On the Conformal Symmetry of Exceptional Scalar Theories
OOn the Conformal Symmetry ofExceptional Scalar Theories
Kara Farnsworth, a , ∗ Kurt Hinterbichler, a , † Ondˇrej Hul´ık, a , b , ‡ a CERCA, Department of Physics,Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106 b Institute of Physics of the Czech Academy of Sciences, CEICO,Na Slovance 2, 182 21 Prague 8, Czech Republic
Abstract
The DBI and special galileon theories exhibit a conformal symmetry at unphysical values ofthe spacetime dimension. We find the Lagrangian form of this symmetry. The special conformaltransformations are non-linearly realized on the fields, even though conformal symmetry isunbroken. Commuting the conformal transformations with the extended shift symmetries, wefind new symmetries, which when taken together with the conformal and shift symmetriesclose into a larger algebra. For DBI this larger algebra is the conformal algebra of the higherdimensional bulk in the brane embedding view of DBI. For the special galileon it is a real form ofthe special linear algebra. We also find the Weyl transformations corresponding to the conformalsymmetries, as well as the necessary improvement terms to make the theories Weyl invariant, tosecond order in the coupling in the DBI case and to lowest order in the coupling in the specialgalileon case. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] F e b ontents Introduction
Soft theories are effective field theories whose amplitudes go to zero as one of the external momentaare scaled to zero, scaling with some power of this momentum as we approach zero. Amongsoft theories consisting of a single scalar φ , there are two exceptional theories that scale with thehighest possible power, a power higher than expected given the number of derivatives per field inthe Lagrangian [1, 2]. These are the Dirac-Born-Infeld (DBI) theory, whose action is given in (2.1),and the special galileon [1, 3], whose action is given in (3.1). Both theories depend on a singledimensionful coupling constant α , and are otherwise completely fixed.These two theories, among others [4], have a privileged place among scalar effective field theories;they naturally appear in the Cachazo-Yuan-He (CHY) representation [5–7] and act as nodes in aweb of theories related by the double copy and other procedures [8, 9] (see [10] for a nice visualrepresentation of this web). Understanding their properties and what makes these theories uniqueis a step towards being able to uncover how fundamental these tools are and their ability to relategauge and gravitational theories.DBI and the special galileon exist for any spacetime dimension d , however in [11] it was notedthat if we consider these theories at unphysical values of d , d = 0 for DBI and d = − α becomes dimensionless and the theory is scale invariant. In fact, eachterm in the Lagrangian is separately scale invariant, as classical scale invariance is automatic andtrivial once there are no dimensionful couplings. In [11] it is shown that these theories also havefull conformal invariance in addition to scale invariance, and that the conformal invariance is non-trivial and fixes the structure of the Lagrangian to be that of DBI/special galileon. For example,it was argued that the stress tensor on flat space could be improved to be traceless, and thatthe amplitudes satisfy conformal Ward identities, only if the structure of the theory was that ofDBI/special galileon.However, conformal symmetry as we usually understand it is linearly realized on the fields, anda linear symmetry can never relate terms with different powers of the field, as would be requiredto fix the full non-linear structure of these theories. We therefore expect that this conformalsymmetry is realized in a novel non-linear way. On the other hand, conformal symmetry shouldstill be unbroken, so that the usual Ward identities on the amplitudes are satisfied. Here we resolvethis puzzle and find the field transformations that leave the Lagrangians invariant and fix theirnon-linear structure, without spontaneously breaking conformal symmetry. The transformationsinclude non-linearities in the special conformal generators (shown in (2.6) for DBI and (3.3) for thespecial galileon), but preserve the φ = 0 vacuum and satisfy the same conformal algebra as theirlinear counterparts.Another defining feature of these theories is that they possess non-trivial shift symmetries, wherethe field shifts by powers of the spacetime coordinate x µ . The shift symmetries for DBI have upto one power of x µ (shown in (2.11)), and the shift symmetries for the special galileon have up totwo powers of x µ (shown in (3.4)). These shift symmetries also fix the full non-linear structure of3he theory, and are responsible for the enhanced soft limits.These shift symmetries do not close with the conformal symmetries under commutation, andinstead produce new symmetries. In the DBI case, we find a new symmetry (2.12) which has 2powers of x µ , and in the special galileon case we find several new symmetries (3.5) which haveup to 4 powers of x µ . These new symmetries combined with the conformal symmetries and shiftsymmetries then close to form a larger algebra. In the DBI case, this larger algebra is the conformalalgebra of one dimension higher. In Section 2.3, we explain this from the brane embedding pointof view; the DBI theory can be thought of as the worldvolume theory of a d dimensional braneembedded in a d + 1 dimensional bulk, and from this perspective, the symmetries all descend fromconformal symmetries of the bulk. In the special galileon case, the full symmetry algebra is a realform of the special linear algebra.Finally, we consider coupling these theories to a metric and look for a local Weyl symmetrythat descends to the new non-linear global conformal symmetries when the metric is frozen to flatspace. This Weyl symmetry requires new non-linear terms in the scalar field transformation inorder to reproduce the non-linear terms in the global special conformal transformations. We findthe improvement terms necessary to add to the theories so that they are Weyl invariant, to lowestorder in the coupling for the special galileon, and to second order in the coupling for DBI. Conventions:
