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Sciences, of Academy Chinese colo hsc n srnm,QenMr nvriyo Lo of University Mary Queen Astronomy, and Physics of School .Mn tde aebe de- been have studies Many ]. 3 9 EC/eateto hsc,Cs etr eev Univers Reserve Western Case Physics, of CERCA/Department , 10 .Teeaeraosto reasons are There ]. 11 00 uldAe lvln,O 40,U.S.A. 44106, OH Cleveland, Ave, Euclid 10900 8 , —e,frexample, for ]—see, 8 16 weebt met- both (where ] ieEdRa,Lno,E N,U.K. 4NS, E1 London, Road, End Mile – 20 hc would which ] ,3, 2, 21 – 23 † uHn Xing, Yu-Hang .Since ]. bigravity a lmnt h Dgotblwtesrn coupling strong the below ghost [ one liter- BD appropriately, scale the the metrics in two eliminate out the can pointed combining by been has ature: exception an theless, xlrn h hs-reeso hsmte opig[ coupling matter this of ghost-freeness the exploring ln cl.Ti a rfudcneune ntephe- cou- the on strong consequences the profound below has ghost This BD scale. the pling avoids that metric osbefrteeetv opst erco e.[ Ref. of not metric composite is effective this the that for revealed possible have considerations additional BD the re-introduces generically [ field cosmological ad- ghost matter richer an a a of to cou- a pling out admits such it built constructing metrics Unfortunately, metrics, coupling phenomenology. both the matter of of a one mixture that to coupling possible minimal is a with tions tests. observational mechanically, towards quantum them propagating and before theoretically classically are both couplings which consistent, detail to important is it nti ae,w rv httennmnmlcoupling non-minimal [ the Ref. that in prove proposed we scale. originally coupling paper, also strong this the are above In which even or couplings below matter ghost-free non-minimal other are 23 [ condition and vielbein formulation symmetric vielbein the the the may relaxing to this switching to that by owing achieved argued been be redressed has be It coupling would mechanism. 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No Ostrogradski ghosts in the decoupling limit choice of matter couplings in these theories to at most two free parameters for each matter sector. We will first impose the condition that the non- minimal matter coupling (1) does not give rise to Os- trogradski instabilities [32] for the scalar mode in the II. UNIQUENESS OF THE COMPOSITE decoupling limit. The precise meaning of this limit can METRIC be found, for example, in Ref. [2]. But for our purposes this means that we focus on the scalar Stückelberg mode, We will be working in D dimensions and consider a taking the limit matter coupling of the form

(I) (I) (I) gµν ηµν , (7) matter = m(G , ψ , ∂ψ ), (1) → L L µν a b a a aµ I fµν ∂µφ ∂ν φ ηab, with φ = x η ∂µπ , (8) X → − (I) where Gµν are effective, composite metrics for the I-th where π is the scalar Stückelberg mode. In this limit, we matter sector denoted collectively as ψ(I). Oneor bothof have the two metrics gµν and fµν are assumed to have a stan- µ µ µ = Π = ∂ ∂ν π, (9) dard kinetic term, corresponding to massive gravity or Kν ν bigravity respectively. We shall consider a strictly local composite metric, by which we mean that we only con- whilst the tensor and vector modes are suppressed. We will use ηµν to lower the indices in this subsection. Con- sider point-wise operations, including inverting gµν and µ µ (I) sequently, we will use ν and Πν interchangeably in this fµν , in constructing Gµν (x) out of gµν (x) and fµν (x), (I) K but not their derivatives or non-local operations. The subsection. Gµν are now simply functions of Πµν (and dRGT graviton potential is given by ηµν ). To avoid the Ostrogradski ghost [32], we shall require D the contribution to the π equation of motion coming from µ1 µ2 µs = αsUs( ), with Us( )= µ2 , (2) to not contain higher order derivatives, either in U K K K[µ1 K ···Kµs] matter s=0 Lπ or in the matter fields. Since the contributions from X (I) (I) (I) where µ is defined by different m(Gµν , ψ , ∂ψ ) contain different matter Kν fields, theseL different contributions do not cancel each ρ ρ σ σ fµν = gµρ(δ )(δ ), (3) other in the π equation of motion. This implies that we σ − Kσ ν − Kν µ can focus on one matter sector, and, omit the index I with the branch choice such that ν 0 when gµν µ K → → here and afterwards. Thus, the contributions to the π fµν . ν can be viewed as the deviation of fµν from gµν . K µ equation of motion arising from m(Gµν ,ψ,∂ψ), Without loss of generality, we can choose gµν and ν as L (I) K the elementary building blocks to construct Gµν . Then µν ∂Gµν π = ∂ρ∂σ √ GT the most general composite metric is given by E − ∂Π  ρσ  G(I) = G(I)N , (4) µν ∂Gµν µν ∂Gµν µν µν = ∂ρ∂σ(√ GT ) +2∂(ρ(√ GT )∂σ) N − ∂Πρσ − ∂Πρσ X µν ∂Gµν with + (√ GT ) ∂ρ∂σ , (10) − ∂Πρσ N (I)N (I) 2 n ρ G = gµρ p ([ ], [ ], ...)( ) , (5) µν N,n K K K ν should not contain higher order derivatives, where G is n=0 X the determinant of Gµν and the energy momentum tensor where [] is the trace of the matrix enclosed, from the I-th matter sector is given by (I) 2 pN,n([ ], [ ], ...) are arbitrary functions of the various K K µν 2 ∂ m(Gµν ,ψ,∂ψ) traces of matrix µ, ( 2)ρ ρ σ and so on. Note that T = − L . (11) ν ν σ ν √ G ∂Gµν (I) K K ≡ K µK ρ p is (N n)-th order in . Since µν gµρ is − (N,n) − Kν K ≡ Kν symmetric in its indices, the indices µ and ν in Eq. (5) It is usually assumed that the matter sectors are diffeo- are symmetrized implicitly. morphism invariant separately, so we have the energy The strategy of our proof is to impose two consistency momentum conservation for each sector (I) conditions at different steps to restrict the form of Gµν µν µ νρ ∂ν (√ GT )+ Γνρ √ GT =0 , (12) to that of Ref. [21]: − − µ (I) 2 ρ ρ 2 2 ρ where Γνρ is the Christoffel coefficients associated with Gµν = gµρ α(I)δν +2α(I)β(I) ν + β(I)( )ν , (6) µν K K the metric Gµν . Now, since T contains terms with where α(I) and β(I) are constant. When β(I)
= 0 or first (or higher) derivatives of the matter fields, the first β = α , a minimal matter coupling is reproduced. term of Eq. (10) contains terms with third (or higher) (I) − (I) 3 derivatives of the matter fields, which cannot be canceled In other words, Eq. (21) has to be an identity. We will by other terms. Therefore, as a necessary condition, we make use of this identity to constrain the form of Gµν . impose the first term of Eq. (10) to vanish identically: Now, since this is an identity, different orders of Π, and thus different orders of , should cancel separately, µν ∂Gµν K ∂ρ∂σ(√ GT ) =0. (13) so it is sufficient to consider the N-th order terms of the ∂Π − ρσ general ansatz For later convenience, we define N µν µν ρσ = ∂ρ∂σ(√ GT ) , (14) N 2 n ρ T − Gµν = gµρ pN,n([ ], [ ], ...)( )ν , (23) µν µν K K K ρ = ∂ρ(√ GT ) , (15) n=0 T − X µν √ µν 2 N−n = GT , and (16) where pN,n([ ], [ ], ...) are of order ( ). T − K K O K ρσ ∂Gµν As an identity, Eq. (21) should be solved by any µν = . (17) G ∂Πρσ configuration of π. To further simplify our discus- µν µν sion, it is sufficient to choose a diagonal configura- Since ρ and ρσ are first and second derivatives of µ T T tion for Πν . (An alternative point of view, which also µν respectively, their numerical values at a specific, ar- µ T works for our purposes, is that Πν can always be di- bitrarily chosen point in spacetime would be independent agonalized via an appropriate coordinate transformation from that of µν in the absence of Eq. (12). That is, by T around a given point.) Suppose λ0, λ1, , λD−1 are choosing the matter configuration appropriately in the µ { ··· αα } the diagonal