Momentum space parity-odd CFT 3-point functions
Sachin Jain, Renjan Rajan John, Abhishek Mehta, Amin A. Nizami, Adithya Suresh
PPrepared for submission to JHEP
Momentum space parity-odd CFT 3-point functions
Sachin Jain, a Renjan Rajan John, a Abhishek Mehta, a Amin A. Nizami, c AdithyaSuresh a a Indian Institute of Science Education and Research, Homi Bhabha Road, Pashan, Pune 411 008,India c Department of Physics, Ashoka University, India
E-mail: [email protected] , [email protected] , [email protected] , {abhishek.mehta,s.adithya}@students.iiserpune.ac.in Abstract:
We study the parity-odd sector of 3-point functions comprising of scalar op-erators and conserved currents in conformal field theories in momentum space. We usemomentum space conformal Ward identities as well as spin-raising and weight-shifting op-erators to fix the form of these correlators. We discuss in detail the regularisation ofdivergences and their renormalisation using specific counter-terms. a r X i v : . [ h e p - t h ] J a n ontents (cid:104) J µ OO (cid:105) odd (cid:104) J µ J ν O (cid:105) odd (cid:104) J J O (cid:105) odd (cid:104) J J O (cid:105) odd correlators 185.2.1 (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd (cid:104) J J J (cid:105) odd in three-dimensions 207 (cid:104)
T T O (cid:105) odd in three-dimensions 22 (cid:104)
T T O (cid:105) for free fermionic theory 227.2 (cid:104) T T O (cid:105) odd using parity-odd spin-raising and weight-shifting operators 247.3 (cid:104) T T O (cid:105) odd using parity-odd spin-raising and weight-shifting operators 25 (cid:104) J J J (cid:105) odd – i – Embedding space formalism 29B Parity-odd two-point functions 30
B.1 Four and Higher dimensions 30B.2 Three-dimensions 30
C Schouten Identities 31D Computation details for (cid:104)
T T O (cid:105) D.1 Transverse and trace Ward identities 33D.2 Details of longitudinal part 34
E Parity-even spin-raising and weight-shifting operators 34F Embedding space parity-odd correlation functions in four-dimensions 35
F.1 (cid:104)
J J T (cid:105) odd (cid:104)
T T T (cid:105) odd Conformal Field Theories (CFTs) have wide applicability in diverse areas of physics, andare central to our understanding of quantum field theory in terms of RG flows. WhileCFTs are well studied in position space and Mellin space, they are relatively less studiedin momentum space. Recent works on aspects of momentum space CFTs include [1–37].CFTs in momentum space find applications in cosmology [38–47], condensed matter physics[48, 49], study of anomalies [50–54], Hamiltonian truncation methods for strongly coupledfield theories [55, 56] and, of course, the conformal bootstrap program [17, 21, 26, 57].From the perspective of perturbative field theory which is naturally formulated inmomentum space, it is of interest to study CFTs in the same setting. Flat space scatteringamplitudes are, via AdS/CFT, directly related to the flat space limit of CFT correlatorsin momentum space [58] . Studying momentum space CFT correlators can therefore shedlight on the structure of flat space amplitudes. Interestingly, evidence for the double copystructure - which exists for flat space amplitudes - was seen directly in momentum spaceCFT 3-point correlators in [16, 27]. An important simplification in momentum space isthat 4-point conformal blocks can be constructed from 3-point functions in a relativelystraightforward manner in momentum space [26, 36]. Another significant application is inthe cosmological setting where the CMB bispectrum which is a measure of non-gaussianityis given by the 3-point function in momentum space.Three-point functions of scalar and spinning operators have a simple, well-known formin position space - this is most easily seen by going to the embedding space. However, There are analogous, though somewhat less straightforward relations in Mellin space [59, 60] and posi-tion space [61–63]. – 1 –heir momentum space analogues are quite complicated. For example, the scalar 3-pointcorrelator in momentum space is solved in terms of triple- K integrals which can involvedivergences [2]. A careful treatment would require the regularisation and renormalisationof these divergences [2, 4] . The story of spinning correlators is even more complicated[2, 3, 9, 12].Parity-odd correlation functions in momentum space have received very little attention.In three-dimensions they appear naturally. Consider, for example, the free fermion theoryin three-dimensions. The scalar primary operator with the lowest dimension is given by ¯ ψψ ,and it is odd under parity. A correlator with an odd number of insertions of this operatorwill be parity-odd. Another place where parity-odd structures arise naturally is CFTs witha broken parity. A prime example of such CFTs is Chern-Simons matter theories [64–71].In [66], it was argued that in such theories : (cid:104) J s J s J s (cid:105) = α (cid:104) J s J s J s (cid:105) Free-Boson + β (cid:104) J s J s J s (cid:105) Free-Fermion + γ (cid:104) J s J s J s (cid:105) odd (1.1)where J s is the spin s conserved current, the subscript odd indicates a parity-odd contri-bution to the correlator and α, β, γ are theory dependent constants. In three-dimensions we cannot obtain the parity-odd part of the correlator from the free theory and we need touse CFT techniques.There are other instances where parity-odd contribution might be interesting. Forexample, although parity-odd correlators are as yet unobserved, the CMB bispectrum - ameasure of non-gaussianity - could contain parity-odd contributions. Such a contributioncan arise, for example, from an inflationary action with higher derivative corrections suchas cubic Weyl tensor terms . Such terms could source contributions to the primordialgraviton bispectrum, as first discussed in [38] using the spinor-helicity technique. In [73]it was shown that the parity-violating (odd) contribution in the non-gaussianity of theprimordial gravitational waves CMB vanishes in an exact de-Sitter background, but existsin inflationary quasi de-Sitter where it is proportional to the slow-roll parameter. Otherstudies of parity-violating CMB bispectrum include [74–76]. Our results on the parity-odd structures for various 3-point functions constrain the momentum space form of anyparity-violating bispectrum that exists.In this paper, we use two complementary approaches to determine the momentum spacestructure of parity-odd CFT 3-point functions. The first method is the more direct one andinvolves solving momentum-space Ward identities. This approach was developed and usedfor the parity-even sector in [2].The second approach utilises spin-raising and weight-shifting operators. These havebeen used in the conformal bootstrap literature [77–79] and more recently in fixing the In position space one could get rid of divergences by working at non-coincident points, but in momentumspace one cannot do this and this leads to UV divergences. In four dimensions, the 3-point function of stress-tensor is given by [72] (cid:104)
T T T (cid:105) = a (cid:104) T T T (cid:105) FB + b (cid:104) T T T (cid:105) FF + c (cid:104) T T T (cid:105)
Maxwell . These are of the form (cid:82) (cid:102)
W W where W denotes the Weyl tensor and (cid:102) W its Hodge-dual. – 2 –orm of cosmological correlators [46, 47]. We will construct parity-odd spin-raising andweight-shifting operators in momentum space and use them on scalar seed correlators togenerate parity-odd spinning correlators.The rest of this paper is organised as follows. In Section 2, besides setting up thenotation and terminology, we outline the two different techniques that we use in this paperto determine parity-odd 3-point functions. We also briefly discuss the divergences that arise,and their regularisation. In Section 3, we give an overview of the possible parity-odd 3-pointstructures in embedding space in various dimensions. In Section 4, we use momentum spaceconformal Ward identities to fix the parity-odd part of (cid:104) J J O (cid:105) . In Section 5, we constructparity-odd spin-raising and weight-shifting operators in momentum space and use them todetermine spinning correlators by their action on simple scalar seed correlators. We showthat the results for (cid:104)
J J O (cid:105) obtained using spin-raising and weight-shifting operators matchthe results for the same obtained in Section 4. In Section 6, we compute the parity-odd3-point function of the spin-one conserved current using weight-shifting and spin-raisingoperators. In Section 7, we compute the form of the odd part of (cid:104)
T T O ∆ (cid:105) for chosenconformal dimensions of O . In the free fermion theory, where ∆ = 2 , we match the resultsobtained using spin-raising and weight-shifting operators with the answer obtained by anexplicit computation in the free theory. In Section 8, we construct parity-odd spin-raisingand weight-shifting operators in four-dimensions and use them to construct the parity-oddpart of the non-trivial correlator (cid:104) J J J (cid:105) in four dimensions. We conclude with a discussionin Section 9. In the appendices we elaborate on various technical details. In Appendix A, wereview essential details on the embedding space formalism. In Appendix B, we discuss thebasics of momentum space two-point functions. In Appendix C, we present the Schoutenidentities relevant to us. In Appendix D, we give some computational details of the resultspresented in Section 7. Appendix E gives the form of various parity-even spin-raising andweight-shifting operators. In Appendix F, we argue on grounds of permutation symmetrythat certain 3-point correlators with spinning operators in four-dimensions vanish.
Determining correlation functions is a significantly harder task in momentum space than inposition space. For parity-odd correlators this gets even more tedious. We will now discusstwo different approaches to determining momentum space correlators. We also discusscertain subtleties and limitations associated with the two approaches.
