JHEP12(2020)030 Springer , decoupling April 28, 2020 3 : Λ October 27, 2020 December 4, 2020 September 6, 2020 : : : Received Revised Accepted Published a,b Published for SISSA by https://doi.org/10.1007/JHEP12(2020)030 [email protected] , cutoff. We construct explicitly the and Andrew J. Tolley 3 a / 1 ) Pl M 2 m = ( 3 . 3 Λ Justinas Rumbutis 2004.07853 a The Authors. c Classical Theories of Gravity, Effective Field Theories, Scattering Amplitudes,

We consider the double copy of massive Yang-Mills theory in four dimensions, , [email protected] massive gravity decoupling limit theory. Although it is known that the double copy of 10900 Euclid Ave, Cleveland, OHE-mail: 44106, U.S.A. [email protected] Theoretical Physics, Blackett Laboratory, ImperialLondon, College, SW7 2AZ, U.K. CERCA, Department of Physics, Case Western Reserve University, 3 b a Open Access Article funded by SCOAP ArXiv ePrint: double copy procedures do notscaling commute of and their we clarify kinematic why factors. this is the case inKeywords: terms of the Duality in Gauge Field Theories are consistent with a limit of this theory and showΛ that it is equivalent toa a nonlinear bi-Galileon extension sigma of model thetheory is standard is a a special more Galileon, general Galileon the theory. decoupling This limit demonstrates of that massive the Yang-Mills decoupling limit and whose decoupling limit isleading terms a in nonlinear the low sigmahas energy model. been effective theory integrated out. of The a Thespin-2, heavy latter obtained Higgs massive double may model, spin-1 copy and in be effective a which field regardedLagrangian massive the theory spin-0 Higgs contains as up field, a the and massive to we fourthinteractions construct match explicitly order those its of interacting in the fields. dRGT massive gravity We theory, and find that that all the up interactions to this order, the spin-2 self Abstract: Arshia Momeni, Massive gravity from double copy JHEP12(2020)030 24 26 10 ] is a relation between the scattering 2 , 22 21 1 [ 23 11 12 10 – 1 – field in 4d double copy B 6 9 7 8 10 14 5 21 20 1 21 17 decoupling limit 3 B.1 Lie algebraB.2 generators of the Polarizations gaugeB.3 group Construction of gravity states from mYM Λ 4.1 Degrees of4.2 freedom Double copy4.3 construction of scattering Double amplitudes copy4.4 of three-point amplitudes Double copy of four-point amplitudes 2.1 Three-point amplitude 2.2 Four-point amplitude 3.1 Three-point amplitude amplitudes of two different theories.that The in BCJ a relation, gauge or theory, colour-kinematics oneso duality, can that states always they represent satisfy kinematic factors ancolour of analogue factors scattering relation by amplitudes to kinematic thedescribing factors gauge other in group theories. a colour factors. given theory, Replacing leads the to new scattering amplitudes 1 Introduction The Bern-Carrasco Johansson (BCJ) C Dualization of the massive D Double copy of the 4-pointE scattering amplitude Decoupling in limit the of decoupling massive limit Yang-Mills amplitude B Conventions 5 6 Discussion A Contact terms 4 Double copy of massive Yang-Mills 3 dRGT massive gravity Contents 1 Introduction 2 Massive Yang-Mills JHEP12(2020)030 3 ]. / m 1 51 ) – Pl 30 M 2 ], where m 58 ]. = ( ] and relations 3 19 29 – Λ – scale [ 16 ], however the case 24 5 / 1 56 ) – ]. An explicit nonlinear Pl 54 60 M ]. Recently the double copy , 4 15 59 m – 13 = ( 5 Λ – 2 – ]. However the ‘double copy’ paradigm has been ] as well as extended gravitational relations such as 3 ]. 