components of Πν . Then, the ββ (α = neighborhood of a specific point, the numerical values of ρσ 6 µν µν µν β, no summation for αα and ββ) component of µν = ρσ, ρ and can be assigned independently at G Tthat point,T subjectT to the following constraints 0 gives e µν = σν Γ µ σν ∂ Γ µ, (18a) ∂ ρν ρ σν ρ σν N αα = N αα = ηαα GN 0. (24) T µν −T νρ µ − T ββ ββ ββ ν = Γνρ . (18b) G G ∂λα ≡ T −T αα e N N To simplify our discussion, we choose a matter configu- Since η = 1, we have ∂G /∂λα = 0. Thus, G µν µν ββ ββ ration where ρ and vanish at a spacetime point. ± N must be independent of λα(α = β). Since G is N-th This can alwaysT be achievedT as follows: Suppose there 6 ββ ′µν µν order in Π, it must be of the form are two matter configurations where ρ p = ρ p ′µν µν T | T | and p = p, where p is a spacetime point; That GN = C λN , (25) T µν| T | µν ββ β β is, ∆ ρ p = 0 and ∆ p = 0; Then, one takes T | µν T | ∆ π = 2∂ρ∂σ [ ∆ ∂Gµν /∂Πρσ] as our starting π. E − T E where Cβ is a constant. It follows from Lorentz invariance After this, at that point the constraint system reduces to that the only possible form of Cβ should be Cηββ, with a condition that is much easier to handle: N µ N µ C being a constant. Since we have (Π )ν = λµ δν (no µν summation for µ) for the diagonal configuration chosen, ρν =0. (19) T we must have To extract the conditions on Gµν encoded in Eq. (13), µν N N ρ N we need to project out the traces of ρσ. That is, we Gµν = C ηµρ(Π )ν = C (Π )µν . (26) µνT need a projector such that ( ) ρν vanishes for un- µν P PT constrained ρσ. This reduces the system of Eqs. (13) Again, by Lorentz invariance, these relations must be also T µ and (19) to a single equation: satisfied by the non-diagonal components of Πν . N ρ ρσ µν Therefore, there is only one term, pN gµρ( ) , at N- µν ( ) ρσ =0. (20) ν G PT th order that survives the consistency checkK in the de- ρσ One can shift the projector to act on µν instead coupling limit, and we end up with µν P G and leave ρσ to be a generic tensor. Then, getting T µν N N ρ rid of the generic tensor ρσ, Eq. (20) reduces to the Gµν = G = gµρ pN ( ) . (27) T µν K ν requirement: N N X X ρσ ρσ µν = ( )µν =0. (21) G PG where pN now are constant. In summary, we have re- In D dimensions, such a projector is explicitly given by2 duced our ansatz in Eq. (5) to the one in Eq. (27) by e requiring that the non-minimal matter coupling does not ρσ ρσ 4 (ρ σ)γ µν = µν δ give rise to Ostragradski ghosts in the decouling limit. G G − D +2 (µGν)γ 2 ρ σ αβ e + δ(µδν) αβ . (22) (D + 2)(D + 1) G 1. Example: Lowest orders

Before moving on to the next step of the proof, it is 2 See Appendix A for details on the derivation. instructive to give a concrete example to illustrate how 4 these abstract arguments work in essence. Consider the to most general composite metric, up to 2nd-order in : 2 ρ ρ 2 2 ρ K Gµν = gµρ α δ +2αβ + β ( ) . (33) 2 ν ν ν Gµν =gµν + a [ ]gµν + a µν + b [ ]gµν K K 1 K 2K 1 K 2 ρσ First, notice thatp0 should be non-zero (positive
defi- + b [ ] gµν + b [ ] µν + b µρg σν , (28) 2 K 3 K K 4K K nite if the signature of the metric is taken into account), µ where ai and bi are constants and ν is to be evaluated otherwise the effective metric Gµν becomes singular in µ K µ in the decoupling limit as Πν . The identity in Eq. (21) the limit ν gµν →fµν 0. Therefore, we can re-write can be straightforwardly calculated K | → ρ ρ Gµν = p0 gµρ (δν + Pν ( )) (34) ρσ ρσ 2a1 ρ σ 8b1 +4b3 (ρ σ) K µν = a1η ηµν δ(µδν) Π(µδν) G − D +1 − D +2 ρ ′ N ρ ρσ ρσ ρσ = p0 gµρ δν + pN ( )ν , (35) +2b1Π ηµν +2b2[Π]η ηµν + b3η Πµν K e N=1 ! X 4b1 4b2(D +2)+2b3 ρ + − [Π]δ δσ ′ −1 (D + 1)(D + 2) (µ ν) where pN = p0 pN . Note that pN are constant here. Then the determinant of ansatz (34) can be re-cast as =0 . (29) D When D > 2, all the terms in