In the first approach, following [1, 2] and [4, 9, 10, 12, 14] where parity-even 3-point functionswere determined, we start with an ansatz of the form (cid:80) m A m ( k i ) T m for the correlator. Here T m are all possible tensor structures that are allowed by symmetry and A m are form factorswhich are functions of the momenta magnitudes ( k i ). The form factors are constrained bypermutation symmetries (if any) of the correlator and by momentum space Ward identities.The latter lead to partial differential equations which can then be solved to determine theform factors, up to undetermined constants that depend on the specific theory. In Section– 3 –, we use this method to fix the parity-odd part of certain 3-point functions in momentumspace. An excellent mathematica package that we found useful in these computations is[80].Let us now describe the momentum space Ward identities associated with dilatationsymmetry and special conformal transformations. We will denote the n -point Euclidean correlation function of primary operators O , . . . , O n by (cid:104)O ( k ) . . . O n ( k n ) (cid:105) . We suppress the Lorentz indices of the operators for brevity. Thecorrelator with the momentum conserving delta function stripped off is denoted as : (cid:104) O ( k ) . . . O n ( k n ) (cid:105) ≡ (2 π ) d δ (3) ( k + . . . + k n ) (cid:104)(cid:104) O ( k ) . . . O n ( k n ) (cid:105)(cid:105) . (2.1)An n -point correlator with scalar or spinning operator insertions satisfies the followingdilatation Ward identity [2] : − ( n − d + n (cid:88) j =1 ∆ j − n − (cid:88) j =1 k αj ∂∂k αj (cid:104)(cid:104) O ( k ) . . . O n ( k n ) (cid:105)(cid:105) . (2.2)This constrains the correlator to have the following scaling behaviour : (cid:104)(cid:104) O ( λ k ) . . . O n ( λ k n ) (cid:105)(cid:105) = λ − [ ( n − d − (cid:80) ni =1 ∆ i ] (cid:104)(cid:104) O ( k ) . . . O n ( k n ) (cid:105)(cid:105) . (2.3)The special conformal Ward identity on an n -point correlator with both scalar and spinningoperators is [2] : n − (cid:88) j =1 (cid:34) j − d ) ∂∂k κj − k αj ∂∂k αj ∂∂k κj + k κj ∂∂k αj ∂∂k jα (cid:35) (cid:104)(cid:104) O ( k ) . . . O n ( k n ) (cid:105)(cid:105) + 2 n − (cid:88) j =1 n j (cid:88) k =1 (cid:32) δ µ jk κ ∂∂k α jk j − δ κα jk ∂∂k j µjk (cid:33) (cid:104)(cid:104) O µ ...µ r ( k ) . . . O µ j ...α jk ...µ jrj j ( k j ) . . . O µ n ...µ nrn n ( k n ) (cid:105)(cid:105) (2.4)In the second line of the RHS of the above equation, the indices of the generator mix with thespin indices of the correlator. In principle, one can solve this equation and get the desiredcorrelator [2]. However, for parity-odd structures in three-dimensions, the computation getscomplicated and has not yet been done.We will always be working with correlation functions with the momentum conservingdelta function stripped off. From here on we will drop the double angular brackets notationto avoid clutter and use single angular brackets everywhere. Triple- K integrals arise as solutions to primary conformal Ward identities which are secondorder differential equations [2]. Along with the three momenta, they are expressed in terms– 4 –f four other parameters : I α { β β β } ( k , k , k ) ≡ (cid:90) ∞ dx x α (cid:89) j =1 k β j j K β j ( k j x ) (2.5)where K β j is a modified Bessel function of the second kind. While the integral is wellbehaved at its upper limit, it is convergent at x = 0 only if [2, 9] : α + 1 − | β | − | β | − | β | > (2.6)When the integral is divergent one can regulate it using two parameters u and v [2, 9] : I α { β β β } → I α + u(cid:15) { β + v(cid:15),β + v(cid:15),β + v(cid:15) } (2.7)The regularised triple- K integral is convergent except when [2, 9] : α + 1 ± β ± β ± β = − n, n ∈ Z ≥ (2.8)for any choice of signs. When (2.8) is satisfied, the integral is singular in the regulator (cid:15) and we will denote the divergence by the choice of signs ( ± ± ± ) for which (2.8) is satisfied.Divergences of the type ( − − − ) are called ultra-local and they occur when all thethree operators are co-incident in position space. In momentum space, this manifests asthe divergent term being analytic in all three momenta squared. Such divergences must, ingeneral, be removed using counter-terms that are cubic in the sources, and they give riseto conformal anomalies.Divergences of the type ( − − +) and its permutations are called semi-local divergences.In position space, this is a divergence that occurs when two of the operators in the correlatorare at co-incident points. In momentum space, the divergence is said to be semi-local whenthe O (1 /(cid:15) ) term is analytic in any two of the three momenta squared. In general, thesedivergences must be removed by counter-terms that have two sources and an operator. Suchterms lead to non-trivial beta functions.Divergences of the kind (+ + +) and (+ + − ) are non-local and they occur even whenall three operators are at separated points in position space. In momentum space, such adivergence is analytic in at most one of the momenta squared. This is not a physical diver-gence and arises because the triple- K integral representation of the correlator is singular.In this case no counter-term exists and the divergence is removed by imposing the conditionthat the constant multiplying the triple- K integral vanishes as an appropriate power of (cid:15) . As we discussed above, divergences of the kind ( − − +) and ( − − − ) that correspond toultra-local and semi-local divergences are removed using suitable counter-terms. In the caseof parity-even correlators this has been extensively studied in [2, 4, 9, 12].We will now list a few potential counter-terms that could turn out to be useful in ourstudy of parity-odd correlators. For ultra-local divergences, for example we have : (cid:90) d x F ( A ) (cid:3) n φ, (cid:90) d x C µν R µν (cid:3) n φ, (cid:90) d x C µν R ∇ µ ∇ ν (cid:3) n φ (2.9)– 5 –nd for semi-local divergences : (cid:90) d x (cid:15) µνλ F µν J λ (cid:3) n φ, (cid:90) d x A µ J µ (cid:3) n φ, (cid:90) d x F µν J µ D ν φ, (cid:90) d x C µν T µν (cid:3) n φ (2.10)where F ( A ) is the Chern-Simons form in three-dimensions given by, F ( A ) = (cid:15) µνλ (cid:18) A µa ∂ ν A λa + 23 f abc A µa A νb A λc (cid:19) , (2.11) C µν is the Cotton-York tensor given by, C µν = ∇ ρ (cid:18) R σµ − R g σµ (cid:19) (cid:15) ρσν , (2.12)and R µν and R are the Ricci tensor and the Ricci scalar respectively.In the above list of possible counter-terms (2.10) we have included certain parity-eventerms such as (cid:82) d x A µ J µ (cid:3) n φ and (cid:82) d x F µν J µ D ν φ . These counter-terms could give riseto the 2-point function of currents, which has a parity-odd contribution (cid:104) J µ ( p ) J ν ( − p ) (cid:105) ∝ (cid:15) µνρ p ρ . The second method of computing correlation functions in momentum space hinges on thetechnique of weight-shifting and spin-raising operators. In position space, this techniquewas initiated in [78] and extensively developed in [79]. In this approach, starting fromcertain seed correlators, the action of conformally covariant weight-shifting and spin-raisingoperators generates the desired correlator.To describe this method in some detail, let us consider a spinning correlator (cid:104) J s J s J s (cid:105) . The first step is to count the number of independent tensor structures associated with thiscorrelator. For parity-even correlators this number in position space is given by [77] : N +3 d ( l , l , l ) = 2 l l + l + l + 1 − p ( p + 1)2 (2.13)where p = max (0 , l + l − l ) . The second step is to consider a seed correlator of the form (cid:104) O ∆ O ∆ J s (cid:105) and find out N +3 d ( l , l , l ) ways to reach (cid:104) J s J s J s (cid:105) . This involves actingupon the seed correlator with various spin-raising and weight-shifting operators.In momentum space, we are constrained in our choice of seed correlators because corre-lators of the form (cid:104) J ( l ) O ∆ O ∆ (cid:105) , where J ( l ) is a spin- l conserved current, are non-zero onlywhen ∆ = ∆ . A more convenient approach was recently advocated in [46, 47] to com-pute (parity-even) spinning cosmological correlators where instead of starting from the seed (cid:104) O ∆ O ∆ J s (cid:105) , one starts from (cid:104) O ∆ O ∆ O ∆ (cid:105) , and apply spin-raising and weight-shiftingoperators such that the resulting correlator saturates the Ward-Takahashi identity. SeeSection 4.2.2 of [47] for an example. – 6 – .3.1 Subtleties with the weight-shifting and spin-raising operator approach In momentum space, one must consider the types of divergences in the seed and targetcorrelators. It is not always possible to reach a target correlator starting from a seedcorrelator although a naive application of the spin-raising and weight-shifting operatorsmight suggest so. This is most easily understood in the case of scalar correlators. Asa concrete example of such a situation, consider the following two correlators in three-dimensions : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 1 k k (2.14) (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = − log (cid:18) k + k + k µ (cid:19) (2.15)where µ is the renormalisation scale. Although it might seem like we can use the weight-shifting operator W ++12 (defined in (E.4)) to go from the first correlator to the second, thisis clearly not possible as (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) violates scale invariance whereas the seedcorrelator (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) does not, i.e. W ++12 (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) (cid:54) = (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) (2.16)The above example tells us that weight-shifting operators fail to reproduce the correct cor-relators when the divergence type changes from non-local to semi-local or ultra-local. Theconditions for various types of divergences, in terms of scaling dimensions of the operatorinsertions, are given by : ( − − − ) ∆ + ∆ + ∆ = 2 d + 2 k ( − − +) ∆ + ∆ − ∆ = d + 2 k (+ + − ) − ∆ − ∆ + ∆ = 2 k (+ + +) ∆ + ∆ + ∆ = d − k where k , k , k , k ≥ . We can see that the only time the divergence structure changes iswhen k i = 0 . For the non-local cases in three-dimensions, these correspond to the followingfor the seed correlator : ∆ = ∆ + ∆ (+ + − ) (2.17) ∆ = 3 − ∆ − ∆ (+ + +) (2.18)When either of these conditions is satisfied by the seed correlator, the action of W ++12 doesnot reproduce the correct result. However, W −− works as it can be checked that it doesnot change the type of divergence. Parity-odd structures for three-point correlators can exist in simple theories such as thefree fermion theory in three-dimensions and in CFTs which do not have a parity symmetry.