12 60 – 4 and so we may naturally expect massive gravity in some form ], efficient gravitational wave calculations [ 1 23 ]. The origin of this relation can be understood from the string – 2 20 ] for an extensive review of recent work in this area. ]. 57 53 , 52 ], at least to quartic order. In fact we will find that the free coefficients in the 60 Remarkably, we find that the double copy paradigm automatically leads to a theory Since a massive spin-1 particle has 3 degrees of freedom in four dimensions, the double In this paper we initiate the application of the double copy paradigm to the scattering The physical applications of double copy extend beyond calculations of scattering am- The first and most important example is the relationship between Yang-Mills theory See for example [ 1 Gabadadze-Tolley (dRGT) model [ in which the interactionsgravity of [ the massive spin-2 field are described by the dRGT massive spin-2 particles are well knownto to lead be to highly a constrained. breakdown of Genericis perturbative the unitarity interactions at spin-2 are the mass. expected Specialscale tunings can which be is made that the raise highesteffective this theory scale possible exhibiting to scale this the in scale is four the dimensions so-called [ ghost-free massive gravity or de Rham- of massive gravity theories, to arise from the double copy procedure. copy theory contains 9particle, propagating a states, single which massive decompose spin-1 particle into and a a single massive massive scalar. spin-2 The interactions of massive spontaneously broken gauge symmetrieswhere have both been of studied thewith in copies only [ of massive gauge gaugebroken bosons) gauge theory symmetries has have (by completely not virtueprocedure broken been of is gauge explored. the still mass symmetry valid, broken On for (i.e. diffeomorphism the the symmetries. bosons) gravitational imply The side, if latter the are the in double the copy purview amplitudes of massive Yang-Mills theory,Mills i.e. the coupled low to energy a effectivebreaks heavy field the Higgs theory gauge field of symmetry (with Yang- On in the the a Higgs gauge way integrated theory that out) side,and all the which act of spontaneously is of the a spontaneously gauge breaking major bosons symmetries is component acquire well the of understood same the mass. standard model. Double copy of gauge theories with effective field theories [ between classical solutions inThe different double theories copy (known has asbackgrounds been [ classical shown double to apply copy) for [ scattering amplitudes around more general relations between two non-gravitational theoriesDBI for or special example non-linear Galileonthat theories sigma between [ model super Yang-Mills and and supergravityparadigm theories was [ extended for gauge theories with massive matterplitudes fields in [ Minkowski spacetime. For example, double copy is used for UV considerations of and gravity amplitudes [ theory point of viewand by looking considering at how theencapsulated open low by and energy the closed effectivefound KLT string field relations to amplitudes theories [ be are of related, more the general, two string and theories. there are This known is examples of extensions of double copy JHEP12(2020)030 1 , u ± but ˆ , n (1.1) (1.2) 0 2 arises , 1 m 0 /m , Σ , 1  0 )+ 2 u m ˆ n by themselves u − s,t,u i c . Here ˆ n u 0 Σ( , 2 0 + 3 → m 2 → t m − m ˆ n m u t m c − = t ). Since in the massive case u + comes from helicity i 2 E.15 s n , n m t ˆ n ˆ n s − c 2 1 s are finite as m  i ) and ( take the form ˆ n 2 )+ 1 cancels out of the gauge theory amplitudes Λ E.14 = 0 and so in the limit Σ , demonstrating the natural decoupling limit = u behaviour in s,t,u Σ ].  ), ( n 3 – 3 – = 0 2 m Σ( 61 + u 2 u − m t /m c n 3 E.13 m 1 n u − = + c m ]. However, the latter are nevertheless more compli- − t + u u t ), ( c . The term s n 62 0 + = + n , t + ˆ s 2 t E.12 c 60 → t n – m n t m , n ]. Even projecting onto the scalar sector, the massive gravity ] and that the decoupling limits of massive gravity theories are + ˆ 58 c − s s , we may regard this as a natural double copy relation between t ˆ n It is known that double copy of a non-linear sigma model is the ˆ n 63 12 2 , , + 2 1 m m/g 62 2 11 , s m 7 )+ n ) by the double copy of the leading helicity-zero interactions of the mas- – s 3 − c 5 Λ = (triple crossing symmetric) and we have Λ s ) 2 s,t,u  are given in eqs. ( 2 m Σ( i g 2 ˆ n s, t, u 3 = 4 = m for fixed m u Σ( − and 0 s + The origin of this is that there are terms needed in the kinematic factors to satisfy This connection is emphasized when we recognize that the leading helicity-0 inter- mYM 4 t strongly suggesting that this is the controlling scale of the EFT at all orders. Since Σ This was for example explicitly proposed in [ = → 2 A s 3 + n by virtue of the colourscaling. relation s satisfy colour-kinematics duality. The interactions since the polarizationthat tensor for for helicity-1 is a finite massive as helicity-0 gluon scales as where in a manner similarwhich to is the crucial generalized to gaugefor understanding transformations why in its the contribution massless is case, finite. a fact The explicit expressions squaring these kinematic factors, theybutions no longer that cancel are and finite give additionalsatisfy in non-zero colour-kinematics the contri- duality decoupling limit. To be precise, the kinematic factors which decoupling limit is not equivalentlimit to procedure a does special Galileon, not and commute so with we the find double that copycolour-kinematics the procedure. decoupling duality that arecel singular out in of the the decoupling gauge limit amplitudes. but nevertheless However when can- we construct the gravity amplitudes by nonlinear sigma model. special Galileon [ also Galileon-like theories [ cated and include inthe particular decoupling non-trivial limit vector [ scalar interactions that survive even in actions of a massivem graviton are dominatedsive in spin-1 the gluon. decouplingmodel, Since limit as the (defined encoded decoupling in by limitthe the taking of interactions Goldstone massive for equivalence Yang-Mills the theorem, is we helicity-0 a may spin-2 reasonably nonlinear states expect sigma are that determined by the double copy of the find that the interactionsΛ of the additionalmassive spin-1 Yang-Mills and is spin-0 itself states anparticle, are EFT namely with also the attwo highest the highest possible scale cutoff cutoff effective for theories. a coloured spin-1 dRGT Lagrangian are fixed by the double copy prescription to this order. We further JHEP12(2020)030 . 3 ), 4 6= 0 ..., 1.4 (1.3) (1.4) i + ˜ n . 0 i 6 3  and our P Λ 2 ΣΣ 0 u A m ˆ n survives as a u − ˆ ···∼ n term is absent. Σ u + 0 6 3  + ) Λ ΣΣ 2 2 0 t m m ˆ n will necessarily be finite t − − ˆ n u ] which allows for a consistent t = 0) + 65 and clarify its inequivalence , )+( 2 2 5 m 0 s , then describe the double copy 64 ( m m , i ˆ n s 3 ˆ n − term is suppressed by virtue of the − ˆ 29 n t , 5 s 24 term enters in the gravity amplitudes  )+( /m 6 3 6 2 1 1 Λ m contains a brief explanation of why giving /m + cutoff scale which is known to be the highest − 1 C 0 s 3 ( / 6 3 1  – 4 – ]. Having determined the gravity Lagrangian up ΣΣ ) Λ 6 and we have in full as an exact statement − 0 Pl 58 and we will recover the special Galileon amplitudes in the m Σ M 2 = Pl 2 ΣΣ m = 0  M m . Appendix − scale. Specifically this will show