The spacetime dimension is d . We use the mostly plus metric. The curvatureconventions are those of [12]. The DBI action for a single scalar φ in d spacetime dimensions is S = (cid:90) d d x − α (cid:112) α ( ∂φ ) . (2.1)It depends on a single coupling constant, α .In any d , this action is manifestly invariant under the standard translations and Lorentz trans-formations, P µ φ = − ∂ µ φ, J µν φ = ( x µ ∂ ν − x ν ∂ µ ) φ , (2.2)which satisfy the commutation relations of the Poincare algebra,[ J µν , J σρ ] = η µσ J νρ − η νσ J µρ + η νρ J µσ − η µρ J νσ , [ J µν , P ρ ] = η ρµ P ν − η ρν P µ , [ P µ , P ν ] = 0 . (2.3)4 .1 Conformal symmetry The DBI coupling constant in (2.1) has mass dimension [ α ] = − d . As was noted in [11], if weconsider the unphysical value d = 0, the coupling becomes dimensionless, and the theory becomesscale invariant under the standard dilation symmetry Dφ = − ( x µ ∂ µ + ∆) φ , where ∆ is the scalingdimension of the field. Scale invariance requires the field φ to have dimension ∆ = d − , so when d = 0 we have ∆ = −
1, and the action (2.1) is invariant under Dφ = − x µ ∂ µ φ + φ . (2.4)This scale symmetry, like the Poincare transformations (2.2), is linear in the fields, and each termin the expansion of the square root in the action (2.1) in powers of α is separately scale invariantwhen d = 0 .Conformal symmetry also includes special conformal transformations, and the standard linearaction of these on a field of weight ∆ is K µ φ = (cid:0) − x µ x ν ∂ ν + x ∂ µ − x µ ∆ (cid:1) φ. (2.5)In [11] it was argued that conformal symmetry should fix the square root structure in the DBIaction when d = 0; a generic scalar effective theory in d = 0 with one derivative per field would bescale invariant, but not conformally invariant, and only the particular square root structure of DBIwould have full conformal invariance. However this cannot be achieved with the standard linearspecial conformal transformations (2.5), since they cannot relate terms with differing powers of φ in order to fix the square root structure.The non-linear special conformal transformation which does accomplish this is K µ φ = (cid:0) − x µ x ν ∂ ν + x ∂ µ + 2 x µ (cid:1) φ + αφ ∂ µ φ. (2.6)This transformation contains a new non-linear piece which depends on α , and when α = 0 itreduces to the standard linear special conformal transformation (2.5) with ∆ = −
1. The non-lineartransformation (2.6) is a symmetry of the DBI action when d = 0, and it fixes the square rootstructure of the action. Note that despite the non-linearities, the transformation (2.6) preservesthe vacuum φ = 0, so we can still say that conformal symmetry is unbroken . Usually non-lineartransformations are associated with broken symmetries, but in these cases there is always a leadingterm in the transformation that is independent of the fields so that the vacuum is not preserved.Here we have no such term, the transformation starts at linear order in φ . In checking invariance for this and other symmetries below in d = 0, we proceed as in dimensional regularization,manipulating everything in general d , setting d = 0 only at the end and never using any dimensionally dependentidentities. A special conformal transformation similar to (2.6) occurs in the worldvolume theory of a flat brane probing anAdS bulk, but in this case conformal symmetry is broken [13].
5e can now compute the commutators of the scale symmetry (2.4) and non-linear special con-formal symmetries (2.6) with the Poincare symmetries (2.2) and we get[ P µ , D ] = P µ , [ K µ , D ] = − K µ , [ K µ , P ν ] = − η µν D − J µν ) , [ J µν , K ρ ] = η ρµ K ν − η ρν K µ , [ K µ , K ν ] = [ J µν , D ] = 0 , (2.7)along with the Poincare commutators (2.3). These are the standard commutation relations of theconformal algebra. They hold independently of α , so despite the non-linear term in (2.6), we havethe usual conformal algebra. In computing these we have not fixed the spacetime dimension d ,though we did need to fix ∆ = − J − , = D, J − ,µ = ( P µ − K µ ) , J ,µ = ( P µ + K µ ) , and assemble the conformal generatorsinto a d + 2 dimensional anti-symmetric matrix J AB with A, B = − , , , · · · , dJ AB ≡ D ( P ν − K ν ) − D ( P ν + K ν ) − ( P µ − K µ ) − ( P µ + K µ ) J µν . (2.8)Then the commutation relations (2.7), (2.3) become (cid:2) J AB , J CD (cid:3) = g AC J BD − g BC J AD + g BD J AC − g AD J BC , (2.9)where g AB ≡ − η µν , (2.10)showing that the conformal algebra is so (2 , d ).Since the field φ has conformal dimension ∆ = − d = 0, it might be thought thatit satisfies the standard scalar unitarity bound ∆ ≥ d − [14, 15]. This unitarity bound comes fromdemanding positivity of the second level descendent state P | ψ (cid:105) , given that the primary | ψ (cid:105) haspositive norm. But there is also a bound ∆ ≥ P µ | ψ (cid:105) . When d ≥ d = 0,∆ = − The DBI action (2.1) is also invariant in any d under the extended shift symmetries, C φ = 1 , B µ φ = x µ + α φ∂ µ φ, (2.11)6here x µ is the spacetime coordinate. These are spontaneously broken, since they do not preservethe vacuum φ = 0. Any theory with one derivative per field has the simple shift symmetry C , butthe extended shifts B µ , which include terms linear in the spacetime coordinate x µ , are α dependentand fix the square root structure of the action in any d .We can now ask how these shift symmetries interplay with the conformal symmetries (2.2), (2.4),(2.6) by computing the commutators. In fact, taken together, the conformal symmetries and shiftsymmetries (2.11) do not quite close under the commutators. By computing [ K µ , B ν ], we find anew scalar symmetry, N φ = − x + α (cid:0) φ − x µ φ∂ µ φ (cid:1) . (2.12)This new transformation is also a symmetry of DBI when d = 0, and it starts with a quadratic shiftin the spacetime coordinate. (For example, we can quickly see that the kinetic term ∼ ( ∂φ ) isinvariant under the lowest order shift N (0) φ ∼ x : N (0) (cid:0) ( ∂φ ) ) (cid:1) ∼ ∂ µ φ∂ µ ( N (0) φ ) ∼ ∂ µ φx µ , whichintegrates by parts to ∼ dφ , which vanishes when d = 0.)After including N , the algebra closes. The complete set of commutators, in addition to theconformal algebra (2.7), (2.3), is[ J µν , B ρ ] = η ρµ B ν − η ρν B µ , [ J µν , C ] = [ J µν , N ] = 0 , (2.13)[ N, D ] = − N , [ B ν , D ] = 0 , [ C, D ] =
C , (2.14)[ P µ , N ] = − B µ , [ P µ , B ν ] = η µν C , [ P µ , C ] = 0 , (2.15)[ K µ , C ] = − B µ , [ K µ , B ν ] = η µν N , [ K µ , N ] = 0 , (2.16)[ B µ , B ν ] = αJ µν , [ C, B µ ] = − αP µ , [ N, B µ ] = − αK µ , [ C, N ] = 2 αD . (2.17)As before, we do not need to fix d = 0 in computing these, though we have used ∆ = − N, B µ , C transform under Lorentz trans-formations simply as their tensor indices indicate, with spins 0 , , D to the right,with eigenvalues − , , P µ and K µ arelike raising and lowering operators respectively between the different D eigenstates. These relations(2.13), (2.14), (2.15), (2.16) are all the commutators between elements of the conformal algebra7nd shift symmetries, and are independent of α . They can be summarized in this figure: (2.18)The DBI symmetries thus form a finite dimensional Verma module under the conformal algebra,where the new symmetry N is like a conformal primary and B µ , C are descendants. This shortfinite dimensional module, which occurs when a scalar conformal primary has dimension − , is nothing but the fundamental vector representation of the so (2 , d ) conformalalgebra. To see this, define S − = − ( C − N ) , S = − ( C + N ) , S µ = B µ , and assemble theDBI symmetries into a d + 2 dimensional vector S A , S A = − ( C − N ) − ( C + N ) B µ . (2.19)Then the commutators (2.13), (2.14), (2.15), (2.16) become (cid:2) J AB , S C (cid:3) = g AC S B − g CB S A , (2.20)with the conformal generators packaged into J AB as in (2.8) with g AB as in (2.10). This is thestatement that S A is a vector under the conformal algebra so (2 , d ). The relations (2.17), which givethe commutators between the shift symmetries, are proportional to α , and can now be summarizedas (cid:2) S A , S B (cid:3) = αJ AB . (2.21)To identify the full algebra, we assemble J AB and S A into an d + 3 dimensional anti-symmetricmatrix J AB with A , B = − , , , · · · , d, d + 1, J AB = (cid:32) J AB − √ α S B √ α S A (cid:33) , α > . (2.22) This is a type I shortening condition in the language of [16], which occur for scalar primaries when they haveweight 0 , − , − , · · · . The trivial case with weight 0 would describe a simple shift symmetric field with conformalsymmetry. The next case with weight − (cid:2) J AB , J CD (cid:3) = g AC J BD − g BC J AD + g BD J AC − g AD J BC , (2.23)where g AB = (cid:32) g AB
00 1 (cid:33) , α > . (2.24)These are the commutation relations of so (2 , d + 1). In writing (2.22), (2.24), we have assumed that α >
0, which is the “right sign” from the standard UV completion point of view [17]. If instead wehave the “wrong sign” α < d + 3 metric (2.24) will have a minus sign in the lowerright corner, and the full algebra will be so (3 , d ).The smallest symmetry algebra of DBI that includes the conformal symmetry of [11] and theshift symmetries is thus isomorphic to the conformal algebra of a space of dimension d + 1. Thebreaking pattern is to the d dimensional conformal algebra, since the shifts (2.11), (2.12) are theonly ones that do not preserve the vacuum φ = 0, so (2 , d + 1) → so (2 , d ) , α > . (2.25)In fact, this is isomorphic to the symmetry breaking pattern of the AdS DBI theory, the theory ofa d + 1 dimensional AdS brane in a d + 2 dimensional AdS bulk [19, 20] (or the k = 1 non-abelianAdS scalar shift theory, in the language of [21]). The DBI theory (2.1) can be thought of as a gauge fixed worldvolume theory of a d dimensional flatbrane probing a d + 1 dimensional flat bulk [22]. The fact that the full symmetry algebra includingthe conformal and shift symmetries is the conformal algebra of one dimension higher suggests thatthey have their origin as conformal symmetries of the bulk. Here we will see that this is indeed thecase. (In this subsection we’ll assume α > X A ( x ), A = 1 , · · · , d + 1, describing theembedding of a d dimensional worldvolume into a d + 1 dimensional bulk. The fixed bulk metric is G AB ( X ), and the induced metric on the brane is¯ g µν ( x ) = ∂X A ∂x µ ∂X B ∂x ν G AB ( X ( x )) . (2.26)The action is S = − (cid:90) d d x √− ¯ g, (2.27)and is manifestly invariant under worldvolume diffeomorphisms ξ µ ( x ), under which the X A trans-form as scalars δ ξ X A = ξ µ ∂ µ X A . 