Poincaré constraints on the gravitational form factors for massive states with arbitrary spin
aa r X i v : . [ h e p - t h ] M a y Poincar´e constraints on the gravitational form factorsfor massive states with arbitrary spin
Sabrina Cotogno ∗ , C´edric Lorc´e † , and Peter Lowdon ‡ CPHT, CNRS, Ecole polytechnique, IP Paris, F-91128 Palaiseau, France
Abstract
In this work we analyse the constraints imposed by Poincar´e symmetry on the gravitationalform factors appearing in the Lorentz decomposition of the energy-momentum tensor matrixelements for massive states with arbitrary spin. By adopting a distributional approach, weprove for the first time non-perturbatively that the zero momentum transfer limit of the leadingtwo form factors in the decomposition are completely independent of the spin of the states. Itturns out that these constraints arise due to the general Poincar´e transformation and on-shellproperties of the states, as opposed to the specific characteristics of the individual Poincar´egenerators themselves. By expressing these leading form factors in terms of generalised partondistributions, we subsequently derive the linear and angular momentum sum rules for stateswith arbitrary spin. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
The matrix elements of local operators are of central importance in characterising the non-perturbative structure of any quantum field theory (QFT). In the case of the energy-momentumtensor (EMT) these matrix elements encode a wide variety of different phenomena, from thequantum corrections which arise in the gravitational motion of particles, to the distribution ofmass and angular momentum within hadrons [1–6]. Although there is a significant breadth ofliterature on these objects, most of these studies have chosen to focus on particular cases wherethe states have lower spin (generally spin 0, , or 1 [2–11]) or correspond to specific particles,as opposed to analysing the constraints imposed for arbitrary states. Whilst this approach hasproven to be successful phenomenologically, it potentially risks obscuring the underlying prop-erties governing these constraints, preventing one from separating model-specific and generalQFT effects.An important feature of EMT matrix elements, like any other local matrix elements, is that theycan be decomposed into a series of Lorentz structures. The coefficients of these terms, known asthe gravitational form factors (GFFs), are constrained by the symmetry properties of the EMT,together with its conservation and the physical requirement that the states are on shell. Al-though this structure has been understood for many years, the subsequent form factor analyseshave generally contained technical difficulties such as in the handling of boundary terms and theconstruction of well-defined normalisable states, leading to incorrect conclusions, as discussedin detail in [12]. In [13] it was demonstrated in the spin- case that these difficulties can becircumvented by taking into account the distributional characteristics of the Poincar´e chargeoperators and matrix elements, avoiding the necessity to define the wave-packet structure of thephysical states themselves. These characteristics arise as a consequence of the fact that in localformulations of QFT fields are defined to be operator-valued distributions which satisfy a seriesof physically motivated axioms, including locality and relativistic covariance [14–16]. Since theseaxioms are assumed to hold independently of the coupling regime, this framework allows one toderive genuine non-perturbative constraints in a purely analytic manner. The main conclusionof [13] was that the zero momentum transfer limit of the leading two GFFs in the spin- EMTmatrix element decomposition are completely constrained by the Poincar´e transformation andon-shell properties of the states. This raises an important question: does this characteristic con-tinue to hold for higher spin states, and if so, how is this limit affected by the spin of the states?The main goal of this work will be to address this question. As a by-product, by relating theseleading GFFs to generalised parton distributions (GPDs), which can be in principle be accessedin processes such as deeply virtual Compton scattering (DVCS) [17–19], the generalisation ofthe well-known spin- sum rules can be analysed for arbitrary spin states.The remainder of this paper is structured as follows: in Sec. 2 we define the leading termswhich appear in the decomposition of the EMT matrix elements for massive states of arbitraryspin. Using this decomposition in Sec. 3 we then apply the procedure developed in [13] to theangular momentum and boost matrix elements, and outline the subsequent constraints on theGFFs. In Sec. 4 we generalise this approach to the covariant Lorentz generators, and discussthe implications of these results in Sec. 5. Finally, in Sec. 6 we conclude by summarising ourkey findings. Gravitational form factors for arbitrary spin states
In order to analyse the constraints imposed on the GFFs appearing in the decomposition ofthe EMT matrix elements for states of arbitrary spin, one must first outline how these statesare defined. Due to the distributional nature of quantised fields it follows that any definitemomentum eigenstate | p i is in fact a distributional -valued state [16]. Normalisable states | g i = R d p g ( p ) | p i are constructed by integrating | p i with test functions (or wave-packets) g ( p ), chosento belong to the space of Schwartz functions of fast decrease S ( R , ). As will be discussed later,this choice of test functions also plays an important role in the definition of charge operators.For the purposes of the analysis in this paper we will be concerned only with massive physicalon-shell states. Since | p i is a priori defined for any four-momentum p ∈ R , , one can imposethis requirement by considering eigenstates which are restricted to the upper hyperboloid Γ + M = { p ∈ R , : p = M , p > } as follows | p, m ; M i = δ (+) M ( p ) | p, m i ≡ π θ ( p ) δ ( p − M ) | p, m i , (1)where M is