up as a non-zero spin-2, 2 0 ∼ 0 u 3 B m = = ( n  Λ u u 2 3 − u 0 n n Λ Σ = Σ u m n + ˆ u − + t n u n 2 0 t + + ˆ m ) which are singular. Indeed in the decoupling limit, the gauge theory n 2 t t 0 s − n n m ˆ n 1.1 t t interaction. Similarly the naive − n . However they no longer sum to zero. Using these in a double copy 2 t 2 + ± + 2 , 0 s 2 /m 0 s 0 m i n , m n s n 0 s − , n and dRGT massive gravity theories in section − does not contribute to the gauge theory amplitudes, first taking the decoupling n 0 s s = ˆ 2 Σ   i The paper is organised as follows: first we briefly introduce massive Yang-Mills in It is worth noting that if we give up strict colour-kinematics duality in the massive case, By contrast, when we square to construct the gravity amplitudes, ˜ n 2 2 Pl Pl It is of course technically true that if we only compute amplitudes in which the spin-1 helicity-1 po- 1 1 3 M M larizations are set todecoupling limit. zero, But then this isno an relation inconsistent to procedure the fromexist massive the gravity point an theory of extension whose view of of decouplingremoval the of the limit gravity additional is recipe theory, and degrees a along has of special the freedom. Galileon. lines There discussed may in however [ and the resulting theory has possible cutoff for massive spin-2to fields quartic [ order, weto specify a the decoupling special limit Galileon.conventions in are give section The in precise appendix quartic interactions are given in appendix the massive case in the same manner as thesection massless. prescription and give theIn action particular obtained we find from that squaring the massive colour-kinematics Yang-Mills duality holds in for section 2-2 scattering amplitudes prescription will give awhose gravity decoupling amplitude limit given correctly by reproduces thewe the have special second no Galileon. term reason on However, to since Indeed, the trust there that r.h.s. is the of no double ( clear copythat recipe prescription throughout to is this generalize meaningful paper this in to we this higher context. assume amplitudes. that It the is colour-kinematics for duality this holds reason in tact in in the decoupling limit,kinematic and factors these ( do notkinematic correspond factors to come purely the from decoupling helicity-0 gluons limit by of the Goldstone the equivalence above theorem. then an acceptable choice ofare kinematic factors that reproduce the gauge theory amplitudes Since limit of them (giving a(giving non-linear a sigma model) special and Galileon) performing willThe the lead kinematic double factors to copy inferred a procedure from different the decoupling result limit in which the and hence it contributes athelicity the kinematic relation contact term. For instancein the the naive combination leading JHEP12(2020)030 → and (2.1) (2.2) (2.3) µ where where 5 ) D i ) x ( x we do the ( V a 1 E φ − a ) x T ( i 2Λ U √ h → , which is the Goldstone ) ) x 1 , ( ) − µ V V m/g A µ ) = exp µ x we complement section ( A ( Λ = . VD ) D V µ x tr ( 2 D = 0 ( V m a µ tr 2 1 φ 2 D 1 Λ − − ) ) − x µν ) ( F µν is the controlling scale. Taking the decoupling V – 5 – i µν F 2 Λ g F µν ( √ F is the gauge transformation parameter. The unitary tr ( 1 4 ) → tr x − µ 1 4 ( a A − = ξ = ]. and . Then the leading terms in the effective Lagrangian in unitary 66 mY M is the covariant derivative and i L ) m mY M µ x L ( A a 2 ξ . This Lagrangian is manifestly gauge invariant under ig √ a T In preparing this work for submission we became aware of results obtained − under which the Stückelberg fields transform as m/g i 2Λ µ ) √ ∂ x ( h = U . The structure of the effective Lagrangian is best understood