n γ ¯ δ αδ ¯ P µ ¯ P ν π + κ / α ¯ K µ ¯ K ν n α n δ ¯ δ βγ ¯ P µ ¯ P ν π − κ / α ¯ K µ ¯ K ν n α n β ¯ δ γδ ¯ P µ ¯ P ν π − κ / α ¯ K µ ¯ K µ n α n β n γ n δ ¯ P ν ¯ P ν π + κ / α ¯ K µ ¯ K µ n γ n δ ¯ δ αβ ¯ P ν ¯ P ν π + κ / α ¯ K µ ¯ K µ n β n γ ¯ δ αδ ¯ P ν ¯ P ν π + κ / α ¯ K µ ¯ K µ n α n δ ¯ δ βγ ¯ P ν ¯ P ν π + κ / α ¯ K µ ¯ K µ n α n β ¯ δ γδ ¯ P ν ¯ P ν π − κ / α ¯ K µ n α n β n γ n δ ¯ P µ ¯ P ν ¯ P ν π (E.33)+ κ / α ¯ K µ n γ n δ ¯ δ αβ ¯ P µ ¯ P ν ¯ P ν π + κ / α ¯ K µ n β n γ ¯ δ αδ ¯ P µ ¯ P ν ¯ P ν π + κ / α ¯ K µ n α n δ ¯ δ βγ ¯ P µ ¯ P ν ¯ P ν π − κ / α ¯ K µ n α n β ¯ δ γδ ¯ P µ ¯ P ν ¯ P ν π − κ / αn α n β n γ n δ ¯ P µ ¯ P µ ¯ P ν ¯ P ν π + κ / αn β n γ ¯ δ αδ ¯ P µ ¯ P µ ¯ P ν ¯ P ν π + κ / α ¯ K α ¯ K γ ¯ K δ n β ˆ P π + κ / α ¯ K α ¯ K γ ˆ Kn β n δ ˆ P π − κ / α ¯ K α ¯ K µ ¯ K µ n β n γ n δ ˆ P π − κ / α ¯ K δ ¯ K µ ¯ K µ n β ¯ δ αγ ˆ P π + 7 κ / α ¯ K µ ¯ K µ ˆ Kn β n δ ¯ δ αγ ˆ P π − κ / α ¯ K γ ¯ K µ ¯ K µ n β ¯ δ αδ ˆ P π − κ / α ¯ K α ¯ K µ ¯ K µ n β ¯ δ γδ ˆ P π + κ / α ¯ K γ ¯ K δ n β ¯ P α ˆ P π + κ / α ¯ K β ¯ K γ n δ ¯ P α ˆ P π + κ / α ¯ K γ ˆ Kn β n δ ¯ P α ˆ P π + κ / α ¯ K µ ¯ K µ n β n γ n δ ¯ P α ˆ P π − κ / α ¯ K µ ¯ K µ n δ ¯ δ βγ ¯ P α ˆ P π – 64 – κ / α ¯ K µ ¯ K µ n β ¯ δ γδ ¯ P α ˆ P π + κ / α ¯ K α ¯ K γ n δ ¯ P β ˆ P π − κ / α ¯ K µ ¯ K µ n δ ¯ δ αγ ¯ P β ˆ P π + κ / α ¯ K α ¯ K δ n β ¯ P γ ˆ P π + κ / α ¯ K α ¯ K β n δ ¯ P γ ˆ P π + κ / α ¯ K α ˆ Kn β n δ ¯ P γ ˆ P π − κ / α ¯ K µ ¯ K µ n α n β n δ ¯ P γ ˆ P π − κ / α ¯ K µ ¯ K µ n δ ¯ δ αβ ¯ P γ ˆ P π − κ / α ¯ K µ ¯ K µ n β ¯ δ αδ ¯ P γ ˆ P π − κ / α ¯ K δ n β ¯ P α ¯ P γ ˆ P π + κ / α ¯ K β n δ ¯ P α ¯ P γ ˆ P π + κ / α ˆ Kn β n δ ¯ P α ¯ P γ ˆ P π + κ / α ¯ K α n δ ¯ P β ¯ P γ ˆ P π + κ / α ¯ K α ¯ K γ n β ¯ P δ ˆ P π − κ / α ¯ K µ ¯ K µ n β ¯ δ αγ ¯ P δ ˆ P π − κ / α ¯ K γ n β ¯ P α ¯ P δ ˆ P π − κ / αn β ¯ P α ¯ P γ ¯ P δ ˆ P π − κ / α ¯ K γ ¯ K µ n α n β n δ ¯ P µ ˆ P π − κ / α ¯ K α ¯ K µ n β n γ n δ ¯ P µ ˆ P π + κ / α ¯ K γ ¯ K µ n δ ¯ δ αβ ¯ P µ ˆ P π − κ / α ¯ K δ ¯ K µ n β ¯ δ αγ ¯ P µ ˆ P π − κ / α ¯ K β ¯ K µ n δ ¯ δ αγ ¯ P µ ˆ P π + κ / α ¯ K µ ˆ Kn β n δ ¯ δ αγ ¯ P µ ˆ P π − κ / α ¯ K γ ¯ K µ n β ¯ δ αδ ¯ P µ ˆ P π − κ / α ¯ K α ¯ K µ n δ ¯ δ βγ ¯ P µ ˆ P π + κ / α ¯ K α ¯ K µ n β ¯ δ γδ ¯ P µ ˆ P π + κ / α ¯ K µ n β n γ n δ ¯ P α ¯ P µ ˆ P π − κ / α ¯ K µ n δ ¯ δ βγ ¯ P α ¯ P µ ˆ P π + κ / α ¯ K µ n β ¯ δ γδ ¯ P α ¯ P µ ˆ P π − κ / α ¯ K µ n δ ¯ δ αγ ¯ P β ¯ P µ ˆ P π − κ / α ¯ K µ n α n β n δ ¯ P γ ¯ P µ ˆ P π − κ / α ¯ K µ n β ¯ δ αδ ¯ P γ ¯ P µ ˆ P π − κ / α ¯ K µ n β ¯ δ αγ ¯ P δ ¯ P µ ˆ P π − κ / α ¯ K γ n α n β n δ ¯ P µ ¯ P µ ˆ P π − κ / α ¯ K α n β n γ n δ ¯ P µ ¯ P µ ˆ P π − κ / α ¯ K δ n β ¯ δ αγ ¯ P µ ¯ P µ ˆ P π + 7 κ / α ˆ Kn β n δ ¯ δ αγ ¯ P µ ¯ P µ ˆ P π − κ / α ¯ K γ n β ¯ δ αδ ¯ P µ ¯ P µ ˆ P π + κ / α ¯ K α n β ¯ δ γδ ¯ P µ ¯ P µ ˆ P π + κ / αn β n γ n δ ¯ P α ¯ P µ ¯ P µ ˆ P π − κ / αn α n β n δ ¯ P γ ¯ P µ ¯ P µ ˆ P π − κ / αn β ¯ δ αδ ¯ P γ ¯ P µ ¯ P µ ˆ P π − κ / αn β ¯ δ αγ ¯ P δ ¯ P µ ¯ P µ ˆ P π + κ / α ¯ K α ¯ K γ n β n δ ˆ P π + κ / α ¯ K µ ¯ K µ n β n δ ¯ δ αγ ˆ P π + κ / α ¯ K γ n β n δ ¯ P α ˆ P π + 7 κ / α ¯ K α n β n δ ¯ P γ ˆ P π + κ / αn β n δ ¯ P α ¯ P γ ˆ P π + κ / α ¯ K µ n β n δ ¯ δ αγ ¯ P µ ˆ P π + κ / αn β n δ ¯ δ αγ ¯ P µ ¯ P µ ˆ P π . – 65 – eferences [1] S. Sachdev, “Quantum Phase Transitions”, Cambridge University Press (2011).[2] D. T. Son, “Toward an AdS/cold atoms correspondence: A Geometric realization of theSchrodinger symmetry,” Phys. Rev. D , 046003 (2008) doi:10.1103/PhysRevD.78.046003[arXiv:0804.3972 [hep-th]].[3] S. Kachru, X. Liu and M. Mulligan, “Gravity duals of Lifshitz-like fixed points,” Phys. Rev.D , 106005 (2008) doi:10.1103/PhysRevD.78.106005 [arXiv:0808.1725 [hep-th]].[4] M. J. Duff, “Twenty years of the Weyl anomaly”, Class. Quant. Grav. , 1387 (1994)doi:10.1088/0264-9381/11/6/004 [hep-th/9308075].[5] S. Deser, “Conformal anomalies: Recent progress”, Helv. Phys. Acta , 570 (1996)[hep-th/9609138].[6] I. Arav, S. Chapman and Y. Oz, “Lifshitz Scale Anomalies”, JHEP , 078 (2015)doi:10.1007/JHEP02(2015)078 [arXiv:1410.5831 [hep-th]].[7] I. Arav, S. Chapman and Y. Oz, “Non-Relativistic Scale Anomalies,” JHEP , 158(2016) doi:10.1007/JHEP06(2016)158 [arXiv:1601.06795 [hep-th]].[8] S. Pal and B. Grinstein, “Weyl Consistency Conditions in Non-Relativistic Quantum FieldTheory,” JHEP (2016) 012 doi:10.1007/JHEP12(2016)012 [arXiv:1605.02748 [hep-th]].