the mass of the state and m is the canonical spin projection in the z -direction. Asa result, even if the test function g ( p ) has support outside of the mass shell, the normalisablestate | g i satisfies the mass shell constraint. Since the norm of the unrestricted eigenstate | p, m i is given by h p ′ , m ′ | p, m i = 2 p (2 π ) δ ( p ′ − p ) δ m ′ m , the above definition implies that the innerproduct of the on-shell states has the following Lorentz-covariant form h p ′ , m ′ ; M | p, m ; M i = (2 π ) δ ( p ′ − p ) δ (+) M ( p ) δ m ′ m . (2)Now it remains to parametrise the EMT matrix elements with respect to these states. Takingthe EMT operator T µν to be symmetric it follows from the conservation of this current, togetherwith the Lorentz covariance and discrete space-time symmetries, that the matrix elements forarbitrary spin can be written [20] h p ′ , m ′ ; M | T µν (0) | p, m ; M i = η m ′ ( p ′ ) O µν ( p ′ , p ) η m ( p ) δ (+) M ( p ′ ) δ (+) M ( p ) , (3)with the Lorentz covariant factor O µν ( p ′ , p ) = ¯ p { µ ¯ p ν } A ( q ) + i ¯ p { µ S ν } ρ q ρ G ( q ) + · · · (4)The · · · indicates contributions with an explicitly higher-order dependence on the four-momentumtransfer q = p ′ − p . We define the average four-momentum: ¯ p = ( p ′ + p ) and the symmetrisation: a { µ b ν } = a µ b ν + a ν b µ . S µν are the Lorentz generators in the chosen spin representation and η m ( p )are the arbitrary spin generalisation of the spinor and polarisation vector in the half-odd andhalf-even spin cases respectively. In particular, in the spin- case one has that: S µν = i [ γ µ , γ ν ], η m ( p ) ∝ u m ( p ), and Eq. (3) agrees with the well-known matrix element parametrisation ofthe nucleon EMT [21]. The parametrisation (3) also assumes that the covariant density matrix[ ρ m ′ m ( p )] AB = [ η m ( p )] A [ η m ′ ( p )] B has the mass-independent normalisationTr[ ρ m ′ m ( p )] = [ η m ( p )] A [ η m ′ ( p )] A = η m ′ ( p ) η m ( p ) = δ m ′ m . (5)Note that the trace is performed in the spin representation space only. A characteristic featureof Eq. (3) is that the arbitrary spin η m ( p ) appear in a purely external manner, and that thecomplexity of this expression is determined by the possible combinations of contracting ¯ p µ and S µν with q µ , whilst respecting the conservation and symmetry of T µν . Eq. (3) also makes it Here we have chosen to define a single form factor G ( q ) for the component involving the Lorentz generator, so G ( q ) = A ( q ) + B ( q ) in comparison with [13] for the spin- case. anifest that the non-perturbative structure is completely encoded in the GFFs.Now that the structure of the EMT matrix elements for arbitrary spin has been determined, inthe proceeding sections we will apply an analogous approach to [13] in order to derive constraintson the GFFs A ( q ) and G ( q ). In [20] these constraints were outlined using a perturbative grav-itational approach together with the Rarita-Schwinger representation. Two other derivationswere also proposed in [22]: one based on the expansion of the EMT in momentum space , andanother using Schwinger’s multispinor formalism together with a non-relativistic expansion. Incontrast to these former works the proof we provide is purely non-perturbative, and relies onlyon the Poincar´e invariance of the QFT and the distributional properties of the matrix elements.Moreover, this approach properly takes into account Wigner rotation effects, and does not re-quire one to consider a non-relativistic expansion or a particular massive spin representation ofthe states. In [13] it was first demonstrated that one can derive constraints on the GFFs by performinga distributional matching procedure. This procedure involves comparing the parametrisationof the matrix elements of the Poincar´e charges with the representation that results from theexplicit action of these charges on the states. Due to the distributional nature of the Poincar´ecurrents, a rigorous definition of the corresponding charges requires integration with a sequenceof appropriate test functions. As will be emphasised in the calculations that follow, taking intoaccount the subtleties of these charge definitions is essential for obtaining consistent form factorconstraints. A more detailed discussion of the motivation behind the various charge definitionscan be found in [13] and references within.
Let us start with the angular momentum operator J i . Its rigorous definition reads J i = 12 ǫ ijk lim d → R →∞ Z d x f d,R ( x ) (cid:2) x j T k ( x ) − x k T j ( x ) (cid:3) , (6)where f d,R ( x ) ≡ α d ( x ) F R ( x ) ∈ S ( R , ), and the test functions α d , F R satisfy the conditions Z d x α d ( x ) = 1 , α d ( x ) d → −−−→ δ ( x ) , (7) F R ( ) = 1 , F R ( x ) R →∞ −−−−→ . (8)This definition guarantees that J i is convergent within matrix elements, and also independent ofthe specific choice of test functions used in the limit . Using this definition and a translation ofthe EMT operator T µν ( x ) = e iP · x T µν (0) e − iP · x , it follows that the angular momentum matrix We observe that this first derivation contains a loophole since a possible contribution of the type ¯ p µ ¯ p ν J ρ q ρ ,allowed owing to Wigner rotation effects [23], has not been considered. This is in line with [12], where the fallacy ofthe expansion used in [22] (and also later in [24]) was pointed out. The independence of the choice of temporal test function α d can in particular be interpreted as the quantumgeneralisation of the time independence of the charge [25]. lement between the states | p, m ; M i can be written h p ′ , m ′ ; M | J i | p, m ; M i = ǫ ijk lim d → R →∞ Z d x f d,R ( x ) x j e iq · x h p ′ , m ′ ; M | T k (0) | p, m ; M i = − iǫ ijk lim d → R →∞ ∂ e f d,R ( q ) ∂q j h p ′ , m ′ ; M | T k (0) | p, m ; M i , (9)with e f d,R ( q ) = R d x e iq · x f d,R ( x ). From the conditions in Eqs. (7) and (8) one has thatlim d → R →∞ e f d,R ( q ) = (2 π ) δ ( q ) , (10)which due to Eq. (9) implies that one must determine the product of derivatives of delta ∂ j δ ( q ) = ∂∂q j δ ( q ) and other factors in order to evaluate the full matrix element. The generalform for this type of covariant distributional expression is derived in Appendix A. In particular,using the parametrisation in Eq. (3) together with Eq. (63) it follows that Eq. (9) can be written h p ′ , m ′ ; M | J i | p, m ; M i = (2 π ) δ (+) M (¯ p ) J im ′ m (¯ p, q ) , (11)with the reduced matrix element J im ′ m (¯ p, q ) = − iǫ ijk ¯ p k h δ m ′ m ∂ j δ ( q ) − ∂ j [ η m ′ ( p ′ ) η m ( p )] (cid:12)(cid:12) q =0 δ ( q ) i A ( q )+ 12 ǫ ijk (cid:2) η m ′ (¯ p ) S jk η m (¯ p ) (cid:3) δ ( q ) G ( q ) . (12)It is important to note that the temporal derivative of δ ( q ) which can potentially appear dueto Eq. (63) drops out of this expression due to the contraction with ǫ ijk ¯ p k . In order to furthersimplify this expression one needs to evaluate the derivative term for arbitrary spin. As provedin Appendix B, it turns out that one has the following closed form expression ∂∂q j [ η m ′ ( p ′ ) η m ( p )] (cid:12)(cid:12)(cid:12)(cid:12) q =0 = i | ¯ p | ǫ jlr ¯ p l [Σ rm ′ m (¯ p ) − Σ rm ′ m ( k )] , (13)where k µ = M g µ is rest-frame four-momentum, and Σ im ′ m (¯ p ) = Tr (cid:2) ρ m ′ m (¯ p )Σ i (cid:3) with Σ i = ǫ ijk S jk the spin matrices in the chosen spin representation. Inserting Eq. (13) into Eq. (12)then gives J im ′ m (¯ p, q ) = (cid:2) Σ im ′ m ( k ) − δ m ′ m iǫ ijk ¯ p k ∂ j (cid:3) δ ( q ) A ( q )+ Σ im ′ m (¯ p ) δ ( q ) (cid:2) G ( q ) − A ( q ) (cid:3) , (14)where we used the fact that helicity (i.e. spin projection along momentum) is invariant underlongitudinal boosts: ˆ¯ p · Σ m ′ m (¯ p ) = ˆ¯ p · Σ m ′ m ( k ), with ˆ¯ p = ¯ p / | ¯ p | .To derive constraints on A ( q ) and G ( q ) one can observe that due to the transformationproperties of the states | p, m ; M i under rotations, the J i reduced matrix elements must havethe general form J im ′ m (¯ p, q ) = (cid:2) Σ im ′ m ( k ) − δ m ′ m iǫ ijk ¯ p k ∂ j (cid:3) δ ( q ) , (15)which is derived in Appendix A. Since Eqs. (14) and (15) are simply different representationsof the same matrix element, the coefficients of these distributions must coincide, which requires hat the following identities hold A ( q ) δ ( q ) = δ ( q ) , (16) A ( q ) ∂ j δ ( q ) = ∂ j δ ( q ) , (17) (cid:2) G ( q ) − A ( q ) (cid:3) δ ( q ) = 0 . (18)Combining these identities implies the constraint A (0) = G (0) = 1 , (19)which proves that the q → Another important point which was raised in [13] is that the constraints on the GFFs are notspecifically related to the conservation of angular momentum, contrary to what is often thoughtin the literature. To emphasise this point it was demonstrated (for spin ) that identical formfactor constraints can also be obtained using the matrix elements of the boost generators K i .Since the calculations in the preceding section concluded that A (0) = G (0) = 1 is a spin-independent constraint, one would therefore expect that the same constraint must also arisefrom the structure of h p ′ , m ′ ; M | K i | p, m ; M i . It turns out that this is in fact the case, as willbe demonstrated in the remainder of this section.Similarly to J i the boost generator is rigorously defined by K i = lim d → R →∞ Z d x f d,R ( x ) (cid:2) x T i ( x ) − x i T ( x ) (cid:3) , (20)and hence the boost matrix element can be written in the form h p ′ , m ′ ; M | K i | p, m ; M i = i lim d → R →∞ ∂ e f d,R ( q ) ∂q i h p ′ ; m ′ ; M | T (0) | p ; m ; M i . (21)The term proportional to h p ′ , m ′ ; M | T i (0) | p, m ; M i vanishes due to the definition of the testfunctions in Eqs. (7) and (8) lim d → R →∞ Z d x e iq · x x α d ( x ) F R ( x ) = 0 . (22)Inserting the parametrisation (3) and using Eq. (63) one can write h p ′ , m ′ ; M | K i | p, m ; M i = (2 π ) δ (+) M (¯ p ) K im ′ m (¯ p, q ) , (23)with the reduced matrix element K im ′ m (¯ p, q ) = i h δ m ′ m (¯ p ∂ i − ¯ p i ∂ ) δ ( q ) − ¯ p ∂ i [ η m ′ ( p ′ ) η m ( p )] (cid:12)(cid:12) q =0 δ ( q ) i A ( q )+ (cid:2) η m ′ (¯ p ) S i η m (¯ p ) (cid:3) δ ( q ) G ( q ) . (24) Technically A ( q ) and G ( q ) are distributions in q , and so are in general not point-wise defined. Nevertheless,one can interpret A (0) and G (0) using a limiting procedure [13]. ue to Eq. (13) one sees that in order to further simplify this relation one requires an explicitexpression for ǫ ijk ¯ p j ¯ p Σ km ′ m (¯ p ). As shown in Appendix B, due to the properties of the covariantdensity matrix ρ m ′ m (¯ p ) one can prove that ǫ ijk ¯ p j ¯ p Σ km ′ m (¯ p ) = −| ¯ p | κ im ′ m (¯ p ) + M ǫ ijk ¯ p j Σ km ′ m ( k ) , (25)where κ im ′ m (¯ p ) = Tr (cid:2) ρ m ′ m (¯ p ) κ i (cid:3) with κ i = S i the boost generator matrices in the chosen spinrepresentation. Upon insertion into Eq. (24) this finally gives K im ′ m (¯ p, q ) = (cid:20) − ǫ ijk ¯ p j ¯ p + M Σ km ′ m ( k ) + δ m ′ m i (¯ p ∂ i − ¯ p i ∂ ) (cid:21) δ ( q ) A ( q )+ κ im ′ m (¯ p ) δ ( q ) (cid:2) G ( q ) − A ( q ) (cid:3) . (26)Just as the rotation transformation properties of the states | p, m ; M i were used to constrain thematrix elements of J i , one can perform an analogous procedure for boosts. In this case the K i reduced matrix elements have the general form K im ′ m (¯ p, q ) = (cid:20) − ǫ ijk ¯ p j ¯ p + M Σ km ′ m ( k ) + δ m ′ m i (¯ p ∂ i − ¯ p i ∂ ) (cid:21) δ ( q ) , (27)which is derived in Appendix A. Comparing this with Eq. (26) one immediately sees that theequality of these expressions implies the same relations as in Eqs. (16), (17) and (18). As antic-ipated, this result emphasises that the form factor constraints are not specific to the propertiesof any one Lorentz generator. In the next section, we will instead consider the constraints im-posed on A ( q ) and G ( q ) by the covariant Poincar´e generators: the four-momentum operator P µ , and the covariant generalisations of J i and K i . Before discussing the covariant generalisation of the rotation and boost operators, consider thesimplest case of the four-momentum operator P µ . Although P µ does not involve an explicitfactor of x α in its definition, P µ is