by reintroducing 2 µ µ ) is the coupling constant. This is the simplest unitary gauge Lagrangian which can Λ = µ D D g 1 are the Stückelberg fields, so that the gauge invariant form of the Lagrangian is A − ) µ The resulting effective theory has a cutoff of at most ) = exp ) x x x A ( ( ( ( a U gauge Lagrangian is recovered by fixing the gauge mode decay constant.this Additional scale, interactions but in for the now effective we action assume could that further lower where U where φ describe a massive coloured spin-1 particle.it Since should the resulting be theory is understoodinfinite not renormalizable, number as of interactions. an Fortr effective instance, we theory, may and further consider toStückelberg a fields this quartic (Goldstone interaction Lagrangian modes) we by replacing may add an gauge are as follows: where 2 Massive Yang-Mills The action of massiveYang-Mills theory Yang-Mills with theory a comesconsider Higgs from field the the in gauge which low symmetryacquire the energy the to Higgs same effective particles be mass, action are broken integrated of in out. such We a way that all of the gauge bosons Note added. by Laura Johnson, Callumdouble Jones copy and interactions Shruti [ Paranjape which also reproduce the quartic equivalent to a massive spin-1in Proca constructing theory, the as interacting wegive Lagrangian. find the the explicit In latter decoupling appendix formulation limitsame more of for useful the gravity the amplitudes, Yang-Mills whilecommute amplitudes in with and appendix the clarify decoupling why limit. the double copy procedure does not a mass to a two form potential (which arises naturally in the massless double copy story) is JHEP12(2020)030 , i ). 2 n ) µ 2.1 (2.7) (2.5) (2.6) (2.4) A µ A ( . . ) ) 1 3 −  · V 2 µ  2 V ∂ p . In the decoupling µ · , and furthermore the ∂ Λ 1 ( labels all internal prop-  µν i tr ˆ η . The resulting kinematic 2 − α . 2 ˆ Λ 1 T i m p , − · + ) ). From a classical perspective, 2 2 4 ˆ 1 + i 2 )  2 α a µ = m 3 p 2.4  A ˆ S · − ν i ∂ i 1 n  2 α i − p c as in . Given this it is not automatic that the + , a ν i − 2 1 ( 2 A i p m µ α m · µν − ∂ by massive ˆ η 3 Q ( i + i  ] and with this in mind we express the tree level – 6 – = 2 α a 2 2 − 2 i p X p 2 i  X p · 1 4 2 1 −  − n − g = ( = abc ) is clearly preferred due to its two derivative nature and it n gf A mY M 2.3 . Our goal is to follow as closely as possible the double copy 2 labels distinct Feynman graphs and 2 L √ i /m fixed ν results in a free massless spin-1 theory and an interacting non-linear ) = p ]. c Λ , µ Λ 3 0 p , 23 b → – 2 g + . i.e. we forgo the introduction of an , 20 }i a = lim i µν k η (1 3 |{ for fixed DL ˆ 1 = are not the same as those that arise in the massless case since they absorb the A 0 L are colour factors i.e. products of the structure constants of the gauge group, − i µν ˆ i n S → ˆ η c }| g f These tree amplitudes are however most conveniently computed in unitary gauge ( k All of our scattering amplitudes are given as the momentum space delta function stripped amplitudes 4 h{ -point scattering amplitudes of this theory as: In terms of polarization andYang-Mills momentum is vectors exactly the same three-point on-shell as vertex that for of massive massless Yang-Mills: of on-shell external momenta now satisfy colour-kinematics duality still holds.up We to will quartic nevertheless order. show that it2.1 continues to hold Three-point amplitude are the kinematic factors, agators in a given graph.the The replacement only of difference massless between propagators thisfactors and the standard doubleinformation copy is from the massive polarization structure encoded in