Analytic constraints on the energy-momentum tensor in conformal field theories
aa r X i v : . [ h e p - t h ] M a y Analytic constraints on the energy-momentum tensorin conformal field theories
C´edric Lorc´e ∗ and Peter Lowdon † CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France
Abstract
In this work we investigate the matrix elements of the energy-momentum tensor for massless on-shell states in four-dimensional unitary, local, and Poincar´e covariant quantum field theories. Inparticular, we demonstrate that these matrix elements can be parametrised in terms of covariantmultipoles of the Lorentz generators, and that this gives rise to a form factor decompositionin which the helicity dependence of the states is factorised. In the remainder of this work weexplore some of the consequences of this decomposition for conformal field theories, includingderiving the explicit analytic constraints imposed by conformal symmetry, and using explicitexamples to show that they uniquely fix the form of the matrix elements. We also use thisdecomposition to gain new insights into the conditions under which general unitary theories areconformal. ∗ [email protected] † [email protected] Introduction
As with any quantum field theories (QFTs), the correlation functions in conformal field theories(CFTs) completely encode the dynamics of these theories. A characteristic property of CFTsis that the conformal symmetry significantly constrains the form of the correlation functions,reducing the classification of these objects to the determination of a series of constant parame-ters. Although the analysis of CFTs has historically focussed on the Euclidean version of thesetheories, in part because of their relative analytic simplicity, in recent years there has been asurge in interest in Minkowskian CFTs, particularly in the context of the analytic bootstrapprogram [1]. Due to the larger number of space-time symmetries, the energy-momentum tensor(EMT) plays a central role in many of the analytic constraints imposed on CFTs. An importantexample are the three-point functions involving the EMT, which have been shown in Euclideanspace [2–5], and more recently for certain cases in Minkowski space [6], to be fully constrainedby the conserved conformal currents, and their corresponding Ward identities.The focus of this work will be the Minkowski matrix elements of the EMT for massless on-shellstates. In particular, we will study these matrix elements in QFTs that are unitary, local, andPoincar´e covariant. By local we mean that all of the fields Φ( x ) in the theory, including theEMT, either commute or anti-commute with each other for space-like separations. Poincar´ecovariance implies that the components of these fields Φ k ( x ) transform as: U ( a, α )Φ k ( x ) U − ( a, α ) = X l D (Φ) kl ( α − )Φ l (Λ( α ) x + a ) , (1)under (proper orthochronous) Poincar´e transformations ( a, α ), where D (Φ) is the correspondingWigner matrix that defines the representation of the field, and Λ( α ) is the four-vector repre-sentation of α [7]. Although the overall structural properties of EMT matrix elements havebeen understood for many years, at least in the case of massive states [8, 9], the explicit spindependence of these objects has only been studied relatively recently. By using the conservationof the EMT, together with its various symmetry properties, one can decompose these matrixelements into a series of form factors with independent covariant coefficients . We refer to thesethroughout as the gravitational form factors (GFFs). As the spin of the states increases, theseobjects become increasingly more complicated due to the larger number of potential covariantstructures. This explains in part why many studies have focussed on calculations for states withlower values of spin, generally 0, , or 1. Once the potential covariant coefficients are known,constraining the corresponding GFFs is therefore essential for understanding the analytic struc-ture of the matrix elements. Since the states appearing in the on-shell matrix elements aredefinite momentum eigenstates, and hence not normalisable, this has led to incorrect physicalconclusions in the literature, as detailed in [18]. This non-normalisability stems from the factthat the matrix elements are distributions, not functions. It was first shown for the massivespin- case that by taking this property into account in the derivation of the GFF constraints,these apparent ambiguities no longer arise [19]. This approach was later generalised to massivestates with arbitrary values of spin [20], as well general spin-state definitions, including masslessstates [21].Most studies of the CFT three-point functions involving the EMT have focussed on cases wherethe other fields have lower values of absolute helicity | h | , since increasing | h | quickly leads tocomplicated expressions. Because these three-point functions are directly related to the masslesson-shell EMT matrix elements, via a projection of the Lorentz components of the fields, it turns By covariant we mean that the components of the coefficients transform in the same manner as the fields [Eq. (1)]under Lorentz transformations. See [10, 11] and [12–17] for some recent examples of gravitational and hadronic studies. ut that this increase in complexity is simply a different realisation of the fact that the numberof independent covariant structures in the form factor expansion increases with | h | . Now whilstit is clear that the total number of these independent structures must be finite for differentvalues of | h | , establishing what these numbers are is non-trivial. In a recent work [22], thisproblem was solved for the EMT matrix elements of massive states with arbitrary spin s . Theessential step in this analysis was the realisation that all covariant structures that can appearin the matrix elements can be generated by contracting the covariant multipoles {M n } of theLorentz generators S µν with the external momenta p ′ and p , and the metric. The advantageof using this multipole basis is that these objects explicitly truncate for each value of s . Inparticular, given states of spin s , one has the constraint: M N = 0 , for N > s, (2)where {M N } are constructed from Lorentz generators that transform under the same represen-tation as the fields creating the states. Not only does this constraint prove that only a finitenumber of independent covariant structures enter into the EMT matrix elements, it also pro-vides a basis from which these structures can be systematically characterised [22]. Since thecovariant multipoles are fundamental objects that exist independently of the specific propertiesof the states, this representation can