– 7 –uch structures change sign under inversion and they always involve the antisymmetricepsilon tensor. The existence, or non-existence, of such correlators in various dimensions iseasily seen in the embedding space formalism [77, 78]. In embedding space, d -dimensionalodd-correlators are characterised by ( d + 2) -dimensional epsilon tensor. Although we re-strict our attention to correlators involving spin-zero, spin-one and spin-two currents, theanalysis below can be easily generalised to parity-odd correlators involving traceless sym-metric operators in various dimensions. We refer the reader to Appendix A for some detailson the embedding space formalism.One of the constraints on correlators with a spinning operator is that they are trans-verse, i.e. the epsilon structure should be invariant under Z i → Z i + βX i . This constrainsthe possible epsilon structures one can have in a given dimension. In three-dimensions, transversality implies that the parity-odd invariants that can exist are: (cid:15) ( Z Z X X X ) , (cid:15) ( Z Z X X X ) , (cid:15) ( Z Z X X X ) (3.1)This immediately implies that the following parity-odd correlation function is zero : (cid:104) J s O ∆ O ∆ (cid:105) odd = 0 (3.2)where O ∆ , O ∆ are scalar operators with dimensions as indicated and J s is a spin s oper-ator. However, correlators of the form (cid:104) J s J s O ∆ (cid:105) odd and (cid:104) J s J s J s (cid:105) odd are non-zero. See[81] for details. In Section 4.1, we will explicitly show that, in momentum space : (cid:104) J µ OO (cid:105) odd = 0 . (3.3)We will also calculate other parity-odd three-point functions in subsequent sections. In four-dimensions, transversality allows only the following parity-odd invariant : (cid:15) ( Z Z Z X X X ) (3.4)This implies that in four-dimensions the following correlators are zero : (cid:104) J s OO (cid:105) odd = 0 , (cid:104) J s J s O (cid:105) odd = 0 (3.5)Correlators are further constrained by symmetry requirements. For example, (cid:104) J J T (cid:105) odd = 0 , (cid:104) T T T (cid:105) odd = 0 . (3.6)See Appendix F for details. While these correlators are zero, (cid:104) J aµ J bν J cρ (cid:105) can be non-zero.– 8 – .3 Five-dimensions and above In five-dimensions, the only parity-odd invariant allowed by transversality is : (cid:15) ( Z Z Z X X X X ) (3.7)The structure of the contracted epsilon tensor makes it clear that we cannot have any oddthree-point function in five-dimensions . At this point it is also clear that we cannot havean odd three-point function in d ≥ , or an odd four-point function in d ≥ . In this section, we will use the direct approach of solving the PDEs for the form factorswhich result from the momentum space conformal Ward identities. We will illustrate thisusing the correlators (cid:104)
J OO (cid:105) and (cid:104)
J J O (cid:105) as examples. This is an extension of the analysisin [2] to the parity-odd sector. We also discuss the divergences that can arise, and thecounter-terms that regulate them. We conclude this section with a small discussion on thedifficulties in using this method to compute correlators involving more general spinningoperators. (cid:104) J µ OO (cid:105) odd Let us consider the parity-odd part of the (cid:104) J µ OO (cid:105) correlator where J µ is a conservedcurrent and O has scaling dimension . A suitable ansatz for the correlator is : (cid:104) O a ( k ) O b ( k ) J µc ( k ) (cid:105) odd = A ( k , k , k ) f abc (cid:15) µk k (4.1)Throughout this paper, we use notations such as (cid:15) µνk and (cid:15) µk k and they stand for theepsilon tensor contracted with the momenta : (cid:15) µνk = (cid:15) µνρ k ρ , (cid:15) µk k = (cid:15) µνρ k ν k ρ (4.2)We have considered J µ to be in the third position as this makes the action of K κ simpler(2.4). Without non-abelian indices the correlator is zero as ( k ↔ k ) exchange symmetrywould require A ( k , k ) = − A ( k , k ) . After acting with K κ , the primary Ward identitiesare given by : ∂ A∂k − ∂ A∂k − k ∂A∂k = 0 ∂ A∂k − ∂ A∂k − k ∂A∂k = 0 (4.3) However, the above odd structure hints at the possibility of odd four-point correlators in five-dimensions.Since there exists no parity-odd three-point function in five-dimensions, the parity-odd four point functionwill be a contact term. – 9 –he above differential equations can be solved in terms of triple- K integrals (2.5) to get : A ( k , k , k ) = c I {
12 12 − } = c k ( k + k + k ) (4.4)The correlator also satisfies an independent secondary Ward identity given by : A + k − k + k k ∂A∂k + 2 k ∂A∂k = 0 (4.5)The right hand side of the above equation is proportional to the 2-point function (cid:104) O ( k ) O ( − k ) (cid:105) .However, the scalar two-point function has no parity-odd contribution and thus the R.H.S.of (4.5) is zero. Substituting (4.4) into (4.5) gives c = 0 = ⇒ A ( k , k , k ) = 0 (4.6)Thus we conclude that : (cid:104) O a ( k ) O b ( k ) J µc ( k ) (cid:105) odd = 0 (4.7)This result can be generalised to scalar operators of arbitrary scaling dimensions. (cid:104) J µ J ν O (cid:105) odd Here we will consider the parity-odd part of the correlator (cid:104) J µ J ν O (cid:105) . We start with thefollowing ansatz for the correlator : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = π µα ( k ) π νβ ( k ) (cid:104) (cid:101) A ( k , k , k ) (cid:15) αk k k β + (cid:101) B ( k , k , k ) (cid:15) βk k k α (cid:105) (4.8)where the orthogonal projector π µν ( p ) is given by : π νµ ( p ) ≡ δ νµ − p ν p µ p . (4.9)The ansatz (4.8) is chosen such that the correlator is transverse with respect to k µ and k ν .Demanding symmetry under the exchange : ( k , µ ) ↔ ( k , ν ) gives the following relationbetween the form factors : (cid:101) A ( k , k , k ) = − (cid:101) B ( k , k , k ) (4.10)Using the definition of projectors (4.9), the ansatz (4.8) expands to the following : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = (cid:101) A ( k , k , k ) (cid:15) µk k (cid:20) ( k ν + k ν )( k + k · k ) k − k ν (cid:21) + (cid:101) B ( k , k , k ) (cid:15) νk k (cid:20) ( k µ + k µ ) − k µ ( k + k · k ) k (cid:21) (4.11)where we have used momentum conservation to choose k and k as the independent mo-menta. – 10 –e now use Schouten identities (C.3) and (C.4) to get rid of the (cid:15) µk k tensor structureand re-express the ansatz in (4.11) as : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = − (cid:15) νk k ( Ak µ − Bk µ − Bk µ ) − (cid:16) (cid:15) µνk + (cid:15) µνk (cid:17) (cid:0) Ak + B ( k · k ) (cid:1) (4.12)where the new form factors A ( k , k , k ) and B ( k , k , k ) are given in terms of (cid:101) A ( k , k , k ) and (cid:101) B ( k , k , k ) as follows : A ( k , k , k ) = (cid:101) A ( k , k , k ) + (cid:101) B ( k , k , k ) + (cid:101) B ( k , k , k ) k · k k B ( k , k , k ) = (cid:101) B ( k , k , k ) − (cid:101) A ( k , k , k ) k · k k (4.13)Note that the exchange symmetry (4.10) continues to hold between A and B : A ( k , k , k ) = − B ( k , k , k ) (4.14)We will now obtain the primary and secondary Ward identities that A ( k , k , k ) and B ( k , k , k ) satisfy, by letting the generator of special conformal transformations K κ (2.4)act on the ansatz (4.12) : K κ (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = (cid:104) − ∂∂k κ − k α ∂∂k α ∂∂k κ + k ,κ ∂∂k α ∂∂k α + 2(∆ − ∂∂k κ − k α ∂∂k α ∂∂k κ + k ,κ ∂∂k α ∂∂k α (cid:105) (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) + 2 (cid:18) δ µκ ∂∂k α − δ κα ∂∂k ,µ (cid:19) (cid:104) J α ( k ) J ν ( k ) O ( k ) (cid:105) (4.15)Note that by choosing k and k as the independent momenta, we got rid of one set ofterms in the generator K κ that mixes with the index structure of the correlator.The primary Ward identities satisfied by A ( k , k , k ) are given by : ∂ A∂k + ∂ A∂k + 2 k k ∂ A∂k ∂k + 2 k k ∂ A∂k ∂k + 2 k ∂A∂k + 8 k ∂A∂k = 0 ∂ A∂k + ∂ A∂k + 2 k k ∂ A∂k ∂k + 2 k k ∂ A∂k ∂k + 8 k ∂A∂k = 0 (4.16)Similarly, the equations for B ( k , k , k ) are given by : ∂ B∂k + ∂ B∂k + 2 k k ∂ B∂k ∂k + 2 k k ∂ B∂k ∂k + 2 k ∂B∂k + 8 k ∂B∂k = 0 ∂ B∂k + ∂ B∂k + 2 k k ∂ B∂k ∂k + 2 k k ∂ B∂k ∂k + 8 k ∂B∂k = 0 (4.17)– 11 –he general solution to both the primary Ward identities can be found in terms of triple- K integrals (2.5). We solve for β , β , β by substituting the triple- K integral into the primaryWard identities, and obtain : A ∝ I α { ∆ − d − , ∆ − d , ∆ − d } B ∝ I α { ∆ − d , ∆ − d − , ∆ − d } (4.18)The unknown α is determined using the dilatation Ward identity. The action of the dilata-tion Ward identity on the ansatz gives the degree of the form factors :deg ( A ) = ∆ + ∆ + ∆ − d − N A deg ( B ) = ∆ + ∆ + ∆ − d − N B (4.19)where N A and N B are the tensorial dimensions of A and B , defined as the number ofmomenta that multiply the form factor in the ansatz. We see from (4.12) and (4.8) that N A = N B = 3 . Similarly, we impose the dilatation Ward identity on the triple- K integraland get : deg ( I α { β j } ) = β + β + β − α − (4.20)This must equal the degree of the form factors A and B (4.19) giving us : α = (cid:88) i =1 ( β i − ∆ i ) + 2 d + 2 (4.21)Thus we obtain : A = c I (cid:80) i =1 ( β i − ∆ i )+2 d +2 { ∆ − d − , ∆ − d , ∆ − d } B = c I (cid:80) i =1 ( β i − ∆ i )+2 d +2 { ∆ − d , ∆ − d − , ∆ − d } (4.22)where c and c are undetermined constants.We now present the explicit expressions for the two form factors for a few values of thescaling dimension of the scalar operator O . When the scalar operator has ∆ = 1 , we have, A ( k , k , k ) = c (cid:114) π k k ( k + k + k ) ,B ( k , k , k ) = c (cid:114) π k k ( k + k + k ) (4.23)For ∆ = 2 : A ( k , k , k ) = c (cid:114) π k ( k + k + k ) ,B ( k , k , k ) = c (cid:114) π k ( k + k + k ) . (4.24)– 12 –hen ∆ = 3 : A ( k , k , k ) = c (cid:114) π k + k + 2 k k ( k + k + k ) B ( k , k , k ) = c (cid:114) π k + k + 2 k k ( k + k + k ) . (4.25)We will now look at the secondary Ward identities to fix the undetermined constants c and c in (4.22).There is one independent secondary Ward identity in this case which leaves just oneindependent, undetermined constant. The identity is given by : k k ∂A∂k + k ∂B∂k = 0 (4.26)Substituting the solutions for the form factors from (4.22) in this equation we get: c = − c (4.27)which is exactly what is expected from symmetry considerations. We saw in equations (4.23), (4.24) and (4.25) that the triple- K integral is convergent for ∆ = 1 , , . For ∆ > , the integral is singular in the regulator and in some cases, we willrequire counter-terms to remove this divergence.The generating functional for the theory is defined as : Z = (cid:90) Dφ exp (cid:18) − (cid:90) d x ( S φ [ A µ , g µν ] + √ gOφ + J µ A µ ) (cid:19) (4.28)where φ and A µ are sources of the scalar operator and the conserved spin-one currentrespectively. For certain classes of divergences, the generating functional is modified bycounter-terms. We classify the values of ∆ into two classes based on the kinds of diver-gences that occur. ∆ = 4 + 2 n : When ∆ = 4 + 2 n where n ∈ { , , , .. } , (2.8) is satisfied for thechoice of signs given by (+ − − ) for n = 0 , i.e., ∆ = 4 . When n > , it is satisfied for thechoice of signs (+ − − ) and ( − + − ) .To remove this singularity, we look at the following parity-odd counter-term from (2.10) S ct = (cid:90) d (cid:15) x µ − (cid:15) (cid:15) µνλ F µν J λ (cid:3) n φ (4.29)where µ is the renormalization scale. After taking suitable functional derivatives, the con-tribution to the correlator from this counter-term is given by (cid:104) J µ ( x ) J ν ( x ) O ( x ) (cid:105) ct = − a ( (cid:15) ) (cid:20) (cid:3) n (cid:16) δ ( x − x ) (cid:15) ρνλ ∂ ρ (cid:104) J λ ( x ) J µ ( x ) (cid:105) (cid:17) − (cid:3) n (cid:16) δ ( x − x ) (cid:15) ρµλ ∂ ρ (cid:104) J λ ( x ) J ν ( x ) (cid:105) (cid:17) (cid:21) (4.30)– 13 – Fourier transform of the above gives : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) ct = − a ( (cid:15) ) (cid:18) k n (cid:15) νk λ π µλ ( k ) k − k n (cid:15) µk λ π νλ ( k ) k (cid:19) = − a ( (cid:15) ) (cid:20) k n k (cid:18) (cid:15) µνk + (cid:15) νk k k µ k (cid:19) − k n k (cid:18) (cid:15) µνk + (cid:15) µk k k ν k (cid:19) (cid:21) µ − (cid:15) (4.31)where we used the following 2-point function : (cid:104) J µ ( k ) J ν ( − k ) (cid:105) = π µν ( k ) k (4.32)Using Schouten identities (C.3) and (C.4), the ansatz for the correlator can be written as (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) = A (cid:16) (cid:15) µνk k + (cid:15) νk k k µ (cid:17) + A (cid:16) (cid:15) µνk k + (cid:15) µk k k ν (cid:17) (4.33)When ∆ = 4 the singular part of the regularised form factors are given by A ( k , k , k ) = 1 k (cid:15) , A ( k , k , k ) = − k (cid:15) (4.34)The contribution of the counter-term (4.31) to the correlator in this case ( ∆ = 4 , orequivalently n = 0 ) is given by : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) ct = − a ( (cid:15) ) (cid:20) k (cid:18) (cid:15) µνk + (cid:15) νk k k µ k (cid:19) − k (cid:18) (cid:15) µνk + (cid:15) µk k k ν k (cid:19) (cid:21) µ − (cid:15) (4.35)Comparing (4.35) and (4.33) along with (4.34) we see that choosing a ( (cid:15) ) = 1 /(cid:15) cancels thesingular part of the correlator. After removing the divergences, the resulting form factor isgiven by : A ( k , k , k ) = c k log (cid:18) k + k + k µ (cid:19) − c k + 3 k ( k + k + k ) k ( k + k + k ) (4.36)The second form factor is obtained by the following exchange : A ( k , k , k ) = − A ( k , k , k ) (4.37)The anomalous dilatation Ward identity takes the form : µ ∂A ∂µ = − c k (4.38) ∆ = 5 + 2 n : When ∆ = 5 + 2 n where n ∈ { , , , .. } , (2.8) is satisfied for thechoice of signs given by ( − − − ) and (+ + − ) . Although we have both an ultra-local anda non-local divergence here, the term at O (1 /(cid:15) ) is non-local in the momenta and therefore The counter-term that we used (4.29) could also contribute to the parity-even part of (cid:104)
JJO (cid:105) since the (cid:104) JJ (cid:105) – 14 –he divergence can be cancelled by multiplying with a constant of O ( (cid:15) ) and then taking thelimit (cid:15) → . In particular, when ∆ = 5 , the divergent term can be calculated to be : A ( k , k , k ) = c ( (cid:15) ) k + k k (cid:15) + O ( (cid:15) ) (4.39)Choosing c to be O ( (cid:15) ) , the resulting form factor is : A ( k , k , k ) = c (1)1 k + k k (4.40)where c (1)1 is O (0) in (cid:15) .It can be easily checked that this form factor satisfies non-anomalous Ward identitiesand that scale invariance is not broken. (cid:104) J J O (cid:105) odd
If we add a colour index to all the three operators in the correlator such that an exchangeof two colour indices gives rise to a minus sign, then the symmetry between the form factorsin (4.8) changes to : (cid:101) A ( k , k , k ) = (cid:101) B ( k , k , k ) (4.41)The solution for (cid:104) J µa ( k ) J νb ( k ) O c ∆ ( k ) (cid:105) , when ∆ = 3 is then given by : (cid:104) J µa ( k ) J νb ( k ) O c ∆ ( k ) (cid:105) odd = f abc (cid:20) (cid:15) νk k (cid:18) k + k + 2 k k ( k + k + k ) k µ + k + k + 2 k k ( k + k + k ) k µ (cid:19) + (cid:15) µνk (cid:18) k + k + 2 k k ( k + k + k ) k + k + k + 2 k k ( k + k + k ) ( k · k ) (cid:19)(cid:21) (4.42)It can be checked that the above solution is symmetric under (1 ↔ exchange upon usingsuitable Schouten identities.In principle one can compute parity-odd correlation functions of higher spin operatorsusing the approach described in this section following [2]. However, it soon gets difficult tofind out the independent tensor structures after the application of the generator of specialconformal transformations, due to non-trivial Schouten identities that relate various tensorstructures. We will now resort to the technique of using weight-shifting and spin-raisingoperators to compute parity-odd correlation functions. In this section we construct parity-odd spin-raising and weight-shifting operators in momen-tum space. We then illustrate how these operators can be used to calculate the parity-oddpart of the (cid:104)
J J O (cid:105) correlator. – 15 – .1 Parity-odd operators
We consider a parity-odd operator which raises the spins of the operators at points 1 and2 and lowers the weight of the operator at point 2. In embedding space, this operator isdefined as (cid:101) D ≡ (cid:15) (cid:18) Z , Z , X , X , ∂∂X (cid:19) (5.1)based on requirements of transversality and interiority[78]. In position space, the operatortakes the form : (cid:101) D = 12 (cid:18) (cid:15) ijk − + z i z j x k D − (cid:15) ijk − + (cid:20) ( x − x )2 z j z k + x j z k ( z · x ) + z j x k ( z · x ) (cid:21) P i (cid:19) (5.2)where (cid:15) ijk − + ≡ (cid:15) ijk . Performing a Fourier transform we get in momentum space : (cid:101) D = − (cid:20) (cid:15) ( z z K − )(∆ − d − k · ∂∂k )+ K − K +12 (cid:15) ( k z z ) + (cid:15) ( k K − z )( z · ∂∂k ) + (cid:15) ( k z K − )( z · ∂∂k ) (cid:21) (5.3)where K +12 and K − are defined in Appendix E.The above operator acts on a momentum space correlator with a momentum conservingdelta function. In its present form, it will be tedious to take the above operator past thedelta function. Consider now the following commutator : [ k µ + k µ + k µ , (cid:101) D ] = −
12 [ (cid:15) ( z z K − ) k µ − K − µ (cid:15) ( k z z ) − (cid:15) ( k K − z ) z µ − (cid:15) ( k z K − ) z µ ] (5.4)The above commutator vanishes on a three-point function due to momentum conservation.Thus we have the following action on 3-point functions : (cid:15) ( z z K − ) k µ − K − µ (cid:15) ( k z z ) − (cid:15) ( k K − z ) z µ − (cid:15) ( k z K − ) z µ = 0 (5.5)We contract the above equation with ∂∂k µ from the right to get the Schouten identity : (cid:15) ( z z K − ) k · ∂∂k + (cid:15) ( z z K − ) − ( K − · ∂∂k ) (cid:15) ( k z z ) − (cid:15) ( k K − z )( z · ∂∂k ) − (cid:15) ( k z K − )( z · ∂∂k ) = 0 (5.6)We use this to rewrite (5.3) as : (cid:101) D = (cid:15) z z k W −− + (cid:15) z k K − ( (cid:126)z · (cid:126)K − ) + (2 − ∆ ) (cid:15) z z K − (5.7)– 16 –here W −− is defined in Appendix E. Note that the operator defined in (5.7) is explicitlytranslation invariant .We will use (cid:101) D in (5.7) to compute (cid:104) JJO ∆ (cid:105) odd , (cid:104) JJJ (cid:105) odd and (cid:104)
TTO ∆ (cid:105) odd 8 startingfrom a scalar-seed. We can also construct (cid:101) D and (cid:101) D to get operators that act on points 2and 3 and points 3 and 1, respectively. We will require them in the computation of (cid:104) JJJ (cid:105) odd as the correlator has cyclic symmetry.We will now construct other parity-odd weight-shifting and spin-raising operators thatare useful. Let us consider the following : (cid:101) D ≡ (cid:15) ( Z , X , ∂∂X , X , ∂∂X ) + (cid:15) ( Z , X , ∂∂X , Z , ∂∂Z )= 12 (cid:8) (cid:15) ijk [ z i x k ( D P j − D P j ) + ( x · z ) x k P i P j − ( x − x )2 z k P i P j ] (cid:9) + 12 (cid:8) (cid:15) ijk [( x · z ) z j ∂∂z k + ( x · z ) z k ∂∂z j + z j z k ( x · ∂∂z )] P i (cid:9) − (cid:15) ijk [ z i z j ∂∂z k ] D . (5.8)The Fourier transform of this operator gives in momentum space the following : (cid:101) D = 12 (cid:8) (cid:15) ijk [ z i K − k (( − ∆ + d + k · ∂∂k ) k j − ( − ∆ + d + k · ∂∂k ) k j )+ ( z · ∂∂k ) K − k k i k j − K − · K +12 z k k i k j ] (cid:9) + 12 (cid:8) (cid:15) ijk [( z · ∂∂k ) z j ∂∂z k + ( z · ∂∂k ) z k ∂∂z j + z j z k ( ∂∂k · ∂∂z )] k i (cid:9) − (cid:15) ijk [ z i z j ∂∂z k ]( − ∆ + d + k · ∂∂k ) (5.9)The above can be written in a translation invariant manner using the methods implementedin the case of (cid:101) D to obtain : (cid:101) D = (2 − ∆ ) (cid:15) ( z k K − ) − (2 − ∆ ) (cid:15) ( z k K − ) + ( z · K − ) (cid:15) ( k k K − ) − (cid:15) ( z k K − )( k · K − ) − (cid:15) ( z k k ) W −− − (cid:15) ijk [( z · K − ) z k ∂∂z j + z j z k ( ∂∂z · K − )] k i + z i z j ∂∂z k ( − ∆ + d )] (5.10)We also introduce the following parity-odd operator : D = ( X i · X j ) (cid:15) ( Z Z X X X )= x ij z · x ) (cid:15) ( z x x ) + x (cid:15) ( z x z ) + x (cid:15) ( z z x )] i, j = 1 , , (5.11) Translational invariance of an operator implies in momentum space that its operation on the momentumconserving delta-function is zero. This happens when the operator is only a function of K − ij in the derivatives,and that is precisely what we have in (5.7). The subscript on operators denotes their bare scaling dimensions. – 17 –hich after a Fourier transform takes the form : D = 2[( z · K − ) (cid:15) ( z K − K − ) + (cid:15) ( z K − z ) W −− + (cid:15) ( z z K − ) W −− ] W −− ij (5.12)This operator is naturally translational invariant.Finally, the polarization vectors can be stripped off from the correlators using theTodorov operator [82] : D µz = (cid:18)