paradigm for massless Yang-Mills theory [ n where This is because thecounterparts, off-shell and vertices the for only massive difference Yang-Millsone is are with the identical structure massless to propagator their is massless replaced by the massive where operators, and theylimit, are these extensions expected just correspond tononlinear to be sigma the model addition suppressed of Lagrangian, further which byfor irrelevant have operators the been example to considered in scale the in [ the double copy context by the effective theory forthe the form of Goldsones the described Lagrangian byis ( ( for the reasonwhat that follows. we Were will we focus toetc. on include it the additional is unitary tree transparent gauge level in interactions amplitudes such the derived as Stückelberg from tr formulation this that form these in correspond to higher order sigma model This encodes straightforwardly the content ofleading the interactions ‘Goldstone for equivalence theorem’ the that helicity-0 the modes of the massive spin-1 particle are determined limit JHEP12(2020)030 4  4 t ·  3 · 2 (2.8) (2.9)  3 p · (2.12) (2.13) (2.14) (2.15) (2.10) (2.11) 1 p 2  2 and the 1  ·   4 3 · 2 ·   2 4 1 p · · 3   3 p 1 2 p = 0 t · · 1  3  4 4 p ·  3 3 ) 4    u 4 · 4 ·  2 p p , · − ·  ·  3  4 1 1 ) · 2 s 3 · +   p 3  2 u  3 1 p · · 3 2 p 3 t 4  2 3 , p  1 2 p − , 1  ·   2 2 . ·  p p − · 4 · · + ) 2  · 1 + 3 3  )  2 4 ·  1 2   · s B p 2 2 4   4  2 p 1 · · 4   2 p 3   and there are 3 possible u ) by defining the colour  · · 1 2  3 · p m + ·   ·  n 3 2 3 2 1 )+ +4 · 3 4  1 1 u p p  s − +3 2.6 1  ·  2 c m p  1 +3 −  ( 1 + − u + ·  − − 1 2 p )+ 2 4 · 2 3 − +4   ) 2 2 4   2 p + t · · +2 4  · ·   = m 3 p · ·  · 3 3 = 4 1 1 2  4 2 i + ·  1 3 p p 3 − p p · t  rather than p · 2 2  ( 4 1 ( ( p p m · 2 n   4 p 3 3 3 3 1 2 t 1 p · ·  m 2    p + c − 4 · p · · ·  3 2 − , u 3 4 m  − 3 · t 2  2 1 1 2 p p )  · . (   · ) 1 3 4 3 · p p p 4 · 2 3 3    p 1 4 2 4 , +  2   2 · · · bde cde = 4 p 3    ade · p p · 4  )) · · ·  2 f f 2 3 2 2 f . Because of that the terms coming from ( 3 2 1 · u    1 1 3 s u 2 p +  2 −  m · 1     2 n abe cae bce − 1 3  · m  1 + + 4 + 4 s f f f  p 1 · +2 +4  − 3 2 c 3 t · – 7 – ( )+  + − +2 − 1 1 2  p · s 2 s = = = = 1    · m 3 2 4 1 − + 2 p · ·  )+     t 1  s u − 1 · p − 4 · · · · 3 3 c s p c c 2  = ( 3 2  2 )+ 1 3 3 3 p p ( ·  g · 4 t p  3 1 2 p p p ·  2 m 3 3    4 1 2 2 · ( 1 p s, t, u − · · ·  p    3 3 4 · p · · · 2 2 2 1 − , t  ) =   4 1 − 2 3 3 1 p p p ) · · d 2  m ) p 4 4 4 4 4 p p p  2 · 4 2 ( 2 4      4 3 4 ·   , 3 2 · · ·  · · p    4 c 1  4  · · · · 3 1 2 3 2   · 3 2    · 3 1 3 p p + , − 2  4   4  1 b  +4 1 3 + +  )+ 2 4  p 4 − − +4 − 4 4 ,  · u (  +4 +4 +4 + 1 3 4 2   · a · 2 4  ·  ·   2 4 4 − 1 − 1 p · · · ·     2 1 (1  2 2 · p · · · 3 3 3 2 4 p p ,  = 3 1 3 2 3 p p p ( p ( + ) · m  A p 4 1 3 3 1 2 s p p p s ( u · 1 2       4 3 3 4 4 4 1 p · · · · · ·        4 · p · · · 1 3 1 2 2 1 · · ·  2 1 2 2 2 p p p p p p 2 2 3 ·  p 3 3 4 2 4 3 p   p p  · 1 3       3 3 3 2 2 2  1 · · · · · ·         · · · · · · · 1 2 2 2 1 1 (4 ( 1       1 1 1 1 1 1 i i 2 2   2  4 2     ( i +4 + − + − − − − − +4 − +2 − − 2 = = = t s u n n n with all incoming momenta.those obtained These from expressions massless Yang-Mills forbetween theory Mandelstam kinematic but variables there factors is are are now locations two very differences: of similar the the to relation poles now are at where the Mandelstam variables are defined as standard: where the kinematic factors are so that polarization states. Our conventions for these are given2.2 in appendix Four-point amplitude We express the four-point amplitudefactors to in be: the form given in eq. ( The difference is that now the on-shell momenta satisfy JHEP12(2020)030 (3.1) (3.2) (3.3) (3.4) (3.5) . The ] in order 57 = 0 2 u m c − + u effective theory t are the free pa- c 3 . The full dRGT 4 . Λ + 5 and ] , κ s 4 2 c 3 K κ m ] 6[ K − and [ − t n , 2 ] U 2 2 . The potential terms can be = 0 n 2 m . K κ 4  m − µ ν · 4 =0 s  X 4 n f p ] + 3[ ]. In our case of massive Yang-Mills 1 g K − 67 ∝ − , g ][ ] 3 u √ 3 q n K 2 Pl  K ) (in fact we do not need to use the relation + M 4 − t 2 + 8[ n , µ ν ] – 8 – ). 