[9] R. Auzzi, S. Baiguera and G. Nardelli, “On Newton-Cartan trace anomalies,” JHEP ,003 (2016) Erratum: [JHEP , 177 (2016)] doi:10.1007/JHEP02(2016)003,10.1007/JHEP02(2016)177 [arXiv:1511.08150 [hep-th]].[10] T. Griffin, P. Hoˇrava and C. M. Melby-Thompson, “Conformal Lifshitz Gravity fromHolography”, JHEP , 010 (2012) doi:10.1007/JHEP05(2012)010 [arXiv:1112.5660[hep-th]].[11] M. Baggio, J. de Boer and K. Holsheimer, “Anomalous Breaking of Anisotropic ScalingSymmetry in the Quantum Lifshitz Model,” JHEP , 099 (2012)doi:10.1007/JHEP07(2012)099 [arXiv:1112.6416 [hep-th]].[12] I. Adam, I. V. Melnikov and S. Theisen, “A Non-Relativistic Weyl Anomaly”, JHEP (2009) 130 doi:10.1088/1126-6708/2009/09/130 [arXiv:0907.2156 [hep-th]].[13] G. Leibbrandt and J. Williams, “Split dimensional regularization for the Coulomb gauge”,Nucl. Phys. B , 469 (1996) doi:10.1016/0550-3213(96)00299-4 [hep-th/9601046].[14] G. Leibbrandt, “The three point function in split dimensional regularization in theCoulomb gauge”, Nucl. Phys. B , 383 (1998) doi:10.1016/S0550-3213(98)00211-9[hep-th/9804109].[15] D. Anselmi and M. Halat, “Renormalization of Lorentz violating theories”, Phys. Rev. D , 125011 (2007) doi:10.1103/PhysRevD.76.125011 [arXiv:0707.2480 [hep-th]].[16] S. Deser and A. Schwimmer, “Geometric classification of conformal anomalies in arbitrarydimensions”, Phys. Lett. B , 279 (1993) doi:10.1016/0370-2693(93)90934-A[hep-th/9302047].[17] L. Bonora, P. Pasti and M. Bregola, “Weyl Cocycles”, Class. Quant. Grav. , 635 (1986).doi:10.1088/0264-9381/3/4/018 – 66 –
18] L. Bonora, P. Cotta-Ramusino and C. Reina, “Conformal Anomaly and Cohomology”,Phys. Lett. , 305 (1983). doi:10.1016/0370-2693(83)90169-7[19] N. Boulanger, “General solutions of the Wess-Zumino consistency condition for the Weylanomalies”, JHEP , 069 (2007) doi:10.1088/1126-6708/2007/07/069 [arXiv:0704.2472[hep-th]].[20] J. Polchinski, ”String Theory” Vol. 1, Cambridge Monographs on Mathematical Physics.[21] D. M. Capper and M. J. Duff, “Trace anomalies in dimensional regularization,” Nuovo Cim.A (1974) 173. doi:10.1007/BF02748300[22] M. J. Duff, “Observations on Conformal Anomalies”, Nucl. Phys. B , 334 (1977).doi:10.1016/0550-3213(77)90410-2[23] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,”doi:10.1017/CBO9780511622632[24] C. Coriano, L. Delle Rose, E. Mottola and M. Serino, “Graviton Vertices and the Mappingof Anomalous Correlators to Momentum Space for a General Conformal Field Theory”,JHEP , 147 (2012) doi:10.1007/JHEP08(2012)147 [arXiv:1203.1339 [hep-th]].