12 + (cid:126)z · ∂∂(cid:126)z (cid:19) ∂∂z µ − z µ ∂ ∂(cid:126)z · ∂(cid:126)z . (5.13)We will now discuss the computation of (cid:104) J J O (cid:105) odd using the technique of spin-raising andweight-shifting operators. (cid:104)
J J O (cid:105) odd correlators
We start with the seed correlator (cid:104) O O O ∆ (cid:105) , where the last operator has an arbitraryscaling dimension ∆ . To get (cid:104) J J O ∆ (cid:105) odd we act on the scalar-seed with the parity-oddoperator (cid:101) D defined in (5.7) : (cid:104) J µ ( k ) J ν ( k ) O ∆ ( k ) (cid:105) odd = D νz D µz (cid:101) D (cid:104) O ( k ) O ( k ) O ∆ ( k ) (cid:105) (5.14) (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd Let us consider the case when ∆ = 2 . We have : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = c log (cid:18) k + k + k µ (cid:19) ( k + k ) − c ( k + k + k ) (5.15)Acting with (cid:101) D and then removing the polarization vectors gives (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = c (cid:15) µk k ( k k ν − k k ν ) k k ( k + k + k ) + c (cid:15) µνk ( k + k − k )2 k ( k + k + k ) (5.16)This matches the answer in (4.12). To see this explicitly, we substitute the solution to theform factors given in (4.24) and (4.27) in the ansatz (4.12) : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = c (cid:15) νk k ( k k µ − k k µ ) k k ( k + k + k ) + c (cid:15) µνk ( k + k − k )2 k ( k + k + k ) (5.17)Using the Schouten identities in (C.3) and (C.4) to replace (cid:15) νk k in the above equation,we match the result obtained using weight-shifting and spin-raising operators in (5.16).Similarly, one can obtain and match the results for (cid:104) J J O (cid:105) odd and (cid:104) J J O (cid:105) odd .We were also able to obtain (cid:104) J J O ∆ (cid:105) for ∆ = 1 , , using the operator in (5.9). However,for ∆ = 3 we obtained the correlator up to a conformally invariant contact term given by (cid:15) ( z z k ) .When the dimension of the scalar operator is greater than or equal to 4, one needs tobe more careful as the correlators are divergent and need to be renormalised (see Section(4.2.1)). – 18 – .2.2 (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd The full seed correlator along with the divergent part is given by : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = c (cid:20) − k − k ( k + k ) + 3 k (7 k + 2 k k + 3 k )+ 4 (4 k + 3 k k + k )+ 6 log (cid:18) k + k + k µ (cid:19) (cid:0) k − k − k (3 k + k ) (cid:1) (cid:21) − c (cid:15) (cid:0) k − k − k (3 k + k ) (cid:1) (5.18)This correlator has two semi-local divergences labelled by (+ − − ) and ( − + − ) . Thegenerating functional is given by : Z = (cid:90) Dφ exp (cid:18) − (cid:90) d x (cid:0) S [ A µ , g µν ] + √ g ( O φ + O φ + O φ − ) + S ct (cid:1)(cid:19) (5.19)where φ , φ and φ − correspond to the sources of O , O and O respectively. Thecounter-term in this case is given by S ct = (cid:90) d (cid:15) x µ − (cid:15) ( a ( (cid:15) ) (cid:3) O φ φ − + a ( (cid:15) ) O (cid:3) φ φ − + a ( (cid:15) ) O φ (cid:3) φ − + a ( (cid:15) ) O φ φ − ) (5.20)The full renormalized correlator is then defined by (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) ren = − (cid:112) g ( x ) (cid:112) g ( x ) (cid:112) g ( x ) δδφ ( x ) δδφ ( x ) δδφ − ( x ) Z = c (cid:34) − k − k ( k + k ) + 3 k (7 k + 2 k k + 3 k )+ 4 (4 k + 3 k k + k )+ 6 log (cid:18) k + k + k µ (cid:19) (cid:0) k − k − k (3 k + k ) (cid:1) (cid:35) (5.21)We now act (cid:101) D from (5.7) on the above renormalised scalar correlator and then strip offthe polarization vectors to obtain : (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = − (cid:15) k k µ k ν A + (cid:15) k k µ k ν A + (cid:15) k µν ( A k · k + A k ) (5.22)where A is as in (4.36) and A ( k , k , k ) = − A ( k , k , k ) . Once again, using theSchouten identities in (C.3) and (C.4), this matches the solution (4.33) and (4.36) obtainedby solving the conformal Ward identities. – 19 – .2.3 (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd The seed correlator here has ultra-local and non-local divergences. The correlator is givenby (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = c (cid:15) (cid:0) k − k k − k + 6 k k + k − k (5 k + k ) (cid:1) + O ( (cid:15) ) (5.23)Unlike the previous case, this divergence is removed by taking the constant c to be O ( (cid:15) ) andthen taking the limit (cid:15) → (similar to (4.39) and (4.40)). Thus we have the renormalisedcorrelator : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) ren = c (cid:0) k − k k − k + 6 k k + k − k (5 k + k ) (cid:1) (5.24)The rest of the calculation is the same as in the case of (cid:104) J J O (cid:105) odd and we get (cid:104) J µ ( k ) J ν ( k ) O ( k ) (cid:105) odd = − (cid:15) k k µ k ν A + (cid:15) k k µ k ν A + (cid:15) k µν ( A k · k + A k ) (5.25)where A is as in (4.40) and A = − A ( k ↔ k ) . It can be easily checked that this matchesthe answer obtained by solving conformal Ward identities (see (4.33) and (4.40)). (cid:104) J J J (cid:105) odd in three-dimensions
We now turn our attention to computing the odd part of the (cid:104)
J J J (cid:105) correlator. The corre-lator is non-zero only when the currents are non-Abelian.We express (cid:104)
J J J (cid:105) odd in terms of transverse and longitudinal parts [2] : (cid:104) J µa J νb J ρc (cid:105) odd = (cid:104) j µa j νb j ρc (cid:105) odd + (cid:20) k µ k (cid:16) f adc (cid:104) J ρd ( k ) J νb ( − k ) (cid:105) − f abd (cid:104) J νd ( k ) J ρc ( − k ) (cid:105) (cid:17) + (( k , µ ) ↔ ( k , ν )) + (( k , µ ) ↔ ( k , ρ )) (cid:21) + (cid:20) (cid:18) k µ k ν k k f abd k α (cid:104) J αd ( k ) J ρc ( − k ) (cid:105) (cid:19) + (( k , µ ) ↔ ( k , ρ )) + (( k , ν ) ↔ ( k , ρ )) (cid:21) (6.1)where (cid:104) j µa j νb j ρc (cid:105) odd denotes the transverse part of the correlator. The ansatz for this partof the correlator can be written as (cid:104) j µa j νb j ρc (cid:105) odd = π µα ( k ) π νβ ( k ) π ργ ( k ) X αβγ (6.2)where X αβγ = A (cid:15) k k α k γ k β + A (cid:15) k k α δ βγ + A (cid:15) k αβ k γ + A (cid:15) k αγ k β + cyclic perm. (6.3)Calculating the form factors by directly solving the conformal Ward identities is quitecomplicated. Here we will instead use spin-raising and weight shifting operators to calculatethem. – 20 –tarting from the seed correlator (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) we can get (cid:104) J J J (cid:105) odd by : (cid:104) J µa ( k ) J νb ( k ) J ρc ( k ) (cid:105) odd = 14 D µz D νz D ρz (cid:101) D D (cid:104) O a ( k ) O b ( k ) O c ( k ) (cid:105) even + cyclic perm.(6.4)where (cid:101) D is defined in (5.7) and D in (E.5). The renormalized seed correlator is : (cid:104) O a ( k ) O b ( k ) O c ( k ) (cid:105) = f abc c (cid:20) log (cid:18) k + k + k µ (cid:19) ( k − k + k ) − k − k + 2 k ( k − k ) + 2 k k + 3 k (cid:21) (6.5)This gives : X αβγ = − k ( k + k + k ) (cid:15) k k α k γ k β − k + k + k ) (cid:15) k k α δ βγ + ( k + k + 2 k ) k ( k + k + k ) (cid:15) k αβ k γ + ( k + 2 k + k ) k ( k + k + k ) (cid:15) k αγ k β + cyclic perm. (6.6)We can now read off the form factors in (6.3) by comparing it with (6.6) : A = − k ( k + k + k ) , A = − k + k + k ) A = k + k + 2 k k ( k + k + k ) , A = k + 2 k + k k ( k + k + k ) (6.7)The (cid:104) JJJ (cid:105) odd correlator obeys the following Ward-Takahashi identity [2] : k µ (cid:104) J µa ( k ) J νb ( k ) J ρc ( k ) (cid:105) odd = f adc (cid:104) J ρd ( k ) J νb ( − k ) (cid:105) odd − f abd (cid:104) J νd ( k ) J ρc ( − k ) (cid:105) odd = − f abc (cid:16) (cid:15) k νρ + (cid:15) k νρ (cid:17) (6.8)Our result does satisfy this identity. To see this, let us contract our result (6.1) for (cid:104) J µa J νb J ρc (cid:105) with k µ : k µ (cid:104) J µa J νb J ρc (cid:105) odd = f abc (cid:15) k k ν ( B k ρ + ( B − B ( k ↔ k )) k ρ )+ f abc (cid:15) k k ρ ( B k ν + B ( k ↔ k ) k ν ) − (cid:16) (cid:15) k νρ B − (cid:15) k νρ B ( k ↔ k ) (cid:17) (6.9)where B = 4(2 k + k + k ) k ( k + k + k ) , B = 2 k + k + k ( k − k ) − k k ( k + k + k ) (6.10)Using the identities in (C.6) and (C.7), we can get rid of those epsilon structures whichhave two momenta contracted with their indices. Doing so we obtain : k µ (cid:104) J µa J νb J ρc (cid:105) = − f abc ( (cid:15) k νρ + (cid:15) k νρ ) (6.11)which matches the desired Ward identity (6.8).– 21 – (cid:104) T T O (cid:105) odd in three-dimensions
We now turn our attention to the (cid:104)