2 2.14 m δ ] + 2[ ] +  60 2 ] K 2.6 + s 2 K ][ n ) = , is satisfied [ ][ 2 K , which appears in the action. It can be written in [ gR K K ) and ( µν f, g − − = 0 = 1 and the terms in the potential are defined as 3[ 6[ f ( 2 µ ν ] 2 √ u − − κ 2.13 n K K 2 Pl 3 4 [ ] ] 2  + M ), ( K K t and ) still obey n = 2.12 + ) = 2 ) = [ ) = [ L is the same for massless and massive theory and colour-kinematics 2.14 = 0 s K K K u 1 ( ( ( n in ( here). It is easy to see that these six terms add to zero, therefore the 2 3 4 n κ 2 U U U u → + = m t ) and ( 0 n κ and = 0 + t 2.13 u s , c s n ), ( + t c In general, kinematic factors of a given scattering amplitude are not unique. They 2.12 + s The squared brackets denoterameters the of traces, the and theory the together with two coefficients the graviton mass where we set as it has aLagrangian straightforward for decoupling a limit single as spin-2 field we can shall then see be in constructed section in unitary gauge as [ unitary gauge in terms of the variables [ This unusual square root metric structure is what is needed to build a 3 dRGT massive gravity In the dRGT theorynon-dynamical of reference massive metric, gravity, the diffeomorphism symmetry is broken by the between value of duality for four-point amplitude still holds in the massive case. theory, it is not immediatelyshows that clear our whether colour this and kinematic isin factors ( still (directly true. calculated from However, usualfact explicit Feynman rules) that calculation this stillonly holds difference between for massive theproportional and massive to massless theory kinematic can factors is be coming understood from by the noticing terms that the to recast the amplitude into the form ( are not invariant underany choice field of redefinitions.c kinematic However factors in of massless four-point Yang-Mills amplitude, theory the for colour-kinematics duality, quartic Yang-Mills vertex now have to be multiplied by JHEP12(2020)030 = (3.8) (3.9) (3.6) (3.7) µναβ . The ε ) x ) are the ( that does A ) 3.2 π + f, g ( constructed out A µ ν 6 with the generalized , it is sufficient to x K d d µν , hence all terms in the . For the purposes of η ...j ...i µν 5 ) = as we shall be largely +1 = +1 x K . k ) is written an a manner k ( j i  δ A µν µν , i.e. in the Lorentzian ! . η 3.2 f k . Possible higher derivative Φ ), so that unitary gauge is 3 0 = 1 β β = µνρσ Λ 2.3 ’s. K d , R 0 ( δ 0 0123 α α Pl ...j β β ε ) . K itself transforms as a tensor, then x K , +1 0 = 0 ( k ν 0 ν ,M ) j α α β β AB k V µ 3 K 5 µνρσαβ 3 η K 0 K 0 0123 Λ ...i µ ∇ 0 Γ µ f, g B ε ν ν 1 in ( α α i , K Φ ε µ K αβ 0 ν K 0 ) d ν 3 µν β 0 β 0  x ∂ K β 0 ...i 0 α ( α 0 A itself is viewed to transform as a tensor. ρσ a 0 ,K ν +1 2 0 Φ φ – 9 – k  i µνα µν µ µν µ µν k ε ε ε may then be split as g ∂ K µν  1 ...i ) 1  i x → ε ( to be Minkowski metric, µναβ µναβ µναβ ∝ g F ε ε ε A µν 3 − Φ f µν f √ M ) = ) = ) = 4 3 K K K ( ( ( 4 2 3 = Λ U U U L ∆ . As long as we are clear that we use one of them with all indices up and the 0 β 0 α 0 ν 0 µ ε 0 ββ η 0 . We will make explicit use of this decomposition in section αα denotes the sum of all diffeomorphism invariant scalar operators