also equally be applied to the matrix elements of masslessstates [21]. As will be outlined in Sec. 2, this has the important implication that the helicitydependence of these matrix elements factorises, or equivalently, that the dependence on theLorentz representation of EMT three-point functions can be written in a manifest way.Before concluding this section we will first discuss a result which plays a central role throughoutthis work, the Weinberg-Witten Theorem [23]. This theorem puts constraints on the potentialstructure of matrix elements involving massless on-shell states, in particular implying that: h p ′ , h ′ | T µν (0) | p, h i = 0 , for | h ′ + h | 6 = 0 , , p ′ − p ) < . (3)In [23] it is further stated that this constraint can be extended by continuity to the point p ′ = p .It turns out though that potential discontinuities can in fact occur when p ′ = p due to thedistributional nature of the matrix elements [24]. However, by making the additional assump-tion that the corresponding QFT is a local theory, this argument can be made consistent [25].Since the EMT operator does not modify the value of | h ′ | or | h | , the constraints from Eq. (3)subsequently lead to the important conclusion: Massless particles of helicity h , where | h | > , cannot possess charges induced by alocal and Poincar´e covariant energy-momentum tensor. As emphasised in [25], this does not mean that consistent Poincar´e or conformal generators donot exist for massless states with higher helicity, only that these generators cannot be written interms of integrals of a local and Poincar´e covariant EMT. In other words, by allowing masslessparticles with | h | > For example, M = 1, M µν = S µν , and M µν,ρσ = { S µν , S ρσ } − ( g µρ g νσ − g µσ g νρ ) S · S + ǫ µνρσ ǫ αβγδ S αβ S γδ define the monopole, dipole, and quadrapole, respectively. See [22] for an in-depth discussion of these objects. he GFF constraints imposed by conformal symmetry and the trace properties of the EMT,and apply these findings to specific CFT examples. In Sec. 5 we combine the results derivedthroughout the paper to make a general connection between the presence of massless particlesand conformal symmetry, and finally in Sec. 6 we summarise our key findings. For the purposes of this paper we are interested in the EMT matrix elements for on-shellmomentum eigenstates. One can covariantly impose this on-shell restriction by defining stateswith mass M : | p, σ ; M i = δ (+) M ( p ) | p, σ i = 2 π θ ( p ) δ ( p − M ) | p, σ i , (4)where | p, σ i is the standard non-covariant momentum eigenstate with internal quantum num-bers σ . The advantage of using the states in Eq. (4) is that they transform covariantly underPoincar´e transformations, which significantly simplifies on-shell matrix element calculations. Inwhat follows, we will restrict ourselves to massless on-shell states. For simplicity we drop thelabel M = 0 and denote these states by | p, h i , where h is the helicity. Since we focus only onunitary QFTs throughout this work, it follows from Eq. (4) and the standard inner product forgeneral eigenstates that: h p ′ , h ′ | p, h i = (2 π ) δ ( p ′ − p ) δ (+)0 ( p ) δ h ′ h . (5)In a previous analysis, which explored the GFF constraints imposed by Poincar´e symmetry [21],it was established under the assumptions that the EMT is symmetric, hermitian, and both parity P and time-reversal T invariant, that the EMT matrix elements for on-shell massless states ina unitary, local, Poincar´e covariant QFT have the following general decomposition: h p ′ , h ′ | T µν (0) | p, h i = η h ′ ( p ′ ) h ¯ p { µ ¯ p ν } A ( q ) + i ¯ p { µ S ν } ρ q ρ G ( q ) + · · · i η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) , (6)where · · · represent contributions with an explicitly higher-order dependence on the four-momentum transfer q = p ′ − p , index symmetrisation is defined: a { µ b ν } = a µ b ν + a ν b µ , and¯ p = ( p ′ + p ). We refer to η h ( p ) as the generalised polarisation tensors (GPTs), which corre-spond to the Lorentz representation index-carrying coefficients that appear in the decompositionof primary free fields with helicity h . For example, since we assume P and T invariance, the GPTin the | h | = case is the Dirac spinor u h ( p ). As discussed in Sec. 1, one can see in Eq. (6) thatthis expression is constructed by contracting covariant multipoles, in this case the monopole anddipole, with all possible combinations of momenta and the metric. Due to the low number ofcovariant indices, it turns out that up to linear order in q there exists only one such combinationfor each multipole which is consistent with the various symmetries of the EMT . As detailedin [21], the constraints arising from Poincar´e symmetry only affect the GFFs with coefficientsthat depend at most linearly on q , which explains why only these leading terms are consideredin Eq. (6). However, in general there are other possible GFFs, with coefficients that potentiallyinvolve contractions with higher multipoles. For massive states, these GFFs were fully classified The difference between these states is that the on-shell factor is included in the definition of | p, σ ; M i , as opposedto the momentum integration measure. In constructing physical states one therefore integrates | p, σ ; M i over d p (2 π ) ,whereas for | p, σ i the measure itself is on shell: d p (2 π ) δ (+) M ( p ) = d p (2 π ) E p dp δ ( p − E p ), with E p = p p + M . In particular: h p ′ , σ ′ | p, σ i = 2 p (2 π ) δ ( p ′ − p ) δ σ ′ σ . This characteristic is also true for massive states [20]. Due to Poincar´e symmetry one finds that: A ( q ) δ ( q ) = δ ( q ), and: G ( q ) δ ( q ) = δ ( q ) [21]. n [22] for arbitrary spin. In the remainder of this section we will discuss the massless case.In general, given a massless field with Lorentz representation ( m, n ), this field creates states withhelicity h = n − m . For example, a left-handed Weyl spinor with representation ( ,
0) givesrise to h = − states. Massless representations of the Lorentz group have significant additionalconstraints imposed upon them, including the fact that all irreducible representations can bebuilt from representations ( m, n ) where either m = 0, n = 0, or both [26]. In particular, itfollows that massless fields which transform covariantly under the vector representation ( , )cannot be irreducible. This can be explicitly seen by the fact that any such field V µ can alwaysbe written as the gradient of a scalar field: V µ = ∂ µ φ , since V µ only defines states with h = 0.More generally, any massless field with Lorentz representation ( m, n ), where both m and n arenon-vanishing, can be written in terms of derivatives of irreducible fields [26]. Since masslessQFTs are constructed from irreducible fields, or their direct sums, the corresponding GPTs ofthese fields, including those in Eq. (6), must also transform under these representations. Aswe will see, this puts significant constraints on the type of GFFs that can appear in the EMTmatrix elements.In Sec. 1 we introduced the Weinberg-Witten Theorem and outlined its implications, namelythat if massless particles with | h | > | h | >