9iven a transformation by a bulk diffeomorphism K A ( X ), δX A = K A ( X ), the induced metricis invariant if K A is a Killing vector of the bulk metric, L K G AB ≡ K C ∂ C G AB + ∂ A K C G CB + ∂ B K C G AC = 0 , (2.28)and this is a global symmetry of the action in any d .Suppose instead that K A is a conformal Killing vector, L K G AB = Φ G AB , (2.29)where Φ = d +1 ∇ A K A . Then under such a diffeomorphism we have δ ¯ g µν = Φ¯ g µν , (2.30)and the change in the action (2.27) is then δS = − (cid:90) d d x δ (cid:0) √− ¯ g (cid:1) = − d (cid:90) d d x √− ¯ g Φ . (2.31)This vanishes when d = 0, which means that the shift by a bulk conformal Killing vector is asymmetry of (2.27) when d = 0.To recover the DBI theory (2.1), we consider a flat bulk metric G AB ( X ) = η AB , and we fix theworldvolume diffeomorphisms by going to the unitary gauge X µ = x µ , X d +1 = φ . The inducedmetric (2.26) becomes ¯ g µν = η µν + ∂ µ φ∂ ν φ , and the action (2.27) becomes the DBI action (2.1). Thetransformation resulting from a bulk diffeomorphism K A must be compensated by a worldvolumediffeomorphism with ξ µ = − K µ in order to preserve the unitary gauge condition, so that theresulting global transformation acting on φ is δ K φ = − K µ ( x, φ ) ∂ µ φ + K d +1 ( x, φ ) . (2.32)When the K A are isometries of η AB , we get global symmetries valid in any d ; the isometries alongthe brane directions µ give the Poincare symmetries (2.2), the translation in the d + 1 directiongives the shift C in (2.11), and the Lorentz boosts mixing the d + 1 and µ directions give theextended shifts B µ in (2.11).We can now ask about the conformal Killing vectors of η AB that are not Killing vectors, whichwill give symmetries only when d = 0. The bulk dilation vector field is K A ( X ) = X A , (2.33)which plugging into (2.32) leads to δφ = − x µ ∂ µ φ + φ, (2.34)which is precisely the dilation in (2.4), including the correct scaling weight ∆ = −
1. The bulkspecial conformal vector fields are K A ( B ) ( X ) = 2 X B X A − X δ AB . (2.35)10ooking at the B = µ components, plugging into (2.32) leads to δ ( µ ) φ = (cid:0) − x µ x ν ∂ ν + x ∂ µ + 2 x µ (cid:1) φ + φ ∂ µ φ, (2.36)which, once the coupling is restored, is precisely the special conformal transformation in (2.6),with the correct scaling weight ∆ = −
1, and including the new non-linear term. Looking at the B = d + 1 component, plugging into (2.32) leads to δ ( d +1) φ = − x + (cid:0) φ − x µ φ∂ µ φ (cid:1) , (2.37)which, once the coupling is restored, is precisely the new transformation in (2.12).This construction makes it clear that new symmetries for d = 0 will also be present in any branetheory where the bulk has conformal Killing vectors which are not Killing vectors. This includesthe AdS and dS DBI theories discussed in [19–21, 24, 25], and more general cases such as thecosmological setup in [26]. We now turn to the special galileon. The action can be written as S sgal = − (cid:90) d d x (cid:98) d +12 (cid:99) (cid:88) n =1 α n − (2 n − ∂φ ) L TD2 n − = (cid:90) d d x −
12 ( ∂φ ) − α
12 ( ∂φ ) (cid:104) ( (cid:3) φ ) − ( ∂ µ ∂ ν φ ) (cid:105) + · · · , (3.1)where the total derivative combinations are L TD n ≡ (cid:80) p ( − p η µ p ( ν ) · · · η µ n p ( ν n ) ∂ µ ∂ ν φ · · · ∂ µ n ∂ ν n φ with the sum running over all permutations of the ν indices with ( − p the sign of the permutation.As with DBI, the theory depends on a single coupling constant, α , and in any d it is manifestlyinvariant under the standard translations and Lorentz transformations (2.2). The galileon coupling constant in (3.1) has dimension [ α ] = − ( d + 2). As was noted in [11],if we consider the unphysical value d = −
2, then the coupling becomes dimensionless, and thetheory becomes scale invariant. In this case the theory will have the standard dilation symmetry Dφ = − ( x µ ∂ µ + ∆) φ , where ∆ is the conformal dimension of the field. The field has dimension∆ = d − , so when d = − −
2, and each term in the action (3.1) is separately invariantunder Dφ = − x µ ∂ µ φ + 2 φ . (3.2)As with DBI, if conformal symmetry is to fix the theory as claimed in [11], the special conformaltransformations should include non-linear parts. The form of the special conformal generators11hich does this is K µ φ = (cid:0) − x µ x ν ∂ ν + x ∂ µ + 4 x µ (cid:1) φ − α ( ∂φ ) ∂ µ φ . (3.3)When α = 0, it reduces to the standard linear special conformal transformation (2.5) with ∆ = − d = −
2, and itfixes the coefficients of the various galileon terms [27] relative to each other into the special galileoncombination. As with DBI, the transformation (3.3) preserves the vacuum φ = 0, so we can still saythat conformal symmetry is unbroken, even though it is non-linearly realized. The transformations(3.2), (3.3) along with the Poincare symmetries (2.2) satisfy the conformal algebra (2.7), (2.3) inany d , despited the non-linear α dependent terms in (3.3). The generators of the galileon shift symmetries are [3] Cφ = 1 , B µ φ = x µ , S µν φ = x µ x ν − d x η µν − α (cid:20) ∂ µ φ ∂ ν φ − d ( ∂φ ) η µν (cid:21) . (3.4)The shifts C , B µ are zeroth and first other in the spacetime coordinate x µ , and they are symmetriesof a generic galileon [27]. The symmetric traceless tensor shifts S µν are second order in x µ , andthey are symmetries of the special galileon that fix the coefficients of the galileon terms relative toeach other.The shift symmetries (3.4) do not close with the conformal symmetries of Section 3.1 under thecommutator. We find 3 new transformations, two scalars and a vector and involving up to fourpowers of x µ , appearing on the right hand side of commutators before everything closes, A µ φ = x x µ + α (cid:2) φ∂ µ φ − ∂ µ φx ν ∂ ν φ − x µ ( ∂φ ) (cid:3) ,T φ = x − α ( ∂φ ) ,N φ = x − α (cid:2) x ( ∂φ ) + 2 x µ x ν ∂ µ φ∂ ν φ − x µ φ∂ µ φ + 8 φ (cid:3) + α ( ∂φ ) . (3.5)The new transformations N, A µ are indeed new symmetries of the special galileon action (3.1) when d = −