nevertheless defined by smearing with the same class of testfunctions as the Lorentz generators P µ = lim d → R →∞ Z d x f d,R ( x ) T µ ( x ) (28)and hence the P µ matrix element can be written as h p, m ′ ; M | P µ | p, m ; M i = lim d → R →∞ e f d,R ( q ) h p ′ , m ′ ; M | T µ (0) | p, m ; M i . (29)Inserting the form factor decomposition of Eq. (3) and applying Eq. (58) it immediately followsthat h ¯ p + q, m ′ ; M | P µ | ¯ p − q, m ; M i = (2 π ) δ (+) M (¯ p ) ¯ p µ δ m ′ m A ( q ) δ ( q ) , (30)where the G ( q )-dependent terms have dropped out due to the explicit q factor. Since theon-shell states are defined to have an inner product as in Eq. (2), and | p, m ; M i are momentumeigenstates satisfying: P µ | p, m ; M i = p µ | p, m ; M i , these relations therefore imply the spin-independent constraint A ( q ) δ ( q ) = δ ( q ) , (31)which is simply A (0) = 1. Due to the non-vanishing ∂ δ ( q ) term one also has the constraint: A ( q ) ∂ δ ( q ) = ∂ δ ( q ). However, thisrelation is essentially trivial because it implies ∂ A (0) = 0, which follows immediately from the fact that A ( q )depends only on q . .1 Pauli-Lubanski matrix element The covariant generalisation of J i , the Pauli-Lubanski operator W µ , is defined by W µ = 12 ǫ µρσλ M ρσ P λ . (32)By definition, the rest-frame matrix element of W µ coincides with J i , up to an overall masscoefficient. Before calculating the matrix element of W µ one must first define the general Lorentzgenerator M µν . Similarly to J i and K i one has that M µν = lim d → R →∞ Z d x f d,R ( x ) (cid:2) x µ T ν ( x ) − x ν T µ ( x ) (cid:3) . (33)The matrix element of W µ can then be written h p ′ , m ′ ; M | W µ | p, m ; M i = − iǫ µρσλ p λ lim d → R →∞ ∂ e f d,R ( q ) ∂q ρ h p ′ , m ′ ; M | T σ (0) | p, m ; M i , (34)which after inserting the parametrisation (3) and applying Eq. (63) gives h p ′ , m ′ ; M | W µ | p, m ; M i = (2 π ) δ (+) M (¯ p ) W µm ′ m (¯ p, q ) , (35)with the reduced matrix element W µm ′ m (¯ p, q ) = S µm ′ m (¯ p ) δ ( q ) G ( q ) , (36)where S µm ′ m (¯ p ) = Tr [ ρ m ′ m (¯ p ) S µ ] with S µ = ǫ µρσλ S ρσ ¯ p λ the covariant spin matrices in thechosen spin representation. The dependence on A ( q ) completely drops out due to the con-traction with ǫ µρσλ ¯ p λ and the explicit ¯ p σ factor multiplying this term. Unlike the rotationand boost generators the Pauli-Lubanski operator acts in a diagonal manner on the momentumcomponent of the states, and so the reduced matrix elements have the general form W µm ′ m (¯ p, q ) = S µm ′ m (¯ p ) δ ( q ) . (37)Equating Eqs. (36) and (37) immediately implies the constraint G ( q ) δ ( q ) = δ ( q ) , (38)which is nothing more than the condition G (0) = 1. The covariant boost B µ is defined by the symmetrised expression B µ = 12 [ M νµ P ν + P ν M νµ ] , (39)and coincides with K i within matrix elements of rest-frame states. The general matrix elementsof B µ can be directly related to those of the rotation and boost operators , and in particularthe corresponding reduced matrix elements B µm ′ m (¯ p, q ) are given by B m ′ m (¯ p, q ) = ¯ p i K im ′ m (¯ p, q ) , (40) B im ′ m (¯ p, q ) = ¯ p K im ′ m (¯ p, q ) + ǫ ijk ¯ p j J km ′ m (¯ p, q ) . (41) Here we use the convention ǫ = +1. The explicit form for the covariant spin matrices in terms of the non-conserved ones Σ im ′ m ( k ) is given by: S µm ′ m ( p ) = (cid:16) p · Σ m ′ m ( k ) , M Σ m ′ m ( k ) + p · Σ m ′ m ( k ) p + M p (cid:17) . Given these definitions of W µ and B µ the general Lorentz generator can be written in the following form: M µν = − P [ { B µ , P ν } − { B ν , P µ } ] − P ǫ µναβ W α P β , where {· , ·} is the anti-commutator. sing Eq. (26) it follows that Eq. (40) can be written B m ′ m (¯ p, q ) = iδ m ′ m ¯ p i (cid:2) ¯ p ∂ i − ¯ p i ∂ (cid:3) δ ( q ) A ( q ) + ¯ p i κ im ′ m (¯ p ) δ ( q ) (cid:2) G ( q ) − A ( q ) (cid:3) = iδ m ′ m ¯ p i (cid:2) ¯ p ∂ i − ¯ p i ∂ (cid:3) δ ( q ) A ( q ) , (42)where the last line follows from the fact that: ¯ p i κ im ′ m (¯ p ) = ¯ p i κ im ′ m ( k ) = 0. Comparing this withthe general boost matrix element representation in Eq. (27) projected on ¯ p therefore implies theconstraints A ( q ) ∂ j δ ( q ) = ∂ j δ ( q ) , A ( q ) ∂ δ ( q ) = ∂ δ ( q ) . (43)Similarly, computing B im ′ m (¯ p, q ) one obtains B im ′ m (¯ p, q ) = M ǫ ijk ¯ p j ¯ p + M Σ km ′ m ( k ) δ ( q ) A ( q ) + δ m ′ m i (cid:2) ¯ p ∂ i − ¯ p i (¯ p · ∂ ) (cid:3) δ ( q ) A ( q )+ (cid:2) ǫ ijk ¯ p j Σ km ′ m (¯ p ) + ¯ p κ im ′ m (¯ p ) (cid:3) δ ( q ) (cid:2) G ( q ) − A ( q ) (cid:3) = M ǫ ijk ¯ p j ¯ p + M Σ km ′ m ( k ) δ ( q ) A ( q ) + δ m ′ m i (cid:2) ¯ p ∂ i − ¯ p i (¯ p · ∂ ) (cid:3) δ ( q ) A ( q ) , (44)where the last line follows from Eq. (88) derived in Appendix B. Comparing this with both thegeneral Lorentz generator matrix elements in Eqs. (15) and (27) one is left with the constraintsin Eq. (43), together with the condition: A ( q ) δ ( q ) = δ ( q ).The calculations in this section explicitly demonstrate that the matrix elements of the covari-antised rotation and boost operators separately determine the constraints on G ( q ) and A ( q )respectively. In other words, choosing this covariant operator basis results in a diagonalisationof the constraints. Overall, it initially appears that the matrix elements of the Lorentz gener-ators, or their covariantised versions, are sufficient to derive all of the form factor constraints.Since these constraints follow from the Lorentz transformations properties of the states, thisseemingly suggests that only Lorentz symmetry is involved. However, in deriving the ma-trix element equations we have also implicitly used the spacetime translation transformation: e iP · x | p, m ; M i = e ip · x | p, m ; M i . This explains why the condition A ( q ) δ ( q ) = δ ( q ), whichfollows from the matrix element of P µ , is also implied by the matrix elements of the variousLorentz generators. Ultimately this means that the total constraints on the GFFs are a resultof the full Poincar´e symmetry, together with the on-shell restriction of the states.