1, this impliesthat these objects must necessarily contain a form factor with either a non-local or non-covariantcoefficient. However, since we restrict ourselves in this work to unitary CFTs which are localand Poincar´e covariant, these matrix elements must in fact vanish [25]. In other words, Eq. (6)is only non-trivial when the states have helicity | h | ≤
1. Another constraint on the type of GFFsappearing in Eq. (6) comes from the covariant multipole bound in Eq. (2). If the GPTs arein the Lorentz representation ( m, n ), it equally follows in the massless case that the multipoles {M N } must vanish for N > m + n ). Since by definition M N contains N products of theLorentz generators S µν , which transform under the same (irreducible) representation as theGPTs: ( m, , n ), or their direct sums, the number of powers of these generators is thereforebounded above by the helicity of the states. In particular: The number of powers of S µν appearing in the EMT matrix elements for masslessstates of helicity h is at most | h | . Combining this with the Weinberg-Witten Theorem constraint | h | ≤
1, one is immediately ledto the conclusion that only GFFs which have coefficients with two or fewer powers of S µν arepermitted. Now we are in a position to write down the full generalisation of Eq. (6). If wecontinue to demand that the GFFs are dimensionless, as in Eq. (6), and also similarly assumethat the EMT is symmetric, hermitian, and both P and T invariant, the non-trivial ( | h | ≤ : h p ′ , h ′ | T µν (0) | p, h i = η h ′ ( p ′ ) h ¯ p { µ ¯ p ν } A ( q ) + i ¯ p { µ S ν } ρ q ρ G ( q )+ ( q µ q ν − q g µν ) C ( q ) + S { µα S ν } β q α q β T ( q ) i η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) . (7)From Eq. (7) one can see that by adopting a parametrisation that uses the covariant multipolesas its basis, this leads to a form factor decomposition of the EMT matrix elements in which the This constraint gives rise to the well-known result that the components of the massless photon field A µ cannottransform covariantly as a vector without violating the positivity of the Hilbert space (unitarity) [27]. We will discussthis characteristic further in Sec. 4.4.3. In particular, for | h | = the coefficient of T ( q ) is no longer independent, and so only A ( q ), G ( q ) and C ( q )can potentially exist, and for h = 0, since ( S µν ) (0 , = 0, only A ( q ) and C ( q ) remain. elicity dependence is factorised: only knowledge of the generator S µν in the Lorentz represen-tation of the GPTs is required to calculate the matrix elements for states of different helicities.It is also interesting to note that by virtue of the fact that q is the only massive Lorentz invari-ant that can appear in massless theories, there cannot exist form factor coefficients other thanthose in Eq. (7) which are compatible with locality, whilst also ensuring the form factors aredimensionless. In the remainder of this paper we will derive some of consequences of Eq. (7). Besides Poincar´e symmetry, CFTs are also invariant under infinitesimal dilations and specialconformal transformations (SCTs). In the case of dilations, the dilation operator D acts onconformal fields in the following manner: i [ D, Φ( x )] = ( x µ ∂ µ + ∆) Φ( x ) , (8)where ∆ is the conformal dimension of the field. For SCTs, the charge K µ instead has theaction: i [ K µ , Φ( x )] = (cid:0) x µ x ν ∂ ν − x ∂ µ + 2 x µ ∆ − ix ν S µν (cid:1) Φ( x ) , (9)where the Lorentz representation indices of both the field and the Lorentz generator S µν havebeen suppressed for simplicity. It turns out that these transformations impose significant con-straints on the properties of the fields. In particular, combining Eq. (9) with the masslessnessof the field implies the following important relation [26]:(∆ − ∂ µ Φ( x ) = iS µν ∂ ν Φ( x ) . (10)For scalar fields ( S µν ) (0 , = 0, and hence ∆ = 1; for spinor fields in the ( ,
0) and (0 , )representations one recovers the Weyl equations when ∆ = ; and for the anti-symmetric tensorfield the substitution of ( S µν ) (1 , ⊕ (0 , results in the Maxwell equations for ∆ = 2. In Eq. (7)the EMT matrix element is expressed in terms of the action of Lorentz generators on masslessGPTs. By inserting the plane-wave expansion for a massless field into Eq. (10), one obtains aconstraint on this action: (∆ − p µ η h ( p ) = ip ν S µν η h ( p ) , (11)where the Lorentz representation indices have again been suppressed for simplicity.Before discussing the specific structure of the currents associated with dilations and SCTs, it isimportant to first outline the additional constraints imposed on the EMT itself. In any QFT itis well-known that the EMT is not unique since one can always add a superpotential term , a termthat is separately conserved, but when integrated reduces to a spatial divergence, and hencedoes not contribute to charges. A prominent example is the pure-spin term that symmetrisesthe canonical EMT. Superpotential terms also play a particularly important role in CFTs, sincethe condition for a theory to be conformal is related to whether or not there exists such a term,which when added to the EMT, renders it traceless. In particular, given a four-dimensionalQFT with a conserved and symmetric EMT, T µν (S) , a necessary and sufficient condition for thistheory to be conformal is that there exists a local operator L µν ( x ) such that [28]: T µ (S) µ ( x ) = ∂ α ∂ β L αβ ( x ) . (12) f this condition holds, it follows that there exists conserved dilation J µ (S) D and SCT currents J µ (S) K ν with the form : J µ (S) D = x ν T µν (S) − ∂ ν L νµ , (13) J µ (S) K ν = (2 x ν x α − g να x ) T µα (S) − x ν ∂ α L αµ + 2 L µν . (14)Under the further assumption that the CFT is unitary, one has that: L µν ( x ) = g µν L ( x ), andhence the EMT trace condition becomes T µ (S) µ ( x ) = ∂ L ( x ) . (15)In the general case that Eq. (12) is satisfied, this implies that there exists superpotential terms,which when added to Eqs. (13) and (14) transform these expressions into the form: J µ (ST) D = x ν T µν (ST) , (16) J µ (ST) K ν = (2 x ν x α − g να x ) T µα (ST) , (17)where T µν (ST) is both symmetric and traceless. For the remainder of this work we will refer to T µν (ST) as the modified EMT . In [21, 22] Poincar´e covariance was used to derive constraints on the GFFs for both massiveand massless states. In this section we derive the corresponding GFF constraints imposed byconformal covariance, as well as from the trace properties of the EMT itself.