2. (For example, we can quickly see that the kinetic term ∼ ( ∂φ ) is invariant under thelowest order shift N (0) φ ∼ x : N (0) (cid:0) ( ∂φ ) ) (cid:1) ∼ ∂ µ φ∂ µ ( N (0) φ ) ∼ ∂ µ φ x x µ , which integrates byparts to ∼ ( d + 2) x φ , which vanishes when d = − T is not a symmetry,however, as we’ll see it always appears in the commutators with a factor of d + 2 which vanisheswhen d = − (cid:104) J µν , S λσ (cid:105) = η µλ S νσ − η νλ S µσ + η µσ S λν − η νσ S λµ , [ J µν , B ρ ] = η ρµ B ν − η ρν B µ , [ J µν , A ρ ] = η ρµ A ν − η ρν A µ , [ J µν , C ] = [ J µν , T ] = [ J µν , N ] = 0 . (3.6)They are eigenstates of the dilation operator,[ C, D ] = 2
C , [ B µ , D ] = B µ , [ S µν , D ] = [ T, D ] = 0 , [ A µ , D ] = − A µ , [ N, D ] = − N , (3.7)and they transform into each other under
K, P ,[ P µ , N ] = 4 A µ , [ P µ , A ν ] = 2 S µν + d + 2 d η µν T , [ P µ , T ] = 2 B µ , (cid:104) P µ , S νλ (cid:105) = η µν B λ + η µλ B ν − d B µ η νλ , [ P µ , B ν ] = η µν C , [ P µ , C ] = 0 , (3.8)[ K µ , C ] = − B µ , [ K µ , B ν ] = − S µν − d + 2 d η µν T , (cid:104) K µ , S νλ (cid:105) = − η µν A λ − η µλ A ν + 2 d A µ η νλ , [ K µ , T ] = − A µ , [ K µ , A ν ] = − η µν N , [ K µ , N ] = 0 . (3.9)13hese relations can be summarized in this figure, (3.10)The shift transformations thus form a finite dimensional Verma module of the conformal algebrawhere N is a conformal primary of weight −
2. This Verma module describes a symmetric tracelesstensor representation of the conformal algebra so (2 , d ). Group the shift generators into a d + 2dimensional symmetric g -traceless matrix, S AB = S BA , S AB g AB = 0, with g as in (2.10), S AB = ( C + N + 2 T ) ( C − N ) − ( A µ + B µ ) ( C − N ) ( C + N − T ) ( A µ − B µ ) − ( A µ + B µ ) ( A µ − B µ ) S µν + D η µν T . (3.11)Grouping the conformal generators as in (2.8), the commutators (3.6), (3.7), (3.8), (3.9) becomesimply (cid:2) J AB , S CD (cid:3) = g AC S BD − g BC S AD + g AD S CB − g BD S CA , (3.12)which is the statement that S AB transforms as a symmetric traceless tensor representation underthe conformal algebra.The commutators of the shift generators with each other give back conformal generators, and14re all proportional to α , (cid:104) S µν , S λσ (cid:105) = − α (cid:16) η µλ J νσ + η νλ J µσ + η νσ J µλ + η µσ J νλ (cid:17) , (cid:104) B µ , S νλ (cid:105) = α (cid:16) η µν P λ + η µλ P ν − d P µ η νλ (cid:17) , (cid:104) A µ , S νλ (cid:105) = − α (cid:16) η µν K λ + η µλ K ν − d K µ η νλ (cid:17) , [ S µν , C ] = [ S µν , T ] = [ S µν , N ] = 0 , [ A µ , A µ ] = [ B µ , B ν ] = 0 , [ A µ , B µ ] = − α ( η µν D − J µν ) , [ A µ , T ] = − αK µ , [ A µ , C ] = 4 αP µ , [ A µ , N ] = 0 , [ B µ , T ] = 2 αP µ , [ B µ , N ] = − αK µ , [ B µ , C ] = 0 , [ N, C ] = 16 αD, [ N, T ] = [
C, T ] = 0 . (3.13)In terms of (3.11) and (2.8), the commutators (3.13) become simply (cid:2) S AB , S CD (cid:3) = − α (cid:0) g AC J BD + g BC J AD + g BD J AC + g AD J BC (cid:1) . (3.14)We can now identify the full algebra. Put together the anti-symmetric conformal generators (2.8)and the symmetric traceless shift generators (3.11) into a traceless matrix as follows M AB ≡ − (cid:18) J AB + 1 √ α S AB (cid:19) , α > . (3.15)Then the commutators (3.14), (3.12), (2.9) can be summarized as (cid:2) M AB , M CD (cid:3) = g CB M AD − g AD M CB , (3.16)which are the commutation relations of sl ( d + 2 , R ). The smallest symmetry algebra of the specialgalileon that includes the conformal symmetry of [11] and all the extended shift symmetries is thusisomorphic to sl ( d + 2 , R ). In writing (3.15) we have assumed α >
0, the “right sign” from the pointof view of positivity bounds [28, 29]. If α <
0, then the algebra would instead be the maximallysplit real form.The breaking pattern is to the d dimensional conformal algebra, since the shifts S AB are the onlyones that do not preserve the vacuum φ = 0, sl ( d + 2 , R ) → so (2 , d ) , α > . (3.17)In fact, this is the same symmetry breaking pattern as the AdS special galileon theory in d + 1dimensions (the k = 2 non-abelian AdS scalar shift theory of [21]).15 Weyl symmetry
A conformal field theory can generally be coupled to a background metric g µν in such a way thatthe conformal symmetry descends from a local Weyl symmetry [30, 31] (there are some exceptionsin non-unitary theories, see e.g. [32–34]). The usual form of the Weyl symmetry for a field φ ofweight ∆ is δg µν = 2 σg µν , δφ = − ∆ σφ, (4.1)where σ ( x ) is the local scalar gauge parameter. This symmetry is present in addition to ordinarydiffeomorphism (diff) symmetry with local parameter ξ µ ( x ), δg µν = ∇ µ ξ ν + ∇ ν ξ µ , δφ = ξ µ ∂ µ φ. (4.2)The global conformal transformations emerge when we go to flat space, g µν = η µν , and restrictto those diff+Weyl symmetries which preserve the flat metric η µν . Such symmetries are those forwhich ξ µ is a conformal Killing vector, ∂ µ ξ ν + ∂ ν ξ ν = d ∂ λ ξ λ η µν , and σ = − d ∂ µ ξ µ . The conformalKilling vector corresponding to dilations is ξ µ = − x µ with σ = 1. Plugging