We now turn to the phenomenological implications of our results, focussing specifically on theapplications to hadronic physics. The quantum interactions between matter and gravity are inprinciple encoded in the GFFs, but in practice they are too weak to be directly measured inexperiment. One way of accessing information about QCD matter is through the generalisedparton distributions (GPDs) [17–19, 21]. In this case one is dealing with a non-local operatoralong the light-like direction n , which enters into the description of deeply virtual Comptonscattering (DVCS) at the amplitude level. The leading-twist quark and gluon GPDs have thefollowing form: V qm ′ m = 12 Z ∞−∞ d z π e ix (¯ p · n ) z D p ′ , m ′ ; M (cid:12)(cid:12)(cid:12) ψ (cid:0) − z n (cid:1) ( γ · n ) W [ − z n, z n ] ψ (cid:0) z n (cid:1)(cid:12)(cid:12)(cid:12) p, m ; M E , (45) V gm ′ m = n α n β x (¯ p · n ) Z ∞−∞ d z π e ix (¯ p · n ) z D p ′ , m ′ ; M (cid:12)(cid:12)(cid:12) F λα (cid:0) − z n (cid:1) W [ − z n, z n ] F λβ (cid:0) z n (cid:1)(cid:12)(cid:12)(cid:12) p, m ; M E , (46) here x is the longitudinal momentum fraction of the parton, and W [ a,b ] denotes a straightWilson line in the adjoint representation joining the spacetime points a and b . The non-localquark and gluon operators which appear within the matrix elements of these definitions ( O qV and O gV ) are related to the quark and gluon EMT operators via the second Mellin moment Z − d x x O qV = 14(¯ p · n ) ψ (0)( γ · n )( i ↔ D · n ) ψ (0) = n µ n ν T µνq p · n ) , (47) Z − d x x O gV = n µ n ν p · n ) F µλ (0) F νλ (0) = n µ n ν T µνg p · n ) , (48)with ↔ D µ = → D µ − ← D µ . Eq. (3) is the most general decomposition for the symmetric and conservedEMT, and the terms we are interested in depend at most linearly on q . To spell out the relationbetween the GFFs in Eq. (3) and the Mellin moments of GPDs we need to restrict ourselves tothe twist-2 part of Eqs. (47) and (48). In particular, neglecting terms with a higher power of q ,the total GPD correlator at twist-2 reads h p ′ , m ′ ; M |O V | p, m ; M i = η m ′ ( p ′ ) (cid:20) H ( x, ξ, t ) + iS αρ n α q ρ ¯ p · n H ( x, ξ, t ) + · · · (cid:21) η m ( p ) δ (+) M ( p ) δ (+) M ( p ′ ) , (49)where ξ = − ( q · n ) / (2¯ p · n ) is the light-front longitudinal momentum transfer, t = q , and O V = O qV + O gV . It follows that one can write the spin-independent relations Z − d x xH ( x, ξ, q ) = A ( q ) + · · · , (50) Z − d x xH ( x, ξ, q ) = G ( q ) + · · · , (51)where · · · denotes possible contributions arising from non-leading GFFs which are multipliedat least by ξ . Using the results derived in this paper one can now generalise Ji’s sum rule [21]such that it holds independently of both the spin and structure of the hadron states. FromEq. (19) it follows that for a state of arbitrary spin with longitudinal polarisation along, say,the z -direction, the total longitudinal linear and angular momentum (summed over quarks andgluons) reads: P z = X a = q,g Z − d x xH a ( x, ,
0) = A (0) = 1 , (52) J z = X a = q,g Z − d x xH a ( x, ,
0) = G (0) = 1 . (53)The totality of the structures that parametrise the EMT cannot be constrained by the actionof the Poincar´e generators alone, and in general contains both asymmetric and non-conservedterms. These terms are crucial in the study of the mechanical properties of hadrons, and receivedifferent contributions from quarks and gluons. In particular, a general expression for Ji’s rela-tion which is valid for quarks and gluons separately would require the inclusion of such additionalterms, as observed in [10, 11]. One approach to derive these terms is to write a parametrisationof the EMT for arbitrary spin states as an expansion in terms of spin multipoles . See [10] for a discussion of the spin-1 case, and [26] for a parametrisation of the vector current case for arbitraryspin. esides the hadronic relevance of the form factor constraints derived in this work, one can alsointerpret these conditions in a gravitational context. In particular, if one considers the situationin which the states correspond to a particle moving in an external (classical) gravitational field,the zero momentum transfer limit of the form factor B ( q ) = G ( q ) − A ( q ) has been argued tocorrespond to the anomalous gravitomagnetic moment (AGM) of the particle [27], by analogyto the case of the anomalous magnetic moment of a charged particle. Due to the constraint inEq. (19) it follows immediately that B (0) = 0, and hence with this interpretation the AGM mustvanish for massive particles of any spin. However, as previously outlined, this constraint arisespurely from the Poincar´e invariance of the theory, and does not in fact rely on any knowledgeof the external gravitational interactions . Einstein’s equivalence principle is therefore notnecessary to derive the constraint B (0) = 0. The purpose of this work was to establish the most general constraints imposed on the formfactors appearing in the Lorentz decomposition of the energy-momentum tensor matrix elementsfor massive states with arbitrary spin. By comparing the form factor representation of the an-gular momentum matrix elements with the representation due to the transformation propertiesof the states under rotations, we were able to prove that the q → A ( q ) and G ( q ) is completely independent of both the spin and internal structureof the states, and in particular that: A (0) = G (0) = 1. Adopting an analogous procedure forthe matrix elements of the boost generators K i , we also established that the structure of theseobjects implies identical constraints to those derived using J i . Together, these results emphasisethat the constraints imposed on the leading gravitational form factors are not specifically relatedto the properties of any one of the Lorentz generators. Besides the standard Lorentz generatorsone can also use the covariantised version of these operators, the Pauli-Lubanski W µ and covari-ant boost generator B µ , to derive constraints in the same manner. It turns out that B µ and W µ separately imply A (0) = 1 and G (0) = 1 respectively. In other words, choosing this covariantoperator basis results in a diagonalisation of the constraints. The main conclusion from thisanalysis is that the spin-independent constraints on A ( q ) and G ( q ) are non-perturbative, andarise purely due to the general Poincar´e transformation and on-shell properties of the states.These results have several immediate implications, including the spin-universality of Ji’s sumrule for generalised parton distributions, and the vanishing of the anomalous gravitomagneticmoment for particles of any spin. Acknowledgements
This work was supported by the Agence Nationale de la Recherche under the projects No.ANR-18-ERC1-0002 and ANR-16-CE31-0019.