By definition, momentum space fields have the following action on the vacuum state: h | e Φ( p ′ ) = P h ′′ η h ′′ ( p ′ ) h p ′ , h ′′ | , which when combined with Eq. (8) implies − i X h ′′ η h ′′ ( p ′ ) h p ′ , h ′′ | D ! = − ∂∂p ′ µ "X h ′′ p ′ µ η h ′′ ( p ′ ) h p ′ , h ′′ | + ∆ X h ′′ η h ′′ ( p ′ ) h p ′ , h ′′ | , (18)where we assume that dilational symmetry is unbroken, and hence: D | i = 0. After acting withEq. (18) on the state | p, h i and projecting on η h ′ ( p ′ ), one can use the orthogonality condition: η h ′ ( p ′ ) η h ′′ ( p ′ ) = δ h ′ h ′′ , together with the on-shell state normalisation in Eq. (5), to obtain thefollowing expression for the matrix element of the dilation operator: h p ′ , h ′ | D | p, h i = i (2 π ) δ (+)0 ( p ) (cid:20) − p ′ µ η h ′ ( p ′ ) ∂η h ∂p ′ µ ( p ′ ) − δ h ′ h p ′ µ ∂∂p ′ µ + δ h ′ h (∆ − (cid:21) δ ( p ′ − p ) . (19)For deriving GFF constraints it is simpler to work with the coordinates (¯ p, q ). To do so, onecan make use of the distributional identity in Eq. (65) of Appendix A, from which it follows: h p ′ , h ′ | D | p, h i = − i (2 π ) δ (+)0 (¯ p ) (cid:20) ¯ p µ η h ′ (¯ p ) ∂η h ∂ ¯ p µ (¯ p ) + δ h ′ h ¯ p µ ∂∂q µ − δ h ′ h (∆ − (cid:21) δ ( q ) . (20) Further background from the early literature on this subject can be found for example in [29–31]. See [32] and references within. Particularly in the CFT literature, the symmetric-traceless EMT is often referred to as the improved
EMT [30]. s in the case of the Poincar´e charges [21,22], GFF constraints can be established by comparingEq. (20) with the matrix element of D obtained using the form factor decomposition in Eq. (7),together with the definition of the dilational current in Eq. (16). A rigorous expression for thismatrix element is defined by: h p ′ , h ′ | D | p, h i = lim d → R →∞ Z d x f d,R ( x ) x ν e iq · x h p ′ , h ′ | T ν (0) | p, h i = − i lim d → R →∞ ∂ e f d,R ( q ) ∂q j h p ′ , h ′ | T j (0) | p, h i = − i (2 π ) ∂ j δ ( q ) h p ′ , h ′ | T j (0) | p, h i , (21)where in order to ensure the convergence of the operator D one integrates with a test function f d,R ( x ) = α d ( x ) F R ( x ) that satisfies the conditions: R d x α d ( x ) = 1, lim d → α d ( x ) = δ ( x ),and F R ( ) = 1, lim R →∞ F R ( x ) = 1, where e f d,R ( q ) is the Fourier transform, and ∂ j indicatesa derivative with respect to q j . To evaluate this expression one therefore needs to understandhow to simplify the final expression, which involves the product of a delta-derivative and aspecific component of the EMT matrix element. These products were already encounteredin [20] when deriving the form factor constraints due to Poincar´e symmetry. Applying thedistributional equality in Eq. (73) of Appendix A to the coefficients of the GFFs appearing inthe ( µ = 0 , ν = j ) component of Eq. (7), together with the masslessness condition: ¯ p j ¯ p j = − ¯ p (for q = 0), one obtains h p ′ , h ′ | D | p, h i = − i (2 π ) δ (+)0 (¯ p ) (cid:20) − ¯ p µ ∂ µ [ η h ′ ( p ′ ) η h ( p )] q =0 δ ( q ) A ( q ) + δ h ′ h ¯ p µ ∂ µ δ ( q ) A ( q ) − i ¯ p j ¯ p η h ′ (¯ p ) S j η h (¯ p ) δ ( q ) G ( q ) (cid:21) . (22)After using the conformal GPT constraint in Eq. (11), together with Eq. (70) in Appendix A,the matrix elements can finally be written h p ′ , h ′ | D | p, h i = − i (2 π ) δ (+)0 (¯ p ) (cid:20) ¯ p µ η h ′ (¯ p ) ∂η h ∂ ¯ p µ (¯ p ) δ ( q ) A ( q ) + δ h ′ h ¯ p µ ∂ µ δ ( q ) A ( q ) − δ hh ′ (∆ − δ ( q ) G ( q ) (cid:21) . (23)Since Eqs. (20) and (23) are different representations of the same matrix element, equating theseexpressions immediately leads to constraints on the GFFs, in particular: A ( q ) δ ( q ) = δ ( q ) , (24) A ( q ) ∂ µ δ ( q ) = ∂ µ δ ( q ) , (25) G ( q ) δ ( q ) = δ ( q ) , (26)which is nothing more than the condition: A (0) = G (0) = 1 . (27)That constraints are only imposed on A ( q ) and G ( q ) follows from the fact that the x -polynomiality order of the conserved currents has a direct bearing on whether the corresponding See [19] for an overview of these test function definitions and their motivation. Although the GFFs are in general distributions of q , one can interpret the values at q = 0 using a limitingprocedure [19]. Eq. (25) follows trivially when A ( q ) is continuous at q = 0 due to the q dependence. harges constrain certain GFFs. Since explicit factors of x in the currents become q -derivativeson the level of the charges, as in Eq. (21), it is these derivatives that can remove powers of q in Eq. (7) and constrain the GFFs at q = 0. Due to the structure of the dilational current[Eq. (16)], the matrix elements of D can therefore only potentially constrain the GFFs whichhave coefficients with at most one power of q . It is interesting to note that these are preciselythe constraints obtained from imposing Poincar´e symmetry alone [21, 22]. Now we perform an analogous procedure for SCTs. Using Eq. (9) together with the fact that thespecial conformal symmetry is unbroken, and hence: K µ | i = 0, one ends up with the followingrepresentation for the K µ matrix elements: h p ′ , h ′ | K µ | p, h i =(2 π ) δ (+)0 ( p ) (cid:20) − η h ′ ( p ′ ) ∂η h ∂p ′ µ ( p ′ ) δ ( p ′ − p ) + 2(∆ − δ h ′ h ∂∂p ′ µ δ ( p ′ − p )+ δ h ′ h (cid:18) p ′ µ ∂∂p ′ α ∂∂p ′ α − p ′ ν ∂∂p ′ ν ∂∂p ′ µ (cid:19) δ ( p ′ − p )+ (cid:18) p ′ µ η h ′ ( p ′ ) ∂η h ∂p ′ α ( p ′ ) ∂∂p ′ α − p ′ ν η h ′ ( p ′ ) ∂η h ∂p ′ ν ( p ′ ) ∂∂p ′ µ − p ′ ν η h ′ ( p ′ ) ∂η h ∂p ′ µ ( p ′ ) ∂∂p ′ ν (cid:19) δ ( p ′ − p )+ (cid:18) p ′ µ η h ′ ( p ′ ) ∂ η h ∂p ′ α ∂p ′ α ( p ′ ) − p ′ ν η h ′ ( p ′ ) ∂ η h ∂p ′ ν ∂p ′ µ ( p ′ ) (cid:19) δ ( p ′ − p ) − iη h ′ ( p ′ ) S µν ∂η h ∂p ′ ν ( p ′ ) δ ( p ′ − p ) − iη h ′ ( p ′ ) S µν η h ( p ′ ) ∂∂p ′ ν δ ( p ′ − p ) (cid:21) . (28)In contrast to the D matrix elements, switching coordinates to (¯ p, q ) in this expression is quitecomplicated due to the appearance of terms that involve more than one p ′ derivative. Never-theless, one can analyse the effect of this coordinate change on each of the non-trivial termsindividually, which is summarised in Appendix A. After applying Eqs. (66)-(69), together withthe GPT relations in Eqs. (70)-(72), one finally obtains: h p ′ , h ′ | K µ | p, h i =(2 π ) δ (+)0 (¯ p ) δ h ′ h (cid:18) ¯ p µ ∂∂q ν ∂∂q ν − p ν ∂∂q ν ∂∂q µ (cid:19) δ ( q ) + (2 π ) ¯ p µ p δ h ′ h ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) − − ∂ µ [ η h ′ ( p ′ ) η h ( p )] q =0 + ¯ p µ ∂ α ∂ α [ η h ′ ( p ′ ) η h ( p )] q =0 − p