T T O (cid:105) odd correlator. Based on symmetry considerationsand conservation, this correlator is expected to take the following form [2] : (cid:104) T µ ν ( k ) T µ ν ( k ) O I ( k ) (cid:105) odd = (cid:104) t µ ν ( k ) t µ ν ( k ) O I ( k ) (cid:105) odd + 2 (cid:20) T µ ν α ( k ) k β + π µ ν ( k ) d − δ α β (cid:21) δ µ α δ ν β (cid:28) δT α β δg α β ( k , k ) O I ( k ) (cid:29) + 2 [( µ , ν , k ) ↔ ( µ , ν , k )] − (cid:20) T µ ν α ( k ) k β + π µ ν ( k ) d − δ α β (cid:21) (cid:20) T µ ν α ( k ) k β + π µ ν ( k ) d − δ α β (cid:21) × (cid:28) δT α β δg α β ( k , k ) O I ( k ) (cid:29) (7.1)where T µνα ( p ) = η αβ T µνβ = η αβ p (cid:20) p ( µ δ ν ) β − p β d − (cid:18) δ µν + ( d − p µ p ν p (cid:19)(cid:21) (7.2)and (cid:104) t µ ν ( k ) t µ ν ( k ) O I ( k ) (cid:105) odd is the transverse part of the correlator. Due to symmetryand transversality, this is expected to take the following form : (cid:104) t µ ν ( k ) t µ ν ( k ) O I ( k ) (cid:105) odd =Π µ ν α β ( k )Π µ ν α β ( k ) (cid:104) A (cid:15) α k k k β k β k α + A (cid:15) α k k k β k β k α + A (cid:15) α k k δ β β k α + A (cid:15) α k k δ β β k α + A (cid:15) α α k k β k β + A (cid:15) α α k k β k β + A (cid:15) α α k δ β β + A (cid:15) α α k δ β β (cid:105) . (7.3)where the traceless-orthogonal projector is given by : Π µναβ ( p ) ≡ (cid:16) π µα ( p ) π νβ ( p ) + π µβ ( p ) π να ( p ) (cid:17) − π µν ( p ) π αβ ( p ) . (7.4)In the following, we first present a direct calculation of (cid:104) T T O (cid:105) in the free-fermion theory forwhich O = ¯ ψψ which is parity-odd in three-dimensions and has scaling dimension ∆ = 2 .We then reproduce the free theory answer using spin and weight-shifting operators. Wealso compute (cid:104) T T O (cid:105) odd with ∆ O = 1 which does not have a free theory analogue . Wedefer the computation of (cid:104) T T O ∆ (cid:105) for general ∆ to future work. (cid:104) T T O (cid:105) for free fermionic theory The free-fermion action in curved space is given by [2] : S = (cid:90) d x e (cid:104) ¯ ψe µa γ a ↔ ∇ µ ψ (cid:105) (7.5) One can get such a parity-odd correlator by coupling a complex scalar field theory to a Chern-Simonsgauge field. See [64–68] – 22 –here e aµ are the vielbeins and the covariant derivative acts as follows on the spinor ∇ µ ψ = ( ∂ µ − i ω abµ Σ ab ) ψ ¯ ψ ← ∇ µ = ¯ ψ ( ← ∂ µ + i ω abµ Σ ab ) (7.6)One may use this action to compute the stress-energy tensor [2] : T µν = 1 √ g δSδg µν = 12 ¯ ψγ ( µ ↔ ∇ ν ) ψ (7.7)which after taking the flat-space limit and a Fourier transform gives the stress-energy tensorin momentum space : T µν ( k ) = 14 (cid:90) d l ¯ ψ ( l )[ γ µ (2 l − k ) ν + γ ν (2 l − k ) µ ] ψ ( k − l ) (7.8)The parity-odd scalar primary in the free-fermion theory is given by : O = ¯ ψψ (7.9)Using these definitions it is straightforward to evaluate (cid:104) T T O (cid:105) for the free-fermionic theory.Since O is parity-odd (cid:104) T T O (cid:105) is also parity-odd. We give the details of the computationin Appendix D and give only the final results here.The form factors that appear in the transverse part of the correlator (7.3) are : A = k + 4 k + k k + k + k ) A = 2 k + 4 k + 3 k k + k + 3 k (2 k + k )6( k + k + k ) A = k ( k + 3 k + k )4( k + k + k ) A = k + 2 k ( k + k ) + 2 k (2 k + k ) + k ( k + 2 k k + 3 k )8( k + k + k ) A i = − A i ( k ↔ k ) . (7.10)To compute the longitudinal part, we require the functional derivative of the fermionicstress-energy tensor. Using the action of the covariant derivative on spinors (7.6) we re-express the stress-energy tensor (7.7) as : T µν = 12 ¯ ψγ ( µ ↔ ∂ ν ) ψ + 116 ω ab ( µ ¯ ψ { γ ν ) , γ ab } ψ (7.11)After some computation (see Appendix D.2) we obtain : δT µν ( x ) δg αβ ( y ) = − i { [ (cid:15) σαµ δ βν + (cid:15) σβµ δ αν + (cid:15) σαν δ βµ + (cid:15) σβν δ αµ ] ∂ σ δ (3) ( x − y ) } O ( x ) (7.12)– 23 –aking a Fourier transform of the above we get : δT µν δg αβ ( k , k ) = 132 [ (cid:15) k αµ δ βν + (cid:15) k βµ δ αν + (cid:15) k αν δ βµ + (cid:15) k βν δ αµ ] O ( k ) − ( k ↔ k ) (7.13)Using the above, one may compute (cid:28) δT µν δg αβ ( k , k ) O ( − k ) (cid:29) = − k
256 [ (cid:15) k αµ δ βν + (cid:15) k βµ δ αν + (cid:15) k αν δ βµ + (cid:15) k βν δ αµ − ( k ↔ k )] (7.14)where we used (cid:104) O ( k ) O ( − k ) (cid:105) = − k in the free-fermion theory. This can now be usedin the reconstruction formula 7.1 to get the full correlator (cid:104) T T O (cid:105) .Having computed the functional derivative, we now give the trace and transverse Wardidentities associated with (cid:104) T T O (cid:105) . They are [2] : (cid:104) T ( k ) T αβ ( k ) O ( k ) (cid:105) = 2 (cid:28) δTδg αβ ( k , k ) O ( k ) (cid:29) (7.15) k µ (cid:104) T µν ( k ) T αβ ( k ) O ( k ) (cid:105) = 2 k µ (cid:28) δT µν δg αβ ( k , k ) O ( k ) (cid:29) (7.16)The expression obtained in (7.14) is traceless in ( µ, ν ) and ( α, β ) . This immediately impliesthat the trace Ward identity is trivial : (cid:104) T ( k ) T αβ ( k ) O ( k ) (cid:105) = 0 (7.17)Contracting (7.14) with k µ gives the transverse Ward identity : k µ (cid:104) T µν ( k ) T αβ ( k ) O ( k ) (cid:105) = k
128 [ − (cid:15) k k α δ βν − (cid:15) k k β δ αν + k β ( (cid:15) k αν − (cid:15) k αν ) + k α ( (cid:15) k βν − (cid:15) k βν )] (7.18)In Appendix (D.1) we show that precisely this Ward identity holds. (cid:104) T T O (cid:105) odd using parity-odd spin-raising and weight-shifting operators In this section we compute the odd part of (cid:104)
T T O (cid:105) using spin-raising and weight-shiftingoperators. We start from the renormalised scalar-seed correlator (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) given by : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 1960 (cid:104) − k − k (7 k − k ) k − k ( k − k ) k ( k + k ) − k (32 k + 77 k ) − k (16 k + 57 k k − k k − k ) − k + k ) (32 k + 41 k k − k k + 137 k )+ 60 log (cid:18) k + k + k µ (cid:19) (cid:0) k + 15 k k + 3 k ( k − k ) + 2 k (3 k k − k ) (cid:1) (cid:105) (7.19)– 24 –he following sequence of operations gives the (cid:104) T T O (cid:105) correlator : (cid:101) D ( a D D + a D D + a H + a W −− S ++12 ) (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) + ( b D D + b D D + b H + b W −− S ++12 ) (cid:101) D (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) + ( k ↔ k , z ↔ z ) (7.20)where a i and b i are coefficients which are fixed by comparing with the result for the corre-lator from the explicit computation in the free fermion theory : a = − , a = −
145 + a , a = −
245 + 8( a + b ) ,b = 0 , b = b + 190 , b = 2 b + 160 (7.21)It can be checked that the operators in (7.20) are not all independent. One has the followinglinear dependence between them : (cid:101) D ( D D + 8 H + W −− S ++12 ) (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 0( D D + 10 H + W −− S ++12 ) (cid:101) D (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 0 (7.22)Using these relations and setting b = one obtains the odd part of (cid:104) T T O (cid:105) correlatorfrom the following : (cid:104) T T O (cid:105) odd = (cid:104) − (cid:101) D ( D D + D D ) + (2 D D − H ) (cid:101) D (cid:105) (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) (7.23) (cid:104) T T O (cid:105) odd using parity-odd spin-raising and weight-shifting operators In this subsection we discuss the parity-odd part of the correlator (cid:104)
T T O (cid:105) . To constructthis correlator we start from the renormalised scalar seed correlator (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) given by : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 1960 k (cid:104) − k − k (7 k − k ) k − k ( k − k ) k ( k + k ) − k (32 k + 77 k ) − k (16 k + 57 k k − k k − k ) − k + k ) (32 k + 41 k k − k k + 137 k )+ 60 log (cid:18) k + k + k µ (cid:19) (cid:0) k + 15 k k + 3 k ( k − k ) + 2 k (3 k k − k ) (cid:1) (cid:105) (7.24)Note that this is related to the scalar-seed (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) in (7.19) which was usedto compute (cid:104) T T O (cid:105) as : (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) = 1 k (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) (7.25)– 25 –he correlator (cid:104) T T O (cid:105) is given by the same sequence of operations (7.23) acting on (cid:104) O ( k ) O ( k ) O ( k ) (cid:105)(cid:104) T T O (cid:105) odd = (cid:104) − (cid:101) D ( D D + D D ) + (2 D D − H ) (cid:101) D (cid:105) (cid:104) O ( k ) O ( k ) O ( k ) (cid:105) (7.26)Note that the spin-raising and weight-shifting operators that appear in the above equationare independent of the third momentum and hence do not act on the k pre-factor in (7.25).As a result we obtain the following relation : (cid:104) T T O (cid:105) odd = 1 k (cid:104) T T O (cid:105) odd (7.27) In this section we will construct and use parity-odd spin-raising operators in four-dimensionsto determine the non-trivial momentum-space correlator (cid:104)
J J J (cid:105) . In four-dimensions, theonly non-zero parity-odd 3-point function is (cid:104)
J J J (cid:105) with a nonabelian current J. Othercorrelators involving operators with spin s ≤ are zero as emphasized in subsection 3.2. In four-dimensions, transversality and interiority allow for only one operator : D ++12 ≡ (cid:15) ( Z Z X X ∂∂X ∂∂X )= 12 (cid:8) − (cid:15) ijkl z j z k x l ( D P i − D P i )+ (cid:15) ijkl [( z · x ) z k x l − ( z · x ) z k x l − (cid:18) x − x (cid:19) z k z l ] P i P j (cid:9) (8.1)where the embedding space result has been converted to ordinary position space. Fouriertransform of the above operator gives : D ++12 = 12 (cid:8) − (cid:15) ijkl z j z k K − l [( − ∆ + d + k · ∂∂k ) k i − ( − ∆ + d + k · ∂∂k ) k i ]+ (cid:15) ijkl [( z · ∂∂k ) z k K − l − ( z . ∂∂k ) z k K − l − K − K +12 z k z l ] k i k j (cid:9) (8.2)Let us now consider the following commutator : [ k µ + k µ + k µ , D ++12 ] = − (cid:15) ( z z K − k ) k µ + (cid:15) ( z z K − k ) k µ − z µ (cid:15) ( z k k K − ) + z µ (cid:15) ( z k k K − ) + K − µ (cid:15) ( z z k k ) (8.3) This is consistent with the fact that the two scalar operators with ∆ = 1 and ∆ = 2 are related by ashadow transformation in three-dimensions. – 26 –hich is zero by momentum conservation. The above commutator gives an operator-basedSchouten identity. We find the following contracted form of the above equation useful inthe present context : − (cid:15) ( z z K − k ) k · ∂∂k + (cid:15) ( z z K − k ) k · ∂∂k − (cid:15) ( z z K − k ) − z · ∂∂k (cid:15) ( z k k K − ) + z · ∂∂k (cid:15) ( z k k K − ) + K − · ∂∂k (cid:15) ( z z k k ) = 0 (8.4)The above contracted form allows us to write a manifestly translation invariant form of thespin-raising operator : D ++12 = 12 (cid:8) (cid:15) ( z z K − k )( − ∆ + 2) − (cid:15) ( z z K − k )( − ∆ + 3) − ( z .K − ) (cid:15) ( z k k K − )+ (cid:15) ( z z K − k )( k · K − ) + (cid:15) ( z z k k ) W −− (cid:9) (8.5)As in (5.7) the operator depends only on K − ij in the derivatives, ensuring translationalinvariance. (cid:104) J J J (cid:105) odd