this into the scalartransformations in (4.1), (4.2) gives the standard dilation transformation δφ = − ( x µ ∂ µ + ∆) φ. Theconformal Killing vector corresponding to special conformal transformations is ξ µ = − b · x x µ + x b µ with σ = 2 b · x , where b µ is a constant vector parametrizing the transformations. Plugging this intothe scalar transformations in (4.1), (4.2) gives the standard linear special conformal transformation δφ = b µ (cid:0) − x µ x ν ∂ ν + x ∂ µ − x µ ∆ (cid:1) φ . To reproduce the flat space non-linear terms in (2.6) after coupling to a general background metric,we need new non-linear terms proportional to α in the Weyl transformation. We will assume thatthe Weyl transformation for the metric, and the diffeomorphisms, retain their standard forms anddo not depend on α . The Weyl transformation for the scalar will be modified. Each power of α comes with 2 derivatives and 2 powers of φ , so the most general possible modification at order α that reduces to (2.4) for σ = 1 and (2.6) for σ = 2 b · x is δφ = σφ + α (cid:18) φ ∂ µ φ∂ µ σ + a φ Rσ + a φ (cid:3) σ (cid:19) + O (cid:0) α (cid:1) , (4.3)with constants a , a .We now want to couple (2.1) to the metric such that this new Weyl transformation is a symmetrywhen d = 0. The most general coupling to the metric, to order α , is S = (cid:90) d d x √− g (cid:20) α − α (cid:112) α ( ∂φ ) + b Rφ (4.4)+ α (cid:0) b R µν φ ∂ µ φ∂ ν φ + b Rφ ( ∂φ ) + b Rφ (cid:3) φ + b W µνλρ φ + b R µν φ + b R φ (cid:1) + O (cid:0) α (cid:1) (cid:21) , W µνλρ is the Weyl tensor and b , . . . , b are constants. Requiring Weyl invariance up to order α when d = 0 then fixes a = 0, b = − / b = 1 / b = 0, b = − a / b = 1 / b = a / a , b free.Thus the Weyl symmetry and invariant action to order α is S = (cid:90) d d x √− g (cid:20) α − α (cid:112) α ( ∂φ ) − Rφ + α (cid:18) R µν φ ∂ µ φ∂ ν φ + 132 R µν φ − a Rφ (cid:3) φ + a R φ + bW µνλρ φ (cid:19) + O (cid:0) α (cid:1) (cid:21) ,δg µν = 2 σg µν , δφ = σφ + α (cid:18) φ ∂ µ φ∂ µ σ + aφ (cid:3) σ (cid:19) + O (cid:0) α (cid:1) . (4.5)where a , b are the two unfixed free parameters. The parameter a can be further removed by thefield redefinition φ → φ + aαRφ , (4.6)as in [30]. This is the only allowed field redefinition at order α that does not transform the scalar,and hence does not modify the DBI form of the Lagrangian, when restricted to flat space. Theparameter b remains arbitrary since it comes in front of the Weyl-squared term, which is separatelyWeyl invariant to lowest order. This term would presumably be completed at higher order intosome invariant of the α -deformed Weyl symmetry, or else forced to vanish if no such invariantexists. Note that the coupling to the metric generically breaks all the shift symmetries.This new Weyl symmetry satisfies the same algebra as the standard Weyl symmetry, namelythat the Weyl transformations are abelian,[ δ σ , δ σ ] = 0 , (4.7)up to order α . In fact this requirement alone is enough to fix the form of the transformation:demanding (4.7) alone for (4.3) when d = 0 fixes a = 0, and then taking into account the possiblefield re-definition (4.6) removes a .In [11], an action linear in the curvatures and including all powers of φ was provided which makesthe stress tensor on flat space traceless. The terms linear in the curvature in the action (4.5) matchthe terms linear in the curvature from [11], up to order α (and when a = 0). However the actionin [11] cannot itself be Weyl invariant in the manner we are requiring, because, as we can see from(4.5), terms quadratic in the curvature are required.Pushing to next order in α we find that the action can be made Weyl invariant when d = 0 to17uadratic order in α , with the addition of the following terms to the action, S (cid:12)(cid:12)(cid:12)(cid:12) α = (cid:90) d d x √− g (cid:26) − α Rφ ( ∂φ ) − α R µν φ ( ∂φ ) ∇ µ φ ∇ ν φ (4.8)+ α φ (cid:20) (cid:0) R µν + R (cid:1) ( ∂φ ) − RR µν ∇ µ φ ∇ ν φ − R µν R νσ ∇ µ φ ∇ σ φ (cid:21) + α W µνρσ (cid:20) φ ∇ µ φ ∇ ρ φ ∇ ν ∇ σ φ − R µρ φ ∇ ν φ ∇ σ φ − φ ∇ σ ∇ ν R µρ − φ ∇ ν R µρ ∇ σ φ (cid:21) − α φ (cid:20) W µνρσ R µρ R νσ + R µν R ντ R τµ − R (cid:21)(cid:27) , along with the additional non-linear terms in the Weyl transformation, δφ (cid:12)(cid:12)(cid:12)(cid:12) α = α S µν φ ∇ µ φ ∇ ν σ. (4.9)Here S µν is the Schouten tensor, defined in general d as S µν = 1( d − (cid:18) R µν − d − Rg µν (cid:19) . (4.10)This action and Weyl transformation will also have ambiguities due to field re-definitions thatvanish in flat space (coming from order α terms in (4.6)), as well as ambiguities from possibleWeyl invariants. Here we have not attempted to parameterize all these ambiguities but havesimply made a choice for which the action and Weyl transformation are manageable, in order toillustrate that there is no obstruction to extending the Weyl symmetry to order α . The first line of(4.8) matches those found in the action linear in the curvatures constructed in [11] by demandingthe theory have a vanishing stress tensor in flat space. In the third line of (4.8) there are additionalterms linear in the curvature that do not appear in [11], but these are proportional to the Weyltensor and so they do not contribute to the flat space stress tensor. Thus what we find is consistentwith [11] up to order α . It would be interesting if the action in [11] could be extended to an actionwhich is fully Weyl invariant to all orders. A similar procedure can be applied in the case of the special galileon in d = −