ν ∂ µ ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 + 2 i∂ ν [ η h ′ ( p ′ ) S µν η h ( p )] q =0 (cid:21) δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) − p µ ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 ∂∂q ν + 2¯ p ν ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 ∂∂q µ + 2¯ p ν ∂ µ [ η h ′ ( p ′ ) η h ( p )] q =0 ∂∂q ν − iη h ′ (¯ p ) S µν η h (¯ p ) ∂∂q ν + 2(∆ − δ h ′ h ∂∂q µ (cid:21) δ ( q ) . (29)Analogously to the case of dilations, one can now compare this expression to the matrix elementderived from the form factor decomposition in Eq. (7). Using the modified form for the SCT urrent in Eq. (17), the matrix element of K µ takes the form h p ′ , h ′ | K µ | p, h i = lim d → R →∞ Z d x f d,R ( x ) (2 x µ x α − g µα x ) e iq · x h p ′ , h ′ | T α (0) | p, h i = (2 π ) (cid:2) ∂ j ∂ j δ ( q ) h p ′ , h ′ | T µ (0) | p, h i − g µk ∂ j ∂ k δ ( q ) h p ′ , h ′ | T j (0) | p, h i (cid:3) , (30)where the inclusion of the test function f d,R ( x ) is again required in order ensure the convergenceof the matrix element. The x -dependent terms are absent from this expression due to thedefinition of f d,R ( x ). In this case, one needs to understand the action of double delta-derivativeterms on the GFF components in order to evaluate Eq. (30). This complicated action is givenby Eq. (74) of Appendix A. Applying this relation, together with Eq. (11), one finally obtainsthe GFF representation h p ′ , h ′ | K µ | p, h i =(2 π ) δ (+)0 (¯ p ) δ h ′ h (cid:18) ¯ p µ ∂∂q ν ∂∂q ν − p ν ∂∂q ν ∂∂q µ (cid:19) δ ( q ) A ( q ) + (2 π ) ¯ p µ p δ h ′ h ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) A ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) − − ∂ µ [ η h ′ ( p ′ ) η h ( p )] q =0 G ( q ) + ¯ p µ ∂ α ∂ α [ η h ′ ( p ′ ) η h ( p )] q =0 A ( q ) − p ν ∂ µ ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 A ( q ) + 2 i∂ ν [ η h ′ ( p ′ ) S µν η h ( p )] q =0 G ( q ) (cid:21) δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) − p µ ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 A ( q ) ∂∂q ν + 2¯ p ν ∂ ν [ η h ′ ( p ′ ) η h ( p )] q =0 A ( q ) ∂∂q µ + 2¯ p ν ∂ µ [ η h ′ ( p ′ ) η h ( p )] q =0 A ( q ) ∂∂q ν − iη h ′ (¯ p ) S µν η h (¯ p ) G ( q ) ∂∂q ν + 2(∆ − δ h ′ h G ( q ) ∂∂q µ (cid:21) δ ( q ) − (2 π ) δ (+)0 (¯ p ) δ h ′ h g µ p (cid:2) A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) (cid:3) δ ( q )+ (2 π ) δ (+)0 (¯ p ) δ h ′ h g µk ¯ p k p ) (cid:2) A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) (cid:3) δ ( q ) . (31)Equating the K µ matrix element representations in Eqs. (29) and Eq. (31) one is then led tothe following conditions: A ( q ) δ ( q ) = δ ( q ) , (32) A ( q ) ∂ µ δ ( q ) = ∂ µ δ ( q ) , (33) A ( q ) ∂ µ ∂ ν δ ( q ) = ∂ µ ∂ ν δ ( q ) , (34) G ( q ) δ ( q ) = δ ( q ) , (35) G ( q ) ∂ µ δ ( q ) = ∂ µ δ ( q ) , (36) (cid:2) A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) (cid:3) δ ( q ) = 0 , (37)which ultimately imply the GFF constraints: A (0) = G (0) = 1 , (38)( ∂ µ ∂ ν A )(0) = 0 , (39) A (0) − − G (0) + 6 C (0) − − T (0) = 0 . (40)So together with the constraints derived in Sec. 4.1 from dilational covariance, SCT covariancealso introduces an additional constraint on the second derivative of A ( q ), and implies that all f the GFFs are in fact related at q = 0 by a specific linear combination depending on theconformal dimension ∆. In Sec. 4.3.2 it will be demonstrated that Eq. (37), and hence Eq. (40),are in fact further strengthened by the assumption made in this section that the SCT currenthas the form in Eq. (17). The trace of the EMT plays an important role in the classification of CFTs. Since on-shell statesare necessarily massless in any CFT, one can use the parametrisation in Eq. (7) to determine thegeneral action of T µµ on these states. In order to keep these calculations as general as possiblewe will first assume that the EMT is symmetric, but not necessarily traceless. After explicitlytaking the trace in Eq. (7) one obtains h p ′ , h ′ | T µµ (0) | p, h i = η h ′ ( p ′ ) h − q A ( q ) + 2 i ¯ p µ S µρ q ρ G ( q ) − q C ( q )+ 2 S µα S βµ q α q β T ( q ) i η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) . (41)To simplify this expression further one can make use of the conformal GPT relation in Eq. (11).The coefficient of G ( q ) becomes:2 i η h ′ ( p ′ ) S µρ η h ( p )¯ p µ q ρ = − i η h ′ ( p ′ ) S µρ η h ( p ) p ′ µ p ρ = − p ′ · p )(∆ − η h ′ ( p ′ ) η h ( p ) = q (∆ − η h ′ ( p ′ ) η h ( p ) , (42)where the first equality follows from the anti-symmetry of S µρ . For the T ( q ) coefficient oneinstead obtains2 η h ′ ( p ′ ) S µα S βµ η h ( p ) q α q β = 2 η h ′ ( p ′ ) S µα S βµ η h ( p )( p α p β + p ′ α p ′ β − p ′ α p β − p α p ′ β ) . (43)Applying Eq. (11) one can see that the first term in Eq. (43) vanishes since: η h ′ ( p ′ ) S µα S βµ η h ( p ) p α p β = − i (∆ − η h ′ ( p ′ ) S µα η h ( p ) p α p µ = 0 . (44)Taking the dual of the second term it follows that this term similarly vanishes. For the thirdterm, one instead finds that: − η h ′ ( p ′ ) S µα S βµ η h ( p ) p ′ α p β = 2 i (∆ − η h ′ ( p ′ ) S µα η h ( p ) p ′ α p µ = − p ′ · p )(∆ − η h ′ ( p ′ ) η h ( p ) = q (∆ − η h ′ ( p ′ ) η h ( p ) . (45)Using the fact that the Lorentz generators in any field representation satisfy (cid:2) S µα , S νβ (cid:3) = i ( g µβ S αν + g αν S µβ − g µν S αβ − g αβ S µν ) , (46)and hence: S µα S βµ = S βµ S µα − i S αβ , the last term can then be written − η h ′ ( p ′ ) S µα S βµ η h ( p ) p α p ′ β = − η h ′ ( p ′ ) (cid:2) S βµ S µα − i S αβ (cid:3) η h ( p ) p α p ′ β = − η h ′ ( p ′ ) (cid:2) − iS βµ p µ p ′ β (∆ −
1) + 2(∆ − p ′ · p ) (cid:3) η h ( p )= q (cid:2) (∆ − + 2(∆ − (cid:3) η h ′ ( p ′ ) η h ( p ) . (47)After combining all of these results, the T ( q ) coefficient takes the form2 η h ′ ( p ′ ) S µα S βµ η h ( p ) q α q β = (cid:8) q (cid:2) (∆ − + 2(∆ − (cid:3) + q (∆ − (cid:9) η h ′ ( p ′ ) η h ( p )= 2∆(∆ − q η h ′ ( p ′ ) η h ( p ) . (48) nserting Eqs. (42) and (48) into Eq. (41), one finally obtains the following expression for thetrace matrix element: h p ′ , h ′ | T µµ (0) | p, h i = − q h A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) i η h ′ ( p ′ ) η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) . (49)Eq. (49) demonstrates an important structural feature of CFTs: although the GFFs in Eq. (7)have coefficients with different q dependencies, taking the trace results in an expression with anoverall q coefficient. The relevance of this feature will be discussed in more detail in Sec. 5. As established in Sec. 4.2, the requirement of SCT covariance implies that the correspondingGFFs are linearly related to one another at q = 0. In deriving this constraint we implicitlyassumed that the SCT current has the form in Eq. (17), and hence the EMT is both symmetricand traceless. We will now demonstrate that the tracelessness of the EMT leads to a strength-ening of the constraint in Eq. (40). By demanding that T µµ ( x ) = 0, it follows from Eq. (49)that q h A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) i δ (+)0 ( p ′ ) δ (+)0 ( p ) = 0 . (50)Switching to the variables (¯ p, q ), this implies the distributional equality (cid:20) (¯ p − q ) | ¯ p + q || ¯ p − q | − (cid:21) h A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) i = 0 . (51)Since the coefficient of this expression vanishes only at q = , the linear combination of formfactors has the general solution A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) = C δ ( q ) , (52)where C is some arbitrary distribution in q . However, in order for this expression to be com-patible with Eq. (37) it must be the case that C ≡