[25] H. Osborn and A. C. Petkou, “Implications of conformal invariance in field theories forgeneral dimensions”, Annals Phys. , 311 (1994) doi:10.1006/aphy.1994.1045[hep-th/9307010].[26] J. Erdmenger and H. Osborn, “Conserved currents and the energy momentum tensor inconformally invariant theories for general dimensions,” Nucl. Phys. B (1997) 431doi:10.1016/S0550-3213(96)00545-7 [hep-th/9605009].[27] T. Griffin, P. Hoˇrava and C. M. Melby-Thompson, “Lifshitz Gravity for LifshitzHolography”, Phys. Rev. Lett. , no. 8, 081602 (2013)doi:10.1103/PhysRevLett.110.081602 [arXiv:1211.4872 [hep-th]].[28] S. Weinberg, “High-energy behavior in quantum field theory,” Phys. Rev. (1960) 838.doi:10.1103/PhysRev.118.838[29] C. Itzykson and J. B. Zuber, “Quantum Field Theory,” New York, Usa: Mcgraw-hill (1980)705 P.(International Series In Pure and Applied Physics)[30] J. C. Collins, “Renormalization : An Introduction to Renormalization, TheRenormalization Group, and the Operator Product Expansion,” Cambridge UniversityPress, Cambridge, 1984[31] G. ’t Hooft and M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields”,Nucl. Phys. B , 189 (1972). doi:10.1016/0550-3213(72)90279-9[32] S. Narison, “Techniques of Dimensional Renormalization and Applications to the TwoPoint Functions of QCD and QED”, Phys. Rept. , 263 (1982).doi:10.1016/0370-1573(82)90023-0[33] K. Jensen, “Anomalies for Galilean fields,” [arXiv:1412.7750 [hep-th]].[34] C. Hoyos, B. S. Kim and Y. Oz, “Lifshitz Hydrodynamics,” JHEP (2013) 145doi:10.1007/JHEP11(2013)145 [arXiv:1304.7481 [hep-th]].[35] C. Hoyos, B. S. Kim and Y. Oz, “Lifshitz Field Theories at Non-Zero Temperature,Hydrodynamics and Gravity”, JHEP , 029 (2014) doi:10.1007/JHEP03(2014)029[arXiv:1309.6794 [hep-th]]. – 67 – , 1719 (2014) [cs.SC/1308.3493].[38] D. M. Capper, G. Leibbrandt and M. Ramon Medrano, “Calculation of the gravitonselfenergy using dimensional regularization”, Phys. Rev. D , 4320 (1973).doi:10.1103/PhysRevD.8.4320, 4320 (1973).doi:10.1103/PhysRevD.8.4320