Eq. (29) cannot cancel √ G = p 2 √ g det( 1+ P ( )) (36) − 0 − K each other, so Eq. (29) enforces a1 = b1 = b2 = b3 = 0. D p 2 √ g det(1p + Q( )) (37) Thus, up to 2nd order, the consistency requirement in ≡ 0 − K D the decoupling limit implies D 2 = p √ g Us(Q( )) , (38) G = g + a + b gρσ . (30) 0 − K µν µν 2 µν 4 µρ σν s=0 ! K K K X When D = 2, we can get the same result, but one where needs to take into account the Cayley–Hamilton theo- µ N µ rem when checking cancellations between the terms in Q ( )= qN ( ) , (39) ν K K ν Eq. (29). (The Cayley–Hamilton theorem states that: N=1 suppose that p(λ)=0 is the characteristic polynomial of X ′ matrix A, then substituting A for λ in the polynomial and qN can be expressed in terms of pN by Taylor ex- gives rise to an identity, p(A)=0. To make use of this panding 1+ P ( ) and comparing to the coefficients N µ K identity in Eq. (29), one can differentiate the identity of ( ) . For the requirement (31) to go through, the K νp with respect to A: ∂p(A)/∂A = 0.) When N = D, the following equation

Cayley–Hamilton identity is used directly; when N >D, D D one multiplies p(A)=0 with powers of A to get relevant Us(Q( )) = asUs( ) (40) identities. The diagonalization of Πµ in the last subsec- K K ν s=0 s=0 tion, on the other hand, is a convenient way to avoid the X X complications due to the Cayley–Hamilton identities for should be satisfied for some constant as. We will check N D. what this requirement implies order by order in . ≥ The 0-th order equation can be satisfied byK setting a = 1. At order 1, we have q U ( ) = a U ( ), which 0 1 1 K 1 1 K B. No BD ghost from matter loop corrections gives q1 = a1. At order 2, we have

2 2 q U2( )+ q2U1( )= a2U2( ), (41) Given an effective composite metric Gµν in dRGT mas- 1 K K K sive gravity or bigravity, it is natural to include the cos- which leads to mological term √ GΛ, Λ being constant, in the La- − 2 grangian. If it is not there in the bare Lagrangian, it has q1 = a2, q2 =0 . (42) been shown that matter loop corrections will generically generate a cosmological term for the effective metric [10], At order 3, making use of the fact that q2 =0, we have much like that in general relativity. So, to avoid matter q3U ( )+ q U ( 3)= a U ( ), (43) quantum corrections to re-introduce the BD ghost, we 1 3 K 3 1 K 3 3 K require which leads to D 3 √ G = √ g asUs( ), (31) q1 = a3, q3 =0 . (44) − − K s=0 X This can be extended to arbitrary orders, so that, at where as are constants and Us( ) are defined in Eq. (2). order s, we simply have We will show that this requirementK is sufficient to reduce 1 D : qsU1( )=0. (46) N N K X X 5

That is, by solving Eq. (40) order by order, we can con- starting point, one can reproduce the non-minimal cou- clude that pling in the metric formulation (33), if one imposes the symmetric vielbein condition [27]. On the other hand, q1 = a1, qs = 0 (s> 1) , (47) if one imposes a modified vielbein condition, as in [23], one would end up with a convoluted metric theory that is which leads to physically different from the theory with (33). However, despite being complicated and exotic, the theory with the 2 ρ Gµν = p0gµρ (1 + q1 ) ν . (48) modified vielbein condition has the same decoupling limit K as the theory with the symmetric vielbein condition [27]. Therefore, by requiring the BD ghost does not re-emerge under matter loop corrections, we have reduced the ansatz in Eq. (32) to Eq. (33), as advertised. This is pre- Note added: The uniqueness of the composite metric in cisely the effective composite metric initially proposed in the vielbein formulation has been argued in Ref. [27], Ref. [21], and it emerged here naturally from requiring which appeared when our paper was being finalized. Our absence of Ostrogradski ghosts in the decoupling limit proof in the metric formulation is complementary to the and absence of the BD ghost from matter loop correc- comments in Ref. [27]. tions. ACKNOWLEDGMENTS