0, otherwise one would end up with theill-defined product: δ ( q ) δ ( q ). So by explicitly taking into account the tracelessness of theEMT, this implies that Eq. (40) is in fact a realisation of the more general condition: A ( q ) − − G ( q ) + 6 C ( q ) − − T ( q ) = 0 . (53)Eq. (53) together with Eqs. (38) and (39) collectively summarise the constraints imposed on theEMT matrix elements by Poincar´e and conformal symmetry. In the next section we will demon-strate, using explicit CFT examples, that these constraints are sufficient to completely specifythe form of these matrix elements. This is not necessarily surprising since it is well-known thatthe structural form of correlation functions in CFTs are fixed by the overall symmetry, andin particular, the closely related EMT three-point functions are determined by the conformalWard identities, as discussed in Sec. 1.Before outlining specific examples in the next section, we first draw attention to the obser-vation made in Sec. 4.1 that the x -polynomiality order of the conserved currents determineswhether certain GFFs can be constrained by the corresponding symmetry. This explains whythe dilational charge matrix elements only result in constraints on A ( q ) and G ( q ), whereasthe SCT charge can also constrain C ( q ) and T ( q ), both of which have coefficients involvingtwo powers of q . Since conformal symmetry is expected to completely constrain the structure f any matrix element, and the conformal currents involve at most two powers of x , this impliesthat local and covariant EMT matrix elements in CFTs can only ever contain form factors thathave coefficients with at most two powers of q , otherwise the conformal symmetry would not besufficient to fully constrain the matrix elements. As pointed out at the end of Sec. 2, in localQFTs it turns out that the masslessness of the states alone is actually sufficient to guaranteethat this is indeed the case. This further emphasises the close connection between the presenceof massless particles and the existence of conformal symmetry. We will explore this connectionin more detail in Sec. 5. The simplest example of a unitary CFT is that of a free massless scalar field φ . The states have h = 0, and the Lorentz generators appearing in Eq. (7) are trivial, hence the only form factorsthat can exist are A φ ( q ) and C φ ( q ). Eq. (53) therefore takes the form: A φ ( q ) + 6 C φ ( q ) = 0 . (54)Due to the absence of interactions, and the fact that q is the only dimensionful parameterin the theory, it follows that A φ ( q ) must be constant. Combining this with the constraint inEq. (38) , Eq. (54) immediately implies C φ ( q ) = − . (55)So the conformal symmetry, together with the masslessness of the states, completely fixes thematrix elements of the symmetric-traceless EMT of the scalar field. Another simple example of a unitary CFT is the free massless fermion ψ . Since the parametrisa-tion in Eq. (7) assumes the EMT is both P and T invariant, for consistency ψ must therefore bein the Dirac representation . In this case the states can have h = ± , and the only independentform factors are: A ψ ( q ), G ψ ( q ), and C ψ ( q ), hence Eq. (53) takes the form: A ψ ( q ) − − G ψ ( q ) + 6 C ψ ( q ) = 0 . (56)As in the scalar case: A ψ ( q ) = 1, but also the absence of interactions and Eq. (38) implies: G ψ ( q ) = 1. Combining these conditions with Eq. (56), it follows that: C ψ ( q ) = − (3 − , (57)where the last equality is due to the fact that the corresponding GPTs in Eq. (7) have ∆ = .So although the EMT matrix elements for h = ± states can potentially have more covariantstructures than those with h = 0, the conformal symmetry and masslessness of the states is stillsufficient to completely fix the form of the EMT matrix elements. We assumed for simplicity in Eq. (7) that the EMT is invariant under discrete symmetries. This requirementcould of course be loosened, which would result in more potential form factor structures, and enable one to analyseCFTs with fields in non P or T -symmetric representations, such as Weyl fermions. .4.3 Massless theories with | h | ≥ states As already discussed in Sec. 2, by virtue of the Weinberg-Witten Theorem, Eq. (7) can no longerhold for arbitrary states with | h | >
1. This does not mean that no parametrisation exists, onlythat for a unitary theory this parametrisation cannot be both local and covariant [25]. It isinteresting to note that this theorem does not explicitly rule out the possibility that Eq. (7)is satisfied for theories containing massless states with h = ±
1. A simple example is the the-ory of free photons. This CFT is constructed from the anti-symmetric tensor field F µν , whichby virtue of Eq. (10) satisfies the free Maxwell equations. Due to the Poincar´e Lemma it fol-lows that F µν cannot be fundamental, but instead must involve the derivative of another field: F µν = ∂ µ A ν − ∂ ν A µ . By treating the massless field A µ to be fundamental, the resulting theory isinvariant under gauge symmetry. However, an important consequence of this gauge symmetryis that it prevents A µ from being both local and Poincar´e covariant [33]. The correspondingEMT matrix elements of the free photon states must therefore necessarily either violate localityor covariance, and hence the parametrisation in Eq. (7) cannot hold in general.As is well known, in order to make sense of gauge theories one must either permit non-localand non-covariant fields, such as in Coulomb gauge, or perform a gauge-fixing that preserveslocality and covariance, but allows for the possibility of states with non-positive norm, likeGupta-Bleuler quantisation [33]. In the latter case, it turns out that one can in fact recover amanifestly local and covariant EMT decomposition which coincides with Eq. (7) for the physical photon states. The difference to the lower helicity examples is that although A A ( q ), G A ( q ), C A ( q ), and T A ( q ) are actually non-vanishing, the tracelessness of the EMT does not resultin the constraint in Eq. (53), since A µ is not a conformal field. In general, for massless fieldsthat create states with higher helicity ( | h | >
