PHYSICAL REVIEW LETTERS 122, 251101 (2019) Interactions of Multiple Spin-2 Fields beyond Pairwise Couplings S. F. Hassan1 and Angnis Schmidt-May2 1Department of Physics and The Oskar Klein Centre, Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden 2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany (Received 21 July 2018; revised manuscript received 16 April 2019; published 26 June 2019) Thus far, all known ghost-free interactions of multiple spin-2 fields have involved at most pairwise couplings of the fields, which are direct generalizations of bimetric interactions. We present a class of spin-2 theories with genuine multifield interactions and explicitly demonstrate the absence of ghost instabilities. The construction involves integrating out a nondynamical field in a theory of spin-2 fields with only pairwise ghost-free interactions. The new multivierbein interactions generated are not always expressible in terms of the associated metrics. DOI: 10.1103/PhysRevLett.122.251101 Introduction.—Interacting theories for multiple fields theory by simply adding copies of the bimetric potentials 1 2 ð Þ with spin 0, = , and 1 are well understood and realized Vbi g; f for pairs of metrics, but without forming loops I in nature via the standard model of particle physics, where [10]. An example for four metrics gμν would be ð 1 2Þþ ð 1 3Þþ ð 3 4Þ the multiplets and their mixings are crucial for the viability Vbi g ;g Vbi g ;g Vbi g ;g . So far, these pair- of the theory. In contrast, general relativity is the simplest wise couplings have been the only known ghost-free possible interacting theory of a single spin-2 field. It is a interactions of multiple spin-2 fields. fundamental question as to whether, in analogy with lower An important class of multiple spin-2 theories was spins, consistent theories of multiple spin-2 fields could constructed in Ref. [11] using antisymmetrized products exist. Such theories could have profound implications for of the vierbein fields which appeared to be ghost-free. the understanding of the gravitational force beyond general However, a more detailed analysis in Ref. [12] revealed that relativity, but they have not been easy to construct. generically such multivierbein interactions contained In a covariant setup, spin-2 fields have more components ghosts. It was argued that the ghost-free subset consisted than physically needed, and generic theories do not have only of models where the vierbeins could be traded off for enough symmetries and constraints to remove the unphys- the metrics by virtue of a vierbein symmetrization con- ical components. Some of these, if not eliminated, give rise dition, exactly as in bimetric theory. The only known to ghost instabilities, with an example being the Boulware- models of this type are the pairwise interactions mentioned Deser ghost [1] of a massive spin-2 field. A few years ago above and described in more detail in Ref. [13]. the ghost-free theory of two interacting spin-2 fields was Reference [14] reached a similar conclusion for multiple found [2], which also fulfills some other important con- spin-2 theories in three dimensions. sistency criteria [3,4]. This generalized previous work on a The question is whether there exist interactions between single massive spin-2 field in a fixed background [5–8]. multiple spin-2 fields beyond the pairwise ones, and thus The model is formulated in terms of two symmetric rank-2 beyond the class conjectured in Ref. [12].Hereweshowthat tensors (or “metrics”) gμν and fμν interacting through a this is indeed the case. These are genuine multiple spin-2 ð Þ— “ specific potential Vbi g; f hence the name ghost-free interactions in the sense that, similar to the gluon vertices in bimetric theory.” For a recent review see Ref. [9]. quantum chromodynamics, they lead to direct interactions Theories for more than two spin-2 fields are also strongly among more than two dynamical vierbein fields and cannot be restricted by the absence of ghosts. From the analysis of reduced to pairwise interactions through field redefinitions. Ref. [2], it is easy to see that certain ghost-free theories can Summary of results: In this work we derive a class of be constructed as straightforward extensions of bimetric ghost-free interactions for multiple spin-2 fields by inte- grating out a nondynamical field in a theory with ghost-free bimetric interactions. The result is an interaction term for A N vierbeins ðeIÞμ of the form Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Z N Further distribution of this work must maintain attribution to X ¼ − 4 4 βI ð Þ the author(s) and the published article’s title, journal citation, Smulti M d x det eI ; 1 and DOI. Funded by SCOAP3. I¼1 0031-9007=19=122(25)=251101(6) 251101-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 251101 (2019) involving a mass scale M and arbitrary dimensionless These allow us to expresspffiffiffiffiffiffiffiffiffiffiffiffi the potential in terms of metrics coefficients βI, I ¼ 1; …; N . The kinetic terms of the −1 ¼ −1 ð Þ¼ pbyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi replacing u eI f0 fI in addition to det u vierbein fields have the standard Einstein-Hilbert form. j det f0j. It is then straightforward to verify that the For restricted vierbein configurations the multivierbein arguments for the absence of a ghost in bimetric theory [2] vertex can be expressed in terms of metrics, but this is extend to this case. not always possible. The interactions can involve up to four Multiple spin-2 action: Let us set the coupling m0 ¼ 0 to different vierbeins in each term and are therefore more A freeze the kinematics of the vierbein uμ . We can then general than the pairwise couplings known so far. A eliminate uμ algebraically using its equation of motion to Generating new interactions.—The starting point is a A A obtain an action for the remaining N vierbeins (note that theory for ðN þ 1Þ vierbeins, uμ and ðe Þ , I ¼ 1; …; N , I μ “ ” → ∞ with ghost-free bimetric interactions. Let us denote the massive gravity limit of m0 gives a similar A 0 A B I starting action, but with a fixed background uμ which the corresponding metrics by fμν ¼ uμ ηABuν and fμν ¼ A B cannot be varied and eliminated; see Ref. [16]). Indeed, ðeIÞμ ηABðeIÞν . The action has the following structure: varying the action (2) for m0 ¼ 0 with respect to the inverse μ XN vierbein uA gives an equation with the unique solution S½u; e ¼ S ½fJþS ½u; e : ð2Þ I EH int I N ¼0 3 X J A A uμ ¼ − ðeIÞμ : ð7Þ β0 It includes the Einstein-Hilbert kinetic terms I¼1 Z A pffiffiffiffiffi Using this to eliminate uμ in Eq. (4) gives the new ½ J¼ 2 4 J ð JÞ ð Þ SEH f mJ d x f R f ; 3 interactions for the remaining N vierbeins, Z ðN þ 1Þ ½ ¼− 4 4 ð þ þÁÁÁþ Þ ð Þ where mJ are the Planck masses. Sint contains the Sint eI M d x det e1 e2 eN ; 8 A simplest ghost-free bimetric interactions between uμ and A each one of the ðeIÞμ , 4 4 −3 with M ≡ −54m β0 . These are not reducible to linear combinations of the pairwise bimetric potentials. That the Z XN 4 4 I −1 starting action (2) was ghost-free does not by itself S ½u; e ¼−2m d x det u β0 þ β Trðu e Þ : int I I guarantee the absence of ghosts in the final theory since I¼1 the procedure changes the number of dynamical and ð Þ 4 nondynamical fields. However, we will explicitly confirm −1 μ the absence of ghosts below. ð Þ¼ ð ÞA A In addition to the traces Tr u eI uA eI μ , we could also The N symmetrization constraints (6), with uμ given by include the remaining ghost-free bimetric interactions, but Eq. (7), provide a set of constraint equations for the new we have chosen not to do so. For brevity, we set βI ¼ 1 for multiple spin-2 theory. Equivalently, one can derive these I I ¼ 1; …; N ,byscalingeI → eI=β and redefining the from the multiple spin-2 action (8) using Eq. (5). In matrix Planck masses mI in Eq. (3) accordingly. notation these new Lorentz constraints are Lorentz constraints: Each vierbein contains six Lorentz parameters that drop out of the corresponding metric and XN XN Tη ¼ Tη ¼ 1 … N ð Þ hence appear in the action (2) only through the potential eJ eI eI eJ;J; ; : 9 I¼1 I¼1 terms in Sint. Since these are nondynamical, their equations of motion are constraints. Specifically, these are the Note that if we sum Eq. (9) over J, the resulting equation is antisymmetric parts of the equations of motion for ðe ÞA I μ identically satisfied. Hence, there are N − 1 independent [11,15], precisely six equations per vierbein, matrix relations among the antisymmetric parts of the ð ÞAη ð ÞB δ δ combinations eJ μ AB eI ν . These can be used to elimi- Sint AB ν Sint AB μ η ðeIÞ − η ðeIÞ ¼ 0: ð5Þ nate 6ðN − 1Þ nondynamical components of the vierbeins. δðe ÞA B δðe ÞA B I μ I ν Another six components are removed by the overall A Lorentz invariance of the action. This leaves us with The corresponding equation for uμ is a linear combination 16N − 6N ¼ 10N independent components, which is of Eq.