2, and once againwe find that the theory can be improved to be Weyl invariant, at least to O ( α ). The non-linearconformal transformations (3.2), (3.3) get uplifted to δφ = 2 σφ − α ∂φ ) ∇ µ σ ∇ µ φ , (4.11)in a general background metric. A priori there are many more terms that could enter this transfor-mation, however requiring that the Weyl transformations commute as in (4.7) reduces this number,18nd employing field redefinitions that vanish on flat space analogous to (4.6) removes the rest. Wefind the following action is Weyl invariant under (4.11) in d = −
2, up to order α , S = (cid:90) d d x √− g (cid:26) −
12 ( ∂φ ) − Rφ − α
12 ( ∂φ ) (cid:104) ( (cid:3) φ ) − ( ∂ µ ∂ ν φ ) (cid:105) (4.12) − α R µν φ ∇ µ ∇ τ φ ∇ ν ∇ τ φ + α Rφ ∇ µ ∇ ν φ ∇ µ ∇ ν φ + α R µν φ (cid:3) φ ∇ µ ∇ ν φ − α Rφ ( (cid:3) φ ) − α W µνρσ φ (cid:20) ∇ ρ φ ∇ µ φ ∇ σ ∇ ν φ − φ ∇ ρ ∇ µ φ ∇ σ ∇ ν φ + W τρνσ φ ∇ µ φ ∇ τ φ (cid:21) − α (cid:20) R µν R µτ φ ∇ τ φ ∇ ν φ − R µν ( ∂φ ) φ − R ( ∂φ ) φ − RR µν φ ∇ µ φ ∇ ν φ (cid:21) − α (cid:20) R νρ ∇ σ W µνρσ φ ∇ µ φ − W µνρσ R µρ φ ∇ σ φ ∇ ν φ − R µρ ∇ ν ∇ σ W µνρσ φ (cid:21) + α (cid:20) W µνρσ R µρ R νσ − R µν R µρ R νρ + 14 R µν R + 136 R (cid:21) φ (cid:27) + O (cid:0) α (cid:1) . The terms in the second line are the same as those in [11] found by demanding tracelessness ofthe stress tensor in flat space . In the third line there are additional terms linear in the curvaturethat do not appear in [11], but these can be removed by field re-definitions that vanish in flatspace with σ = 2 b · x , at the expense of making the Weyl transformation (4.11) more complicated.Although admittedly not very illuminating, we present these results both to demonstrate there isno immediate obstruction to Weyl invariance, and in the hope that these actions may be found toarise from a more fundamental understanding of the Weyl invariance of these theories to all ordersin α . We have found the field transformations behind the conformal symmetries for DBI in dimension d = 0 and the special galileon in dimension d = − d + 1 dimensional target space in the embedded brane wayof understanding DBI. In the special galileon case it is a real form of the special linear algebra (itwould be interesting if this could be understood from a geometric picture of the special galileon[29, 35]). We have also found the corresponding Weyl invariances, to second order in the couplingin the DBI case and to lowest order in the coupling in the special galileon case.A natural question is whether theories with higher order shift symmetries [36, 37], δφ ∼ x k for k >
2, could emerge and have conformal symmetry. Continuing the pattern we see here for DBI Our expressions match those in [11] for α = − k = 1) and special galileon ( k = 2), we would expect such a theory to be conformal in dimension d = − k − − k . We would expect the shift symmetries andnew symmetries found by commuting with the conformal generators to form a rank k symmetric-traceless representation of the so (2 , d ) conformal algebra, where the conformal primary has weight − k . However, as shown in [21], the commutators of these generators among themselves cannotclose back to the conformal algebra when k >
2, except in the trivial case where such commutatorsare all abelian, which would presumably correspond to a free theory or one whose structure is notfixed in a non-trivial way by the symmetry (this is consistent with the results of [38, 39] forbiddinga non-trivial algebra for k > d , it would be inter-esting to explore whether the tools of the modern conformal bootstrap [42, 43] could be extendedto these unphysical dimensions and brought to bear in order to non-perturbatively explore thesetheories. An immediate obstacle is that the conformal transformations we have are non-linear andso the fields do not transform as conformal primaries in the usual way, so the form of correlators,crossing symmetry, etc. are all presumably modified. Alternatively, perhaps there are compositeoperators that transform in the usual linear way as conformal primaries, given that the basic field φ transforms non-linearly. If such operators exist, their correlators could be a target for the boot-strap. But there is no guarantee that they should exist, especially given expectations that these softscalar theories are not true local field theories with local operators all the way to the UV [44, 45]. Acknowledgments:
We would like to thank James Bonifacio, Austin Joyce and David Ste-fanyszyn for helpful conversations. KH acknowledges support from DOE grant DE-SC0019143, KHand KF acknowledge support from Simons Foundation Award Number 658908. OH was supportedby the GA ˇCR Grant EXPRO 20-25775X.
References [1] C. Cheung, K. Kampf, J. Novotny, and J. Trnka, “Effective Field Theories from Soft Limitsof Scattering Amplitudes,” Phys. Rev. Lett. (2015) no. 22, 221602, arXiv:1412.4095[hep-th] .[2] C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, “A Periodic Table of EffectiveField Theories,” JHEP (2017) 020, arXiv:1611.03137 [hep-th] .[3] K. Hinterbichler and A. Joyce, “Hidden symmetry of the Galileon,” Phys. Rev. D (2015)no. 2, 023503, arXiv:1501.07600 [hep-th] .[4] J. J. M. Carrasco and L. Rodina, “UV considerations on scattering amplitudes in a web oftheories,” Phys. Rev. D (2019) no. 12, 125007, arXiv:1908.08033 [hep-th] .205] F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in ArbitraryDimensions,” Phys. Rev. Lett. (2014) no. 17, 171601, arXiv:1307.2199 [hep-th] .[6] F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein ToYang-Mills, DBI and NLSM,” JHEP (2015) 149, arXiv:1412.3479 [hep-th] .