1) the Poincar´e Lemma equally applies, and hencesimilarly forces the introduction of non-covariant gauge-dependent fields [25]. In this sense, theexistence of massless particles with | h | ≥ | h | > Although the analysis in Sec. 4 implicitly assumes that the GFFs, and the constraints imposedupon them, correspond to those of the modified current T µν (ST) , the decomposition in Eq. (7)holds for any choice of symmetric EMT, T µν (S) . In what follows, we will use this expression tofurther explore the conditions under which the conformality property in Eq. (15) holds.Firstly, consider a unitary, local, Poincar´e covariant QFT with massless one-particle states, andthat given some choice of T µν (S) the matrix elements of these states satisfy: h p ′ , h ′ | T µ (S) µ (0) | p, h i = − q F ( q ) η h ′ ( p ′ ) η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) , (58)where F ( q ) is a local form factor. Due to translational covariance, it therefore follows that h p ′ , h ′ | T µ (S) µ ( x ) | p, h i = ∂ h e iq · x F ( q ) η h ′ ( p ′ ) η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) i . (59)Although there are several subtleties regarding whether or not this equation can be consistently nverted, we will make the assumption here that this is indeed the case, from which it follows F ( q ) η h ′ ( p ′ ) η h ( p ) δ (+)0 ( p ′ ) δ (+)0 ( p ) = h p ′ , h ′ | ( ∂ ) − T µ (S) µ (0) | p, h i . (60)Since by definition F ( q ) contains no non-local contributions, ( ∂ ) − T µ (S) µ ( x ) must act like astrictly local operator on the one-particle states, and hence: T µ (S) µ ( x ) = ∂ L ( x ) for some (non-unique) choice of scalar operator L ( x ). One can extend this argument to multi-particle states by making use of the action of translations U ( a ) on these states: U ( a ) | p , h ; p , h ; · · · p n , h n i = U ( a ) | p , h i U ( a ) | p , h i · · · U ( a ) | p n , h n i , (61)where the inner product of | p , h ; p , h ; · · · p n , h n i is constructed by taking the weighted sumof all possible products of one-particle inner products, with weight +1 or − U ( a ) = e iP · a , by acting with dda (cid:12)(cid:12) a =0 on Eq. (61) it follows thatthe multi-particle matrix elements of P µ , and hence the EMT, are fixed by the correspondingone-particle matrix elements. To summarise: if the trace condition in Eq. (58) holds for mass-less one-particle states, and one assumes that Eq. (59) is consistently invertible, it follows thatfor any massless multi-particle states, the EMT T µν (S) must satisfy the (unitary) conformalityproperty in Eq. (15).In general, given any theory with massless states, it follows from Eq. (41) that states with h = 0must in fact satisfy Eq. (58), where the corresponding form factor is defined F h =0 ( q ) = 12 (cid:2) A ( q ) + 6 C ( q ) (cid:3) . (62)The difference with h = 0 states is that one necessarily needs to understand how the Lorentzgenerators S µν act on massless GPTs η h ( p ) in order to evaluate the EMT trace matrix elements,in particular those with | h | ≤
1. It turns out though that for free irreducible massless fieldsΦ( x ) with these helicities , the fields satisfy: C Φ ∂ µ Φ( x ) = iS µν ∂ ν Φ( x ) , (63)and hence the corresponding GPTs obey the condition: C Φ p µ η h ( p ) = ip ν S µν η h ( p ) , (64)where C Φ is some constant which depends on the representation of the field. Since Eq. (64) hasthe same form as Eq. (11), and as outlined in Sec. 2, Eq. (7) only involves the GPTs of irre-ducible fields, one can perform an identical calculation to that in Sec. 4.3, similarly arriving atan expression for the one-particle matrix elements of T µ (S) µ with the structure of Eq. (58). Thesearguments demonstrate that Eq. (58) is actually a generic feature of unitary, local, Poincar´ecovariant QFTs with massless on-shell states .If one now assumes that a theory contains only massless multi-particle states, it therefore followsfrom the results outlined in this section, under the various assumptions, that Eq. (15) holds forall states, and hence the theory must be conformally invariant. Although the assumptions meritfurther investigation, this result sheds new light on the connection between the constraint ofhaving a purely massless particle spectrum, and the existence of conformal symmetry. We implicitly assume that such asymptotic massless states exist. The irreducible massless fields are precisely those with Lorentz representations ( m, , n ), or their direct sums. For massive states, one can immediately see that Eq. (58) is violated, since the leading order component inthe form factor expansion ¯ p { µ ¯ p ν } A ( q ) introduces an additional term 2 M A ( q ) to the trace matrix element, whichcannot be written in the local form q F ( q ). Conclusions
It is well-known that conformal symmetry imposes significant constraints on the structure ofconformal field theories (CFTs), in particular the correlation functions. In this work we inves-tigate four-dimensional unitary, local, and Poincar´e covariant CFTs, focussing on the analyticproperties of the energy-momentum tensor (EMT) and the corresponding on-shell matrix ele-ments. By adopting a parametrisation in terms of covariant multipoles of the Lorentz generators,we establish a local and covariant form factor decomposition of these matrix elements for statesof general helicity. Using this decomposition, we derive the explicit constraints imposed on theform factors due to conformal symmetry and the trace properties of the EMT, and demonstratewith specific CFT examples that they uniquely fix the form of the matrix elements. We also usethis decomposition to gain new insights into the conditions under which general unitary theoriesare conformal. Besides the applications outlined in this work, the matrix element decompositioncould also be used to shed light on other aspects of massless QFTs, such as model-independentconstraints like the averaged null energy condition [34–36], and conformal collider bounds [37].Although we have focussed here on the on-shell matrix elements of the EMT, which are pro-jections of the subset of three-point functions involving the EMT, the same covariant multipoleapproach is equally applicable to more general CFT correlation functions, and could enablehelicity-universal representations of these objects to be similarly derived.