A On-shell matrix elements of the Poincar´e generators
A.1 Covariant representation
The simplest on-shell matrix elements occur when calculating the matrix elements of the four-momentum operator. In this case one has distributional relations of the following form T ( p ′ , p ) = δ (+) M ( p ′ ) δ (+) M ( p ) C ( p ′ , p ) δ ( p ′ − p ) , (54) Although the conditon B (0) = 0 for arbitrary spin states has been discussed before [20, 22, 27], until now thisstatement has not been proven in a non-perturbative manner. here C ( p ′ , p ) is some function. In particular, C ( p ′ , p ) corresponds to the coefficients multiplyingthe form factors in Eq. (3). As with any distribution, the key to simplifying Eq. (54) is tounderstand how it acts on a generic test function f . For the purposes of the form factor analysisin this paper we are mainly interested in working with the variables ¯ p = ( p ′ + p ) and q = p ′ − p .In these variables one can write the smeared distribution T (¯ p, q ) = T ( p ′ , p ) in the followingmanner Z d ¯ p d q T (¯ p, q ) f (¯ p, q ) = Z d ¯ p d q δ (+) M (¯ p + q ) δ (+) M (¯ p − q ) C (¯ p, q ) δ ( q ) f (¯ p, q )= Z d ¯ p d q δ (cid:0) ¯ p − ¯ p ⋆ (cid:1) δ (cid:0) q − q ⋆ (cid:1) p + q )(¯ p − q ) C (¯ p, q ) δ ( q ) f (¯ p, q )= Z d ¯ p C (¯ p ⋆ , ¯ p , q ⋆ , q ) f (¯ p ⋆ , ¯ p , q ⋆ , q )4 q (¯ p + q ) + M q (¯ p − q ) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = = Z d ¯ p C ( E ¯ p , ¯ p , , ) f ( E ¯ p , ¯ p , , )(2 E ¯ p ) , (55)where C (¯ p, q ) = C ( p ′ , p ) and one has used that¯ p ⋆ = 12 (cid:20)q (¯ p + q ) + M + q (¯ p − q ) + M (cid:21) , (56) q ⋆ = q (¯ p + q ) + M − q (¯ p − q ) + M , (57)which implies: ¯ p ⋆ (cid:12)(cid:12) q = = p ¯ p + M = E ¯ p and q ⋆ (cid:12)(cid:12) q = = 0. On the level of distributionsEq. (55) implies that the matrix element T can be explicitly written T (¯ p, q ) = 2 π δ (+) M (¯ p ) C (¯ p, p δ ( q ) . (58)The calculation of the rotation and boost generator matrix elements instead requires one toevaluate more complicated distributional relations of the form T j ( p ′ , p ) = δ (+) M ( p ′ ) δ (+) M ( p ) C ( p ′ , p ) ∂∂p j δ ( p ′ − p ) , (59)Performing an identical procedure as before, and applying the definition of the distributionalderivative [28], the smeared distribution T j (¯ p, q ) = T j ( p ′ , p ) is given by Z d ¯ p ∂∂q j C (¯ p ⋆ , ¯ p , q ⋆ , q ) f (¯ p ⋆ , ¯ p , q ⋆ , q )4 q (¯ p + q ) + M q (¯ p − q ) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = (60)Differentiating the denominator and evaluating at q = leads to a vanishing expression, so theonly terms which contribute are the derivatives of the coefficient and the test function. Since ¯ p and q are set to ¯ p ⋆ and q ⋆ respectively, both of which depend on q , this results in additionalterms besides those that arise due to the explicit q -dependence of f and C . Besides the factthat ¯ p ⋆ (cid:12)(cid:12) q = = E ¯ p and q ⋆ (cid:12)(cid:12) q = = 0, it also follows from Eqs. (56) and (57) that ∂ ¯ p ⋆ ∂q j (cid:12)(cid:12)(cid:12)(cid:12) q = = 0 , ∂q ⋆ ∂q j (cid:12)(cid:12)(cid:12)(cid:12) q = = − ¯ p j E ¯ p . (61) fter applying the chain rule together with the above identities one obtains Z d ¯ p (2 E ¯ p ) " ( − ¯ p j E ¯ p ∂C ( E ¯ p , ¯ p , q , q ) ∂q (cid:12)(cid:12)(cid:12)(cid:12) q = q ⋆ + ∂C ( E ¯ p , ¯ p , , q ) ∂q j ) f ( E ¯ p , ¯ p , , q ) − C ( E ¯ p , ¯ p , , q ) ( − ¯ p j E ¯ p ∂f ( E ¯ p , ¯ p , q , q ) ∂q (cid:12)(cid:12)(cid:12)(cid:12) q = q ⋆ + ∂f ( E ¯ p , ¯ p , , q ) ∂q j ) q = , (62)which on the level of distributions implies T j (¯ p, q ) = − π δ (+) M (¯ p )2¯ p " C (¯ p, ∂ j δ ( q ) − C (¯ p,
0) ¯ p j ¯ p ∂ δ ( q ) − (cid:18) ∂C∂q j − ¯ p j ¯ p ∂C∂q (cid:19) q =0 δ ( q ) . (63) A.2 Explicit matrix elements
In order to perform the distributional matching procedure one requires the explicit forms forthe rotation and boost generator matrix elements. In the variables p ′ and p these are given by h p ′ , m ′ ; M | J i | p, m ; M i = (2 π ) δ (+) M ( p ) (cid:20) Σ im ′ m ( k ) + δ m ′ m iǫ ijk p k ∂∂p j (cid:21) δ ( p ′ − p ) , (64) h p ′ , m ′ ; M | K i | p, m ; M i = − (2 π ) δ (+) M ( p ) (cid:20) ǫ ijk p j p + M Σ km ′ m ( k )+ δ m ′ m i (cid:18) p ∂∂p i − p i ∂∂p (cid:19) (cid:21) δ ( p ′ − p ) , (65)which are a covariant generalisation of those derived in [12]. To derive these equations onecan use the fact that states of arbitary spin s transform under (proper orthochronous) Lorentztransformations α as follows [15]: U ( α ) | p, k ; M i = X l D ( s ) lk ( α ) | Λ( α ) p, l ; M i , (66)where D ( s ) is the (2 s + 1)-dimensional Wigner rotation matrix, and Λ( α ) is the four-vectorrepresentation of α . Since we are interested in the matrix elements of J i and K i one mustconsider the specific cases of a pure rotation α = R i and boost α = B i about the i -axis, where: U ( R i ) = e − iβJ i and U ( B i ) = e iξK i . Combining Eq. (66) for a pure rotation together with thedefinition of the norm of the on-shell states in Eq. (2) implies h p ′ , m ′ ; M | J i | p, m ; M i = i (cid:20) ∂∂β h p ′ , m ′ ; M | U ( R i ) | p, m ; M i (cid:21) β =0 = i ∂∂β X l D ( s ) lm ( R i ) h p ′ , m ′ ; M | Λ( R i ) p, l ; M i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = i ∂∂β X l D ( s ) lm ( R i ) (2 π ) δ ( p ′ − Λ( R i ) p ) δ (+) M ( p ′ ) δ m ′ l !