In this section we use the operator in (8.5) to derive the odd part of (cid:104)
J J J (cid:105) . In four-dimensions, this correlator is expected to have the following structure based on transver-sality and momentum conservation : (cid:104) J aµ ( k ) J bν ( k ) J cλ ( k ) (cid:105) odd = π αµ ( k ) π βν ( k ) π γλ ( k )[ A abc (cid:15) αβγk + B abc (cid:15) αβγk + C abc k γ (cid:15) αβk k + D abc k α (cid:15) βγk k + E abc k β (cid:15) γαk k ] (8.6)The above ansatz can be simplified using the Schouten identities in (C.8) to the following (cid:104) J aµ ( k ) J bν ( k ) J cλ ( k ) (cid:105) odd = π αµ ( k ) π βν ( k ) π γλ ( k ) (cid:104) C abc k γ (cid:15) αβk k + D abc k α (cid:15) βγk k + E abc k β (cid:15) γαk k (cid:105) (8.7)Due to cyclic symmetry we obtain the following relations between the form factors C abc ( k , k , k ) = − E bca ( k , k , k ) E abc ( k , k , k ) = − D bca ( k , k , k ) D abc ( k , k , k ) = − C bca ( k , k , k ) (8.8)This shows that there is only one independent form factor. To compute (cid:104) J J J (cid:105) odd in four-dimensions one has to compute : (cid:104)
J J J (cid:105) odd = D ++12 (cid:104) O O J (cid:105) + cyclic perms. (8.9)The correlator (cid:104) O O J (cid:105) has the following ansatz : (cid:104) O ( k ) O ( k ) J µ ( k ) (cid:105) = (cid:104) O ( k ) O ( k ) j µ ( k ) (cid:105) − k µ k ( (cid:104) O ( k ) O ( − k ) (cid:105) + (cid:104) O ( k ) O ( − k ) (cid:105) ) (8.10)– 27 –here (cid:104) OOj (cid:105) is the transverse part, which has the following ansatz (cid:104) O ( k ) O ( k ) j µ ( k ) (cid:105) = A ( k , k , k ) k ν π µν ( k ) (8.11)The form factor A is computed by solving the conformal ward identities, and is given by A ( k , k , k ) = 2 I { } (8.12)We substitute this in the reconstruction formula (8.10) to get (cid:104) O O J (cid:105) . Acting with D ++12 on (cid:104) O O J (cid:105) gives : (cid:104) J J J (cid:105) = D ++12 (cid:104) O O J (cid:105) + cyclic perms. = D ++12 (cid:104) O O j (cid:105) + cyclic perms. (8.13)since D ++12 kills the longitudinal part. We now compute (8.13) to get the following explicitform of the form factor : C abc ( k , k , k ) = − f abc J (cid:2) − (( k · k ) − k · k k ) I { } + (( k · k ) + ( k · k )( k + k )) I { } + ( − k · k k + k k ) I { } (cid:3) (8.14)where J = ( k + k − k ) ( k − k + k ) ( − k + k + k ) ( k + k + k ) . In this paper we computed momentum space parity-odd 3-point functions of scalar op-erators and conserved currents in a CFT. We used conformal Ward identities to fix theparity-odd part of the (cid:104)
J J O (cid:105) correlator. While using this method to fix more complicated3-point functions of spinning operators such as (cid:104)
J J J (cid:105) odd or (cid:104) T T T (cid:105) odd , we faced the dif-ficulty of identifying independent tensor structures after the application of the generatorof special conformal transformations on the ansatz for the correlator. The difficulty arisesfrom Schouten identities that relate the various tensor structures. We leave this problemto a future work.We defined parity-odd spin-raising and weight-shifting operators, and used them tocompute (cid:104)
J J J (cid:105) odd and (cid:104)
T T O ∆ (cid:105) odd for ∆ = 1 , . However, we found it quite complicated togeneralise this analysis to obtain correlators such as (cid:104) J J T (cid:105) odd and (cid:104)
T T T (cid:105) odd . The difficultyarises from the large number of possible paths to reach a specific correlator starting froma seed correlator. Another difficulty was the difference in the singularity structure of theseed correlator and the correlator of interest to us. We leave further investigation for thefuture.Some of the directions that we would like to pursue in the near future are the following.It will be interesting to extend the analysis of [16, 27] and study the double copy structureof parity-odd correlation functions [83]. It will be interesting to generalise the construction– 28 –f momentum space conformal blocks in [36] to the case where parity-odd contributionsare important. In momentum space, conformal blocks are constructed quite simply bytaking products of 3-point functions of primary operators (contrasted against position space,where an infinite sum over conformal descendants is required) [23, 26]. Recently, in thestudy of cosmological correlators [43, 46, 47], the form of tree-level four-point functionsin momentum space was constrained. Tree level spinning correlators such as (cid:104)
J J OO (cid:105) canget parity-odd contributions as they are built out of products of three-point functions. Itwould be interesting to find the explicit form of these and study their physical implications.Another interesting direction to pursue is to understand the parity-odd structure of 3-pointcorrelators with operators of arbitrary spin. To do this, in addition to the techniques used inthis paper, one could use the constraints imposed on the correlators by higher spin equations[32, 35]. Solving higher spin equations requires us to include possible contact terms in thecorrelator. It would be interesting to classify both parity-even and parity-odd contact termsfor a given spinning correlator. One can also study momentum space correlation functionsof spinning operators in supersymmetric theories [84, 85].
Acknowledgments
We thank Nilay Kundu and Vinay Malvimat for discussions and collaboration at an earlystage. The work of SJ and RRJ is supported by the Ramanujan Fellowship. AM would liketo acknowledge the support of CSIR-UGC (JRF) fellowship (09/936(0212)/2019-EMR-I).The work of AS is supported by the KVPY scholarship. We acknowledge our debt to thepeople of India for their steady support of research in basic sciences.
A Embedding space formalism
In this section we briefly review some aspects of the embedding space formalism following[77]. Conformal invariance is most manifest in the embedding space formalism as the d -dimensional Euclidean conformal algebra is the ( d + 2) -dimensional Poincare algebra.The d -dimensional CFT correlator in position space can be written in terms of ( d + 2) -dimensional embedding space. The embeddding space coordinates X M are defined in termsof the position space coordinates x i as follows : X A = ( X + , X − , X i ) = (1 , x , x i ) Z B = ( Z + , Z − , Z i ) = (0 , z · x, z i ) X A = ( X + , X − , X i ) = ( x / , / , − x i ) Z B = ( Z + , Z − , Z i ) = ( z · x, , − z i ) (A.1)where the Z M and z i are the null polarization vectors in the embedding space and positionspace, respectively. The derivative on the embedding space is defined as follows ∂∂X M = − Dδ + M + P i δ iM (A.2)– 29 –here D = ∆ + x i ∂∂x i , P i = ∂∂x i (A.3)The ( d + 2) -dimensional space metric is as follows η MN = − − − (A.4)The embedding space coordinates are defined such that Z · X = X = Z = 0 because theCFT is defined on the null light cone of the AdS space.The spin-raising and weight-shifting operators that we study in this paper are con-structed to have two properties in the embedding space, namely transversality and interi-ority. Transversality requires that under the transformation Z i → Z i + βX i , the operatorsremain invariant, while under interiority, the operator must map null light cone to itself.The operators thus constructed are manifestly invariant under conformal transformations.In the main text we have often used notations such as (cid:15) ( Z , Z , X , X , X ) : (cid:15) ( Z , Z , X , X , X ) = (cid:15) ABCDE Z A Z B X C X D X E . (A.5) B Parity-odd two-point functions
As is well known, scale invariance completely fixes CFT two-point functions. parity-oddstructures can exist for two-point functions of spinning operators.
B.1 Four and Higher dimensions
In four or higher dimensions, it is not possible to have any parity-odd two-point functionof either spin-one or any other spinning symmetric spinning correlator. This is because aparity-odd correlator must necessarily involve the (cid:15) tensor and it is simple to show that itis impossible to have any parity-odd 2-point function of a spin-1 or any symmetric tensoroperator.
B.2 Three-dimensions
In three-dimensions parity-odd two-point functions exist. These come from purely contactterms . We will look at the parity-odd 2-point functions of spin-one and spin two conservedcurrents. In this case, the corresponding position space correlator with separated points vanishes. – 30 – J µ J ν (cid:105) odd The general ansatz for the correlator is given by (cid:104) J µ ( k ) J ν ( − k ) (cid:105) odd = A ( k ) (cid:15) µνk (B.1)The ansatz guarantees that the correlator is transverse to the momentum. Imposing scaleinvariance gives the following differential equation for the form factor A ( k ) : k ∂∂k A ( k ) = 0 (B.2)This implies that the form factor is just a constant in this case and we have : (cid:104) J µ ( k ) J ν ( − k ) (cid:105) odd = c J (cid:15) µνk (B.3)We will now consider the parity-odd 2-point function of the stress-tensor. (cid:104) T µν T ρσ (cid:105) odd We consider the following ansatz for this correlator : (cid:104) T µν ( k ) T ρσ ( − k ) (cid:105) odd = B ( k )∆ µνρσ ( k ) (B.4)where ∆ µνρσ ( k ) is a parity-odd, transverse-traceless projector given by : ∆ µνρσ ( k ) = (cid:15) µρk π νσ ( k ) + (cid:15) µσk π νρ ( k ) + (cid:15) νσk π µρ ( k ) + (cid:15) νρk π µσ ( k ) (B.5)where π µν ( k ) is the same projector used in previous sections. The ansatz guarantees thatthe correlator is transverse and traceless.The dilatation Ward identity gives the following equation for the form factor B ( k ) : (cid:18) k ∂∂k − (cid:19) B ( k ) = 0 (B.6)This can be easily solved to get B ( k ) = c T k (B.7)Therefore, the correlator is given by (cid:104) T µν ( k ) T ρσ ( − k ) (cid:105) odd = c T ∆ µνρσ ( k ) k (B.8) C Schouten Identities