[7] F. Cachazo, P. Cha, and S. Mizera, “Extensions of Theories from Soft Limits,” JHEP (2016) 170, arXiv:1604.03893 [hep-th] .[8] C. Cheung, C.-H. Shen, and C. Wen, “Unifying Relations for Scattering Amplitudes,” JHEP (2018) 095, arXiv:1705.03025 [hep-th] .[9] C. Cheung, G. N. Remmen, C.-H. Shen, and C. Wen, “Pions as Gluons in HigherDimensions,” JHEP (2018) 129, arXiv:1709.04932 [hep-th] .[10] M. Carrillo Gonz´alez, R. Penco, and M. Trodden, “Radiation of scalar modes and theclassical double copy,” JHEP (2018) 065, arXiv:1809.04611 [hep-th] .[11] C. Cheung, J. Mangan, and C.-H. Shen, “Hidden Conformal Invariance of Scalar EffectiveField Theories,” Phys. Rev. D (2020) no. 12, 125009, arXiv:2005.13027 [hep-th] .[12] S. M. Carroll, Spacetime and Geometry. Cambridge University Press, 7, 2019.[13] K. Hinterbichler, A. Joyce, J. Khoury, and G. E. Miller, “DBI Realizations of thePseudo-Conformal Universe and Galilean Genesis Scenarios,” JCAP (2012) 030, arXiv:1209.5742 [hep-th] .[14] G. Mack, “All unitary ray representations of the conformal group SU(2,2) with positiveenergy,” Commun. Math. Phys. (1977) 1.[15] J. C. Jantzen, “Kontravariante formen auf induzierten darstellungen halbeinfacherlie-algebren,”Mathematische Annalen (Feb, 1977) 53–65. https://doi.org/10.1007/BF01391218 .[16] J. a. Penedones, E. Trevisani, and M. Yamazaki, “Recursion Relations for ConformalBlocks,” JHEP (2016) 070, arXiv:1509.00428 [hep-th] .[17] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality,analyticity and an IR obstruction to UV completion,” JHEP (2006) 014, arXiv:hep-th/0602178 .[18] P. Cooper, S. Dubovsky, and A. Mohsen, “Ultraviolet complete Lorentz-invariant theory withsuperluminal signal propagation,” Phys. Rev. D (2014) no. 8, 084044, arXiv:1312.2021[hep-th] .[19] G. Goon, K. Hinterbichler, and M. Trodden, “Symmetries for Galileons and DBI scalars oncurved space,” JCAP (2011) 017, arXiv:1103.5745 [hep-th] .2120] G. Goon, K. Hinterbichler, and M. Trodden, “A New Class of Effective Field Theories fromEmbedded Branes,” Phys. Rev. Lett. (2011) 231102, arXiv:1103.6029 [hep-th] .[21] J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, “Shift Symmetries in (Anti) deSitter Space,” JHEP (2019) 178, arXiv:1812.08167 [hep-th] .[22] C. de Rham and A. J. Tolley, “DBI and the Galileon reunited,” JCAP (2010) 015, arXiv:1003.5917 [hep-th] .[23] K. Hinterbichler, M. Trodden, and D. Wesley, “Multi-field galileons and higher co-dimensionbranes,” Phys. Rev. D (2010) 124018, arXiv:1008.1305 [hep-th] .[24] C. Burrage, C. de Rham, and L. Heisenberg, “de Sitter Galileon,” JCAP (2011) 025, arXiv:1104.0155 [hep-th] .[25] M. Trodden and K. Hinterbichler, “Generalizing Galileons,” Class. Quant. Grav. (2011)204003, arXiv:1104.2088 [hep-th] .[26] G. Goon, K. Hinterbichler, and M. Trodden, “Galileons on Cosmological Backgrounds,”JCAP (2011) 004, arXiv:1109.3450 [hep-th] .[27] A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,”Phys. Rev. D (2009) 064036, arXiv:0811.2197 [hep-th] .[28] C. de Rham, S. Melville, A. J. Tolley, and S.-Y. Zhou, “Massive Galileon Positivity Bounds,”JHEP (2017) 072, arXiv:1702.08577 [hep-th] .[29] D. Roest, “The Special Galileon as Goldstone of Diffeomorphisms,” JHEP (2021) 096, arXiv:2004.09559 [hep-th] .[30] K. Farnsworth, M. A. Luty, and V. Prilepina, “Weyl versus Conformal Invariance inQuantum Field Theory,” JHEP (2017) 170, arXiv:1702.07079 [hep-th] .[31] F. Wu, “Note on Weyl versus Conformal Invariance in Field Theory,” Eur. Phys. J. C (2017) no. 12, 886, arXiv:1704.05210 [hep-th] .[32] G. K. Karananas and A. Monin, “Weyl vs. Conformal,” Phys. Lett. B (2016) 257–260, arXiv:1510.08042 [hep-th] .[33] C. Brust and K. Hinterbichler, “Free (cid:3) k scalar conformal field theory,” JHEP (2017) 066, arXiv:1607.07439 [hep-th] .[34] Y. Nakayama, “Conformal equations that are not Virasoro or Weyl invariant,” Lett. Math.Phys. (2019) no. 10, 2255–2270, arXiv:1902.05273 [hep-th] .[35] J. Novotny, “Geometry of special Galileons,” Phys. Rev. D (2017) no. 6, 065019, arXiv:1612.01738 [hep-th] . 2236] K. Hinterbichler and A. Joyce, “Goldstones with Extended Shift Symmetries,” Int. J. Mod.Phys. D (2014) no. 13, 1443001, arXiv:1404.4047 [hep-th] .[37] T. Griffin, K. T. Grosvenor, P. Horava, and Z. Yan, “Scalar Field Theories with PolynomialShift Symmetries,” Commun. Math. Phys. (2015) no. 3, 985–1048, arXiv:1412.1046[hep-th] .[38] M. P. Bogers and T. Brauner, “Lie-algebraic classification of effective theories with enhancedsoft limits,” JHEP (2018) 076, arXiv:1803.05359 [hep-th] .[39] D. Roest, D. Stefanyszyn, and P. Werkman, “An Algebraic Classification of ExceptionalEFTs,” JHEP (2019) 081, arXiv:1903.08222 [hep-th] .[40] E. Joung and K. Mkrtchyan, “Partially-massless higher-spin algebras and theirfinite-dimensional truncations,” JHEP (2016) 003, arXiv:1508.07332 [hep-th] .[41] J. Bonifacio, K. Hinterbichler, L. A. Johnson, and A. Joyce, “Shift-Symmetric Spin-1Theories,” JHEP (2019) 029, arXiv:1906.10692 [hep-th] .[42] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, “Bounding scalar operator dimensions in4D CFT,” JHEP (2008) 031, arXiv:0807.0004 [hep-th] .[43] D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, NumericalTechniques, and Applications,” Rev. Mod. Phys. (2019) 015002, arXiv:1805.04405[hep-th] .[44] S. Dubovsky, R. Flauger, and V. Gorbenko, “Solving the Simplest Theory of QuantumGravity,” JHEP (2012) 133, arXiv:1205.6805 [hep-th] .[45] L. Keltner and A. J. Tolley, “UV properties of Galileons: Spectral Densities,” arXiv:1502.05706 [hep-th]arXiv:1502.05706 [hep-th]