Acknowledgements
This work was supported by the Agence Nationale de la Recherche under the Projects No. ANR-18-ERC1-0002 and No. ANR-16-CE31-0019. The authors would like to thank Peter Schweitzer,Guillaume Bossard, and Christoph Kopper for useful discussions. Distributional relations
To derive the various constraints in Sec. 4 it is necessary to change coordinates from ( p ′ , p ) to(¯ p, q ). In order to do so, one makes use of the following relations: δ (+)0 ( p ) p ′ µ ∂∂p ′ µ δ ( p ′ − p ) = − δ (+)0 (¯ p ) δ ( q ) + δ (+)0 (¯ p ) ¯ p µ ∂∂q µ δ ( q ) , (65) δ (+)0 ( p ) ∂∂p ′ µ δ ( p ′ − p ) = ¯ p µ p ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) + δ (+)0 (¯ p ) ∂∂q µ δ ( q ) , (66) δ (+)0 ( p ) p ′ α η h ′ ( p ′ ) ∂η h ∂p ′ ν ( p ′ ) ∂∂p ′ µ δ ( p ′ − p ) = η h ′ (¯ p ) ∂η h ∂ ¯ p ν (¯ p ) (cid:20) ¯ p µ ¯ p α p ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) + ¯ p α δ (+)0 (¯ p ) ∂∂q µ δ ( q ) − g µα δ (+)0 (¯ p ) δ ( q ) (cid:21) −
12 ¯ p α (cid:20) ∂η h ′ ∂ ¯ p µ (¯ p ) ∂η h ∂ ¯ p ν (¯ p ) + η h ′ (¯ p ) ∂ η h ∂ ¯ p ν ∂ ¯ p µ (¯ p ) (cid:21) δ (+)0 (¯ p ) δ ( q ) , (67) δ (+)0 ( p ) η h ′ ( p ′ ) S µν η h ( p ′ ) ∂∂p ′ ν δ ( p ′ − p ) = η h ′ (¯ p ) S µν η h (¯ p ) (cid:20) ¯ p ν p ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) + δ (+)0 (¯ p ) ∂∂q ν δ ( q ) (cid:21) − (cid:20) ∂η h ′ ∂ ¯ p ν (¯ p ) S µν η h (¯ p ) + η h ′ (¯ p ) S µν ∂η h ∂ ¯ p ν (¯ p ) (cid:21) δ (+)0 (¯ p ) δ ( q ) , (68) δ (+)0 ( p ) (cid:18) p ′ µ ∂∂p ′ ν ∂∂p ′ ν − p ′ ν ∂∂p ′ ν ∂∂p ′ µ (cid:19) δ ( p ′ − p ) = δ (+)0 (¯ p ) (cid:18) ¯ p µ ∂∂q ν ∂∂q ν − p ν ∂∂q ν ∂∂q µ (cid:19) δ ( q ) + 7¯ p µ p ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) + 6 δ (+)0 (¯ p ) ∂∂q µ δ ( q ) , (69) η h ′ (¯ p ) ∂η h ∂ ¯ p µ (¯ p ) = − ∂∂q µ [ η h ′ ( p ′ ) η h ( p )] q =0 , (70) ∂η h ′ ∂ ¯ p µ (¯ p ) ∂η h ∂ ¯ p ν (¯ p ) + ∂η h ′ ∂ ¯ p ν (¯ p ) ∂η h ∂ ¯ p µ (¯ p ) = − ∂∂q µ ∂∂q ν [ η h ′ ( p ′ ) η h ( p )] q =0 , (71) ∂η h ′ ∂ ¯ p ν (¯ p ) S µν η h (¯ p ) − η h ′ (¯ p ) S µν ∂η h ∂ ¯ p ν (¯ p ) = 2 ∂∂q ν [ η h ′ ( p ′ ) S µν η h ( p )] q =0 . (72)As opposed Eqs. (70)-(72), which follow immediately from the definition of the variables (¯ p, q ),Eqs. (65)-(69) are equalities between distributions, and so to derive them one needs to explicitlydetermine their action on test functions. Since the derivation of these various relations is rathersimilar, we will not repeat them all here but instead focus on proving Eq. (66). Integrating thisexpression with the test function f ( p ′ , p ) = ¯ f (¯ p, q ), and performing a change of variable, oneobtains Z d p ′ d p δ (+)0 ( p ) ∂∂p ′ µ δ ( p ′ − p ) f ( p ′ , p ) = 2 π Z d ¯ p d q δ (cid:16) ¯ p − q − q (¯ p − q ) (cid:17) q (¯ p − q ) ∂∂q µ δ ( q ) ¯ f (¯ p, q )= − π Z d ¯ p d q ∂∂q µ ¯ f (cid:0) ¯ p ⋆ , ¯ p , q (cid:1) q (¯ p − q ) δ ( q ) , here ¯ p ⋆ = q + q (¯ p − q ) . Since the test function now has both an explicit and implicitdependence on q , one must apply the chain rule in order to evaluate the derivative Z d p ′ d p δ (+)0 ( p ) ∂∂p ′ µ δ ( p ′ − p ) f ( p ′ , p )= − π Z d ¯ p " ∂∂q µ (cid:18) q (¯ p − q ) (cid:19) − ¯ f (cid:0) ¯ p ⋆ , ¯ p , q (cid:1) + 12 | ¯ p | d ¯ p ⋆ dq µ ∂ ¯ f (cid:0) ¯ p , ¯ p , q (cid:1) ∂ ¯ p + 12 | ¯ p | ∂ ¯ f (cid:0) ¯ p ⋆ , ¯ p , q (cid:1) ∂q µ q =0 = 2 π Z d ¯ p " g µk ¯ p k | ¯ p | ¯ f (¯ p, − ¯ p µ (2 | ¯ p | ) ∂ ¯ f (¯ p, ∂ ¯ p − | ¯ p | ∂ ¯ f (¯ p, ∂q µ ¯ p = | ¯ p | = Z d ¯ p d q " ¯ p µ p ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q ) + δ (+)0 (¯ p ) ∂∂q µ δ ( q ) ¯ f (¯ p, q ) , which proves the equality in Eq. (66).In both of the GFF constraint calculations one is required to evaluate the product of delta-derivatives with specific components of the EMT matrix element. This amounts to understand-ing how these delta-derivatives act on the coefficients F (¯ p, q ) of the various GFFs. Since thesecoefficients are continuous functions, one has the following identities: δ (+)0 ( p ′ ) δ (+)0 ( p ) F (¯ p, q ) ∂ j δ ( q ) =(2 π ) δ (+)0 (¯ p ) " F (¯ p, p (cid:18) ∂ j − ¯ p j ¯ p ∂ (cid:19) δ ( q ) − p (cid:18) ∂F∂q j − ¯ p j ¯ p ∂F∂q (cid:19) q =0 δ ( q ) , (73) δ (+)0 ( p ′ ) δ (+)0 ( p ) F (¯ p, q ) ∂ j ∂ k δ ( q ) =(2 π ) δ (+)0 (¯ p ) F (¯ p, (cid:20) − p ) (cid:18) ¯ p k ∂∂q j + ¯ p j ∂∂q k (cid:19) ∂∂q + ¯ p k ¯ p j p ) ∂∂q ∂∂q + 12¯ p ∂∂q k ∂∂q j (cid:21) q =0 δ ( q )+ (2 π ) F (¯ p, p ) (cid:20) g kj + ¯ p k ¯ p j (¯ p ) (cid:21) ∂∂ ¯ p δ (+)0 (¯ p ) δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) − ¯ p k p ) ∂ F∂q ∂q j − ¯ p j p ) ∂ F∂q ∂q k + ¯ p k ¯ p j p ) ∂ F∂q ∂q + 12¯ p ∂ F∂q k ∂q j (cid:21) q =0 δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) ¯ p k p ) ∂F∂q − p ∂F∂q k (cid:21) q =0 ∂ j δ ( q ) + (2 π ) δ (+)0 (¯ p ) (cid:20) ¯ p j p ) ∂F∂q − p ∂F∂q j (cid:21) q =0 ∂ k δ ( q )+ (2 π ) δ (+)0 (¯ p ) (cid:20) ¯ p k p ) ∂F∂q j + ¯ p j p ) ∂F∂q k − ¯ p k ¯ p j (¯ p ) ∂F∂q (cid:21) q =0 ∂ δ ( q ) . (74)Both of these relations are proven in a similar manner to Eq. (66), except in Eq. (74) one hasthe added complication of having two nested derivatives, which introduces a significant numberof additional contributions. References [1] D. Poland, S. Rychkov and A. Vichi,
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