Here, we list the Schouten identities used in our calculations in the main text. The mostgeneral form of a Schouten identity in d -dimensions is (cid:15) [ µ µ ...µ d δ ν ] ρ = 0 (C.1)In three-dimensions, this translates to (cid:15) µ µ µ δ νρ − (cid:15) µ µ ν δ µ ρ + (cid:15) µ νµ δ µ ρ − (cid:15) νµ µ δ µ ρ = 0 (C.2)– 31 –otting the indices in (C.2) with momenta lets us relate different epsilon structures thatoccur in the correlation functions calculated earlier. Dotting with k µ , k ρ and k ν gives (cid:15) µ µ k ( k · k ) + (cid:15) µ k k k µ = (cid:15) µ µ k k + (cid:15) µ k k k µ (C.3)Similarly, dotting with k µ , k ρ and k ν (cid:15) µ µ k ( k · k ) + (cid:15) µ k k k µ = (cid:15) µ µ k k + (cid:15) µ k k k µ (C.4)(C.3) and (C.4) were useful in rewriting the (cid:104) J J O (cid:105) ansatz. One can also derive these twoidentities by considering the contraction of three Levi-Civita tensors. (cid:15) µ αk (cid:15) βk µ (cid:15) βρα = (cid:15) µ µ k k ρ + (cid:15) µ k k δ µ ρ (cid:15) βk µ (cid:15) αk µ (cid:15) αβρ = (cid:15) µ µ k k ρ + (cid:15) µ k k δ µ ρ (C.5)Equating the RHS of the two equations after dotting them with k ρ and k ρ respectively, weget back (C.3) and (C.4). Similarly, while checking the transverse identity for (cid:104) J J J (cid:105) , weused the following Schouten identities (cid:15) k k µ k µ = (cid:15) k µ µ ( k · k ) + (cid:15) µ µ k k + (cid:15) k k µ k µ (C.6) (cid:15) k k µ k µ = (cid:15) k µ µ ( k · k ) + (cid:15) µ µ k ( k · k ) + (cid:15) µ k k k µ (C.7)In four-dimensions, we use the following identities to rewrite the ansatz for (cid:104) J J J (cid:105) . ( k · k ) (cid:15) µ µ µ k − k µ (cid:15) k µ µ k − k µ (cid:15) µ k µ k − k µ (cid:15) µ µ k k − k (cid:15) µ µ µ k = 0( k · k ) (cid:15) µ µ µ k − k µ (cid:15) k µ µ k − k µ (cid:15) µ k µ k − k µ (cid:15) µ µ k k − k (cid:15) µ µ µ k = 0 (C.8) D Computation details for (cid:104)
T T O (cid:105)
In this section we give some details of the computation of the (cid:104)
T T O (cid:105) correlator in the freefermion theory (7.5). We use the form of the stress tensor and the scalar operator as givenin (7.8) and (7.9). The correlator of interest in terms of spinor fields is as follows : (cid:104) T µν ( k ) T αβ ( k ) O ( k ) (cid:105) = (cid:90) (cid:104) ¯ ψ ( l ) γ µ (2 l − k ) ν ψ ( k − l ) ¯ ψ ( l ) γ α (2 l − k ) β ψ ( k − l ) ¯ ψ ( l ) ψ ( k − l ) (cid:105) (D.1)The Wick contraction (1¯3)(3¯2)(2¯1) gives : (cid:90) tr ( γ µ γ ρ γ σ γ α γ τ )(2 l − k ) ν (2 l − k ) β l ρ l σ l τ l l l δ ( k − l + l ) δ ( k − l + l ) δ ( k − l + l ) (D.2)where the fermion propagator given by : (cid:104) ¯ ψ α ( k ) ψ β ( k ) (cid:105) = δ (3) ( k + k ) /k ,αβ k (D.3) We are not giving the details of the complex conjugate here as it gives the same result. – 32 –as used. Computing the integrals over l , l and using momentum conservation, we maywrite the integral (D.2) over a single variable as : (cid:20)(cid:90) tr ( γ τ γ σ γ α γ ρ γ µ )(2 l − k ) ν (2 l + k ) β l ρ ( l − k ) τ ( l + k ) σ l ( l + k ) ( l − k ) (cid:21) + symmetrize in ( µ, ν ) and ( α, β ) (D.4)By projecting (D.4) with spin-2 projectors, one can now determine the transverse part ofthe correlator and hence, identify the form factors in (7.3) explicitly. D.1 Transverse and trace Ward identities
Here we provide an explicit verification of the transverse and trace Ward identities satisfiedby (cid:104)
T T O (cid:105) . Contracting (D.4) with k µ we obtain : (cid:90) tr ( 1 /l − /k /l + /k γ α /l /k )(2 l − k ) ν (2 l + k ) β + (cid:90) tr ( 1 /l − /k /l + /k γ α /l γ µ )(2 l − k ) · k (2 l + k ) β + symmetrize in ( α, β ) (D.5)which when simplified using : /l /k /l − /k = 1 /l − /k − /l (2 l − k ) · k = − ( l − k ) + l (D.6)After a little algebra we get k (cid:2) k β ( (cid:15) ανk − (cid:15) ανk ) + k α ( (cid:15) βνk − (cid:15) βνk ) − δ α,ν (cid:15) βk k − δ β,ν (cid:15) αk k (cid:3) (D.7)which precisely reproduces the transverse Ward identity given in (7.18). We now contractthe ( µ, ν ) indices to check the trace Ward identity : (cid:90) tr ( 1 /l − /k /l + /k γ α /l (2 /l − /k ))(2 l + k ) β = (cid:90) tr ( 1 /l − /k /l + /k γ α )(2 l + k ) β + (cid:90) tr ( 1 /l /l + /k γ α )(2 l + k ) β (D.8)In the first term, one may transform the integration variable to l → l − k : (cid:90) tr ( 1 /l + /k /l γ α )(2 l + k ) β + (cid:90) tr ( 1 /l /l + /k γ α )(2 l + k ) β = − i [ (cid:90) (cid:15) l k α l ( l + k ) (2 l − k ) β − (cid:90) (cid:15) l k α l ( l + k ) (2 l + k ) β ] (D.9)– 33 –aking use of the following identities (cid:90) l l µ l ( l + k ) = Ak µ (cid:90) l l µ l ν l ( l + k ) = Bη µν + Ck µ k ν (D.10)where A, B, C are scalars, one can see that (D.9) vanishes and hence show that the correlatorsatisfies (7.17).
D.2 Details of longitudinal part
In this section we compute the function derivative δT µν ( x ) δg αβ ( y ) in the free fermion theory, relevantto Section 7.1. Since the functional dependence on the metric is only via the spin connectionwe have : δT µν ( x ) δg αβ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) g αβ → η αβ = 132 (cid:18) δω abν ( x ) δg αβ ( y ) | g αβ → η αβ ¯ ψ { γ µ , γ ab } ψ + δω abµ ( x ) δg αβ ( y ) | g αβ → η αβ ¯ ψ { γ ν , γ ab } ψ (cid:19) = i (cid:32) δω abν ( x ) δg αβ ( y ) (cid:15) abµ + δω abµ ( x ) δg αβ ( y ) (cid:15) abν (cid:33) O ( x ) (D.11)where δω abν ( x ) δg αβ ( y ) | g αβ → η αβ = δ σb δ aτ (cid:104) ∂ σ ( δ α ( τ δ βν ) δ (3) ( x − y )) + ∂ ν ( δ α ( τ δ βσ ) δ (3) ( x − y )) − ∂ τ ( δ α ( σ δ βν ) δ (3) ( x − y )) (cid:105) (D.12)The second line in (D.11) was obtained by recognising that in three-dimensions, γ ab =[ σ a , σ b ] = 2 i(cid:15) abc σ c . In the limit, g αβ → η αβ , the vierbiens go to e aα → δ aα was also used.Simplifying the above expression to obtain, δω abν ( x ) δg αβ ( y ) (cid:15) abµ = − [ (cid:15) σαµ δ βν + (cid:15) σβµ δ αν ] ∂ σ δ (3) ( x − y ) (D.13)From here we get (7.12). E Parity-even spin-raising and weight-shifting operators
In this section we list out all the parity-even weight-shifting operators used in the maintext of the paper [46, 47].The operator that decreases the scaling dimension of operators at points 1 and 2 is : W −− = 12 (cid:126)K − · (cid:126)K − (E.1)where K − µ = ∂ k µ − ∂ k µ (E.2)– 34 –e also use K + µ = ∂ k µ + ∂ k µ (E.3)We can also define an operator that increases the scaling dimension at 2-points. Althoughthis has a very complicated expression, it simplifies when acting on scalar operators and isgiven by : W ++12 =( k k ) W −− − ( d − )( d − ) k · k + (cid:0) k ( d − )( d − − ∆ + k · K ) + (1 ↔ (cid:1) (E.4) D raises the spin of the operator at point 1 and simultaneously lowers its weight. Thiswas used in the construction of both (cid:104) T T O (cid:105) and (cid:104)
J J J (cid:105) : D =(∆ − (cid:126)k · (cid:126)K ) (cid:126)z · (cid:126)K − ( (cid:126)k · (cid:126)z ) W −− − ( (cid:126)z · (cid:126)K ) ( (cid:126)z · ∂ (cid:126)z ) + ( (cid:126)z · (cid:126)z ) ∂ (cid:126)z · (cid:126)K (E.5)We can similarly define D and D by doing cyclic permutations of the momenta andpolarization vectors in (E.5). For example, D (( k , z ) , ( k , z ) , ( k , z )) = D (( k , z ) , ( k , z ) , ( k , z )) (E.6) S ++12 raises the spin at points 1 and 2 : S ++12 =( s + ∆ − s + ∆ − z · z − ( z · k )( z · k ) W −− + [( s + ∆ − k · z )( z · K ) + (1 ↔ (E.7) S ++23 and S ++13 are once again defined by cyclic permutations of (E.7).The operator H which raises the spin at points 1 and 2 and also lowers the weight atboth the points is given by : H = 2 ( z · K )( z · K ) − z · z ) W −− (E.8)The operator that raises the spin at point 1 and simulataneously lowers the weight at point2 is given by : D = (∆ + s − z · K − ( z · k ) W −− (E.9)A (1 ↔ exchange in this operator gives D . Both of these were used in the constructionof (cid:104) T T O (cid:105) . F Embedding space parity-odd correlation functions in four-dimensions
In this appendix, we show that (cid:104)
J J T (cid:105) odd and (cid:104)
T T T (cid:105) odd are zero in four-dimensions.– 35 – .1 (cid:104) J J T (cid:105) odd
We first write the (cid:104)
J J T (cid:105) odd correlator in a basis of (embedding space) conformally invariantstructures (the notation used is that of [77]) (cid:104) J ( Z , X ) J ( Z , X ) T ( Z , X ) (cid:105) odd = c (cid:15) ( Z Z Z X X X ) V X / X / X / (F.1)Under simultaneous exchange of ( X , X ) and ( Z , Z ) we have V → − V (F.2)Hence, symmetry consideration demands that c = 0 . This implies that the parity-odd (cid:104) J J T (cid:105) correlator vanishes in four-dimensions. In momentum space, it can be a contactterm.
F.2 (cid:104)
T T T (cid:105) odd
We first write the correlator in a basis of conformally invariant structures (cid:104) T ( Z , X ) T ( Z , X ) T ( Z , X ) (cid:105) odd = (cid:15) ( Z Z Z X X X ) (cid:88) n ,n ,n A n ,n ,n V − n − n V − n − n V − n − n H n H n H n X / X / X / = (cid:15) ( Z Z Z X X X )[ A V V V X / X / X / + A V H X / X / X / + A V H X / X / X / + A V H X / X / X / ] (F.3)Under simultaneous Z ↔ Z , X ↔ X exchange we have V → − V , V → − V , H → H , H → H (F.4) (cid:104) T ( Z , X ) T ( Z , X ) T ( Z , X ) (cid:105) odd = − (cid:15) ( Z Z Z X X X )[ A V V V X / X / X / + A V H X / X / X / + A V H X / X / X / + A V H X / X / X / ] (F.5)Therefore, we must have A = 0 , A = − A , A = 0 (F.6)Hence, (cid:104) T ( Z , X ) T ( Z , X ) T ( Z , X ) (cid:105) odd = A (cid:15) ( Z Z Z X X X ) (cid:34) V H X / X / X / − V H X / X / X / (cid:35) (F.7)– 36 –ow, under simultaneous exchange of ( Z , Z ) and ( X , X ) V → − V V → − V H → H H → H (F.8)Therefore, (cid:104) T ( Z , X ) T ( Z , X ) T ( Z , X ) (cid:105) odd = A (cid:15) ( Z Z Z X X X ) (cid:34) − V H X / X / X / + V H X / X / X / (cid:35) (F.9)Hence, we must have A = 0 . Therefore, symmetry considerations force (cid:104) T T T (cid:105) odd to bezero.
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