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = i (cid:20) ∂∂β D ( s ) lm ( R i ) (cid:21) β =0 (2 π ) δ ( p ′ − p ) δ (+) M ( p )+ (2 π ) δ (+) M ( p ) δ m ′ m i (cid:20) ∂∂β δ ( p ′ − Λ( R i ) p ) (cid:21) β =0 , (67) here one has implicitly used the fact that δ (+) M (Λ( R i ) p ) = δ (+) M ( p ). By definition: Σ im ′ m ( k ) = i h ∂∂β D ( s ) m ′ m ( R i ) i β =0 are the (2 s + 1)-dimensional spin matrices. To consistently calculate thesecond term one must use the distributional properties of the Dirac delta. In general, due tothe transformation properties of distributions under linear transformations [28], one has that Z d p δ ( p ′ − Λ( R i ) p ) f ( p ) ≡ | detΛ( R i ) | − Z d ℓ δ ( p ′ − ℓ ) f (Λ − ( R i ) ℓ )= f (Λ − ( R i ) p ′ ) , (68)where f is an arbitrary test function. Expanding the test function around the point β = 0 gives f (Λ − ( R i ) p ′ ) = f ( p ′ ) + β ǫ ijk p ′ j ∂f ( p ) ∂p k (cid:12)(cid:12)(cid:12)(cid:12) p = p ′ + O ( β ) . (69)Combining this expansion together with Eq. (68) one can then explicitly determine how thedistribution i h ∂∂β δ ( p ′ − Λ( R i ) p ) i β =0 acts on test functions Z d p i (cid:20) ∂∂β δ ( p ′ − Λ( R i ) p ) (cid:21) β =0 f ( p ) = i ∂∂β f ( p ′ i ) + β ǫ ijk p ′ j ∂f ( p ) ∂p k (cid:12)(cid:12)(cid:12)(cid:12) p = p ′ + O ( β ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = iǫ ijk p ′ j ∂f ( p ) ∂p k (cid:12)(cid:12)(cid:12)(cid:12) p = p ′ , (70)which implies the following equality: i (cid:20) ∂∂β δ ( p ′ − Λ( R i ) p ) (cid:21) β =0 = iǫ ijk p k ∂∂p j δ ( p ′ − p ) . (71)Combining this relation with Eq. (67) finally proves Eq. (64).In the case of a pure boost α = B i the matrix element is more complicated because the Wignerrotation matrix D ( s ) ( B i ) depends on both ξ and the momentum. Nevertheless, one can demon-strate that i (cid:20) ∂∂ξ D ( s ) m ′ m ( B i ) (cid:21) ξ =0 = ǫ ijk p j p + M Σ km ′ m ( k ) . (72)Performing identical steps as in Eq. (67), it remains to calculate an explicit expression for thedistribution i h ∂∂ξ δ ( p ′ − Λ( B i ) p ) i ξ =0 . In this case f (Λ − ( B i ) p ′ ) = f ( p ′ ) − ξ p ′ ∂f ( p ) ∂p i (cid:12)(cid:12)(cid:12)(cid:12) p = p ′ + ξ p ′ i ∂f ( p ) ∂p (cid:12)(cid:12)(cid:12)(cid:12) p = p ′ + O ( ξ ) , (73)from which it follows i (cid:20) ∂∂ξ δ ( p ′ − Λ( B i ) p ) (cid:21) ξ =0 = (cid:20) ip ∂∂p i − ip i ∂∂p (cid:21) δ ( p ′ − p ) . (74)Combining this with Eq. (72) proves Eq. (65). In [12] the authors derive the form for the infinitesimal Wigner rotation for boosts, from which one can derivethe manifestly spin-representation independent expression in Eq. (72). n order to compare these equations with the on-shell matrix elements one must instead workwith the variables ¯ p and q . Due to the explicit δ ( p ′ − p ) component in the first terms of Eqs. (64)and (65), these expressions are simply proportional to δ (+) M (¯ p ) δ ( q ). The second terms involvingderivatives of δ ( p ′ − p ) are non-trivial though due to the q -dependence of δ (+) M ( p ). Nevertheless,in the case of rotations one can write δ (+) M ( p ) i (cid:20) ∂∂β δ ( p ′ − Λ( R i ) p ) (cid:21) β =0 = − δ (cid:16) ¯ p − q − q (¯ p − q ) + M (cid:17) p − q ) iǫ ijk (¯ p − q ) k ∂∂q j δ ( q )= − δ (cid:16) ¯ p − q (¯ p − q ) + M (cid:17) p iǫ ijk ¯ p k ∂∂q j δ ( q ) , (75)since the term involving the q j -derivative of q k vanishes due to the anti-symmetric tensor. Ifone now integrates this expression with a test function f (¯ p, q ) one ends up with Z d ¯ p iǫ ijk ¯ p k " − ¯ p j E p f ( E ¯ p , ¯ p , q , q ) + ¯ p j E p ∂f (¯ p , ¯ p , q , q ) ∂ ¯ p (cid:12)(cid:12)(cid:12)(cid:12) ¯ p =2 E ¯ p + 12 E ¯ p ∂f ( E ¯ p , ¯ p , q , q ) ∂q j q =0 . (76)The first two terms vanish due to the contraction with ǫ ijk ¯ p k , and hence one can conclude that δ (+) M ( p ) iǫ ijk p k ∂∂p j δ ( p ′ − p ) = − δ (+) M (¯ p ) iǫ ijk ¯ p k ∂∂q j δ ( q ) . (77)The J i matrix element in (¯ p, q ) variables is therefore given by h ¯ p + q, m ′ ; M | J i | ¯ p − q, m ; M i = (2 π ) δ (+) M (¯ p ) (cid:20) Σ im ′ m ( k ) − δ m ′ m iǫ ijk ¯ p k ∂∂q j (cid:21) δ ( q ) . (78)One can perform exactly the same procedure in the pure boost case, except