III. CONCLUSION We would like to thank Claudia de Rham, Kurt Hin- terbichler, Rachel Rosen and Andrew J. Tolley for useful In this letter we have explored a generic class of com- discussions. RHR acknowledges the hospitality of the posite couplings to matter in dRGT massive gravity and Perimeter Institute of Theoretical Physics during stages bigravity, which involve a generic admixture of the met- of this work. RHR’s research was supported by a DOE rics gµν and fµν . We have imposed two diagnostic tests grant de-sc0009946, the STFC grant ST/J001546/1 and to ensure the ghost-freeness of the theory at least be- in part by Perimeter Institute for Theoretical Physics. low the strong coupling scale. First, we have required Research at Perimeter Institute is supported by the Gov- such non-minimal coupling not to give rise to Ostrogradki ernment of Canada through Industry Canada and by the ghosts for the scalar Stückelberg mode in the decoupling Province of Ontario through the Ministry of Economic limit. This has allowed us to discard a big subset of all Development & Innovation. SYZ acknowledges support such couplings. Furthermore, we have imposed that mat- from DOE grant DE-SC0010600, and would like to thank ter loop corrections do not re-introduce the BD ghost, Kavli Institute for Theoretical Physics China at the Chi- which has allowed us to single out one composite met- nese Academy of Sciences for hospitality during part of ric with two free parameters as the unique non-minimal this work. QGH, YHX and KCZ are supported by the coupling to matter. This is precisely the composite met- grants from NSFC (grant NO. 11322545 and 11335012). ric proposed in Ref. [21], and given in the metric lan- guage in Eq. (33). Consequently, cosmological solutions in these theories only depend on a finite choice of healthy Appendix A: Projector P couplings to matter at energy scales comparable to the strong coupling scale. ρσ Consider a generic (2, 2) tensor Λµν where the up and We note that our proof does not assume any specific down two indices are symmetric respectively. We want form of the matter fields in the non-minimal matter cou- ρσ to derive a projector that projects out the traces in Λµν . pling – One can view ψ as a vector field encompassing ρσ ρν That is, for generic Λµν , we need ( Λ)σν =0. Consider- all possible fields. However, we do assume that mat- ing the index structure of Λρσ , theP only trace terms are ter fields are not coupled to derivatives of the metrics µν α(ρ σ) αβ (ρ σ) and in each matter sector all matter fields couple to one Λα(µδν) and Λαβδ(µδν) . Therefore we have universal composite metric. More generically, however, ρσ ρσ α(ρ σ) αβ (ρ σ) one may consider a case where derivatives of the metrics ( Λ)µν =Λµν + aΛα(µδν) + bΛαβδ(µδν) , (A1) also enter the matter couplings and matter fields cou- P ple to the two metrics in a convoluted non-trivial way. where a and b are constants. To determine a and b, we Whether ghost free non-minimal couplings generically ex- impose the condition that the right hand side of Eq. (A1) ist in this case is beyond the scope of this letter, but such vanish identically when σ and ν are contracted. This an exotic theory should exist and be free of the BD ghost leads to, in D dimensions, below the strong coupling scale, based on the results of 3 4 2 Ref. [23, 27] : The non-minimal matter coupling (33) has a = , b = . a very simple representation in the vielbein formulation. −D +2 (D + 1)(D + 2) Taking this vielbein non-minimal matter coupling as the 6

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3 We would like to thank Kurt Hinterbichler for discussions on this.