this time there aretwo derivative components. Changing variables in the expression δ (+) M ( p ) h ip ∂∂p i − ip i ∂∂p i δ ( p ′ − p ) and integrating with a test function gives i Z d ¯ p (cid:20) ∂f ( E ¯ p , ¯ p, q , q ) ∂q i − ¯ p i E ¯ p ∂f ( E ¯ p , ¯ p, q , q ) ∂q (cid:21) q =0 , (79)where the two terms involving ¯ p -derivatives of the test function cancel one another. From thiswe conclude that δ (+) M ( p ) (cid:20) ip ∂∂p i − ip i ∂∂p (cid:21) δ ( p ′ − p ) = − δ (+) M (¯ p ) (cid:20) i ¯ p ∂∂q i − i ¯ p i ∂∂q (cid:21) δ ( q ) , (80)and hence the K i matrix element in (¯ p, q ) variables has the form h ¯ p + q, m ′ ; M | K i | ¯ p − q, m ; M i = (2 π ) δ (+) M (¯ p ) (cid:20) − ǫ ijk ¯ p j ¯ p + M Σ km ′ m ( k )+ δ m ′ m i (cid:18) ¯ p ∂∂q i − ¯ p i ∂∂q (cid:19) (cid:21) δ ( q ) . (81) B Arbitrary spin η m ( p ) identities In this appendix we prove a series of identities involving the arbitrary spin η m ( p ). .1 Proof of Eq. (13) In order to prove Eq. (13) it is important to first recognise that one can write ∂∂q i [ η m ′ ( p ′ ) η m ( p )] = (cid:20) ∂η m ′ (¯ p + q ) ∂q i (cid:21) η m (¯ p ) + η m ′ (¯ p ) (cid:20) ∂η m (¯ p − q ) ∂q i (cid:21) = (cid:20) ∂η m ′ (¯ p + q ) ∂q i (cid:21) η m (¯ p ) − η m ′ (¯ p ) (cid:20) ∂η m (¯ p + q ) ∂q i (cid:21) . (82)The rest and moving frame η m are related by a global boost: η m ( p ) = e i ξ ( p ) · κ η m ( k ), where k µ = M g µ is the rest frame four-momentum, κ i = S i are the standard boost generator matricesin the chosen spin representation, and the boost parameter is given by: ξ ( p ) = ξ ( p ) ˆ ξ ( p ), with ξ ( p ) = sinh − ( | p | /M ) and ˆ ξ ( p ) = p / | p | . Let us first consider the derivative of the exponentialargument in this boost, evaluated at q = 0 ∂∂q i (cid:2) i ξ (¯ p + q ) · κ (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) q =0 = (cid:20) ∂∂q i ξ (¯ p + q ) (cid:21) q =0 ( i ˆ ξ · κ ) + ξ ∂∂q i h i ˆ ξ (¯ p + q ) · κ i(cid:12)(cid:12)(cid:12)(cid:12) q =0 = − ¯ p i | ¯ p | ¯ p ξ ( i ξ · κ ) + i | ¯ p | ǫ ijk ǫ klr ˆ ξ j ξ l κ r = − ¯ p i | ¯ p | ¯ p ξ ( i ξ · κ ) − i | ¯ p | ǫ ijk ¯ p j (cid:2) Σ k , ( i ξ · κ ) (cid:3) , (83)where we used the fact that the boost generators transform as a three-vector under rotations[Σ k , κ l ] = iǫ klr κ r . Because the commutator with Σ k = ǫ kij S ij acts as a derivation, it followsfrom the above relation that the q i -derivative on the full exponential can be written (cid:20) ∂∂q i e i ξ (¯ p + 12 q ) · κ (cid:21) q =0 = − ¯ p i | ¯ p | ¯ p ξ ( i ξ · κ ) e i ξ (¯ p ) · κ − i | ¯ p | ǫ ijk ¯ p j h Σ k , e i ξ (¯ p ) · κ i , (84)and similarly with κ
7→ − κ . Using these expressions together with Eq. (82) one finds that ∂∂q i [ η m ′ ( p ′ ) η m ( p )] (cid:12)(cid:12)(cid:12)(cid:12) q =0 = Tr " ρ m ′ m ( k ) (cid:20) ∂∂q i e − i ξ (¯ p + 12 q ) · κ (cid:21) q =0 e i ξ (¯ p ) · κ − Tr " ρ m ′ m ( k ) e − i ξ (¯ p ) · κ (cid:20) ∂∂q i e i ξ (¯ p + 12 q ) · κ (cid:21) q =0 = ¯ p i | ¯ p | ¯ p ξ Tr [ ρ m ′ m ( k ) ( i ξ · κ )]+ i | ¯ p | ǫ ijk ¯ p j Tr h ρ m ′ m ( k ) n(cid:16) e − i ξ (¯ p ) · κ Σ k e i ξ (¯ p ) · κ (cid:17) − Σ k oi . (85)The first term vanishes because of the trace Tr (cid:2) ρ m ′ m ( k ) κ i (cid:3) = 0, and one is left with ∂∂q i [ η m ′ ( p ′ ) η m ( p )] (cid:12)(cid:12)(cid:12)(cid:12) q =0 = i | ¯ p | ǫ ijk ¯ p j (cid:8) Tr (cid:2) ρ m ′ m (¯ p )Σ k (cid:3) − Tr (cid:2) ρ m ′ m ( k )Σ k (cid:3)(cid:9) . (86) This trace must indeed to vanish, otherwise a state at rest would be characterised by an additional three-vectorbesides the spin vector. .2 Proof of Eq. (25) In order to prove Eq. (25) note that since a state is characterised only in terms of the momentumand Pauli-Lubanski four-vectors, one can in general write [29] η m ′ (¯ p ) S µν η m (¯ p ) = − M ǫ µναβ S αm ′ m (¯ p ) ¯ p β , (87)where S αm ′ m (¯ p ) = Tr [ ρ m ′ m (¯ p ) S α ] with S α = ǫ αρσλ S ρσ ¯ p λ the standard covariant spin matricesin the chosen spin representation, and ǫ = +1. Contracting this relation with the four-momentum leads to η m ′ (¯ p ) S µν η m (¯ p )¯ p µ = 0, and hence ǫ ijk ¯ p j Σ km ′ m (¯ p ) = − ¯ p κ im ′ m (¯ p ) , (88)since κ i = S i and Σ k = ǫ kij S ij . Another consequence of Eq. (87) is that η m ′ (¯ p ) S µν η m (¯ p ) k µ = 1 M ǫ νµαβ k µ S αm ′ m (¯ p ) ¯ p β = − M ǫ νµαβ k µ S αm ′ m ( k ) ¯ p β = η m ′ ( k ) S µν η m ( k )¯ p µ , (89)and hence it follows that M κ im ′ m (¯ p ) = − ǫ ijk p j Σ km ′ m ( k ) . (90)Combining Eqs. (88) and (90) together with (¯ p ) = | ¯ p | + M leads us to ǫ ijk ¯ p j ¯ p Σ km ′ m (¯ p ) = −| ¯ p | κ im ′ m (¯ p ) + M ǫ ijk ¯ p j Σ km ′ m ( k ) . References [1] J. F. Donoghue, B. R. Holstein, B. Garbrecht and T. Konstandin,
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