η 0 are then the Stückelberg fields we need to reintroduce manifest diffeomorphism F νν η ) ) = 0 0 Just as for a massive Yang-Mills field we can write this same Lagrangian in a manifestly x x We use Euclidean coventions so that for flat spacetime All breaking of diffeomorphism invariance can be captured by the tensor ( ( 5 6 µµ η A A − other with all indicesKronecker delta down expressed together as with a determinant of aLagrangian matrix are built diffeomorphism out invariant of when 3.1 Three-point amplitude The three-point amplitude in dRGT massive gravity is as follows: π invariance and play theπ analogue ofcalculating the scattering amplitudes it is sufficient to work with the unitary gauge Lagrangian. not transform appropriately as awrite tensor this is metric the in reference an metric arbitrary coordinate system The four diffeomorphism scalars of its arguments with dimensionless Wilson coefficients. covariant way via the introduction ofin Stückelberg which fields. it Since ( wouldthis be tells us manifestly how covariant to if introduce Stückelberg fields. Since the only part of operators will arise schematically as where In this paperconcerned we with will scattering consider amplitudes inunique Minkowski interactions spacetime. which The lead termsdegrees to in of second ( freedom. order equations Howeverleading from of terms the motion in EFT for an perspective EFT all it expansion, 5 is controlled propagating by natural the to scale view them as the written in terms of the flat space Levi-Civita tensor JHEP12(2020)030 . is +1 µν (4.1) ∗ (3.10) B ) ), massive , for helicities is a massive µν ν ) h 2 = 2+1 i  µν ( × h µ 2 ) i  , the corresponding . Therefore, in order = ( 3 = 0 µν M k ) c i 7  , ( + ν ) j i c is a massless antisymmetric 2-  ) are massive: ( + µ µν ) i φ, 4.1 i c  B ], i.e. whenever three of the colour ⊕ 1 cyclic permutations of 1,2,3 = ( µν + B such that the second term vanishes, i.e. β µν 2 ) ⊕ can be written as 3 p i  α κ 2 ( 1 µν ± p h β – 10 – ν 3 =  , ν is the graviton, . The first term is already proportional to the square  µα 2 A 2  Pl field is a spin-0 field while in massive case it is spin-1. µν ) fields. We see that there is an interesting physical m µν ⊗ h 1 µ φ B /M µ  α 3  A να = 2 + 2 2  β 1 κ p µν 1 α 1  p ) 3 αβ κ . 2 are related by the Jacobi identity,  00 λ ν k is a massive scalar field. In terms of degrees of freedom we now have  . In this paper we will consider four dimensions and write the action µν ) subset of theories from 2-parameter family of massive gravity theories. c 0 3 is a massive 2-form field which is dual to a massive spin-1 field in four 4 (1 + λ µ is a massless scalar field (dilaton). In four dimensions the massless  φ  κ 3 2 00 φ µν µν 1 λ and 0  + λ λ B ( j C c  ) and massive spin-0 ( is the cubic vertex from Einstein-Hilbert term plus the cubic potential term 00 , µ i λ iκ 0 3 . We see that already at cubic level the double copy construction picks a particular c A λ 1 Γ ) must satisfy the colour-kinematics duality [ = . It is expressed as follows: − P we need to sum over the products of different helicities weighted by Clebsch–Gordan coefficients ) 3 In the case of massive Yang-Mills, all the fields in ( 3 = 5 + 3 + 1 2.6 Note that only polarization tensors for helicity 0 K = = M 7 , ( × 1 3 3 λ µν in ( factors, ±  difference between the field contenttheories: of the in double the copy massless of case massless the and massive4.2 Yang-Mills Double copy constructionIn of order scattering to amplitudes double copy massless Yang-Mills theory the representation for the amplitude spin-2 field, dimensions and 3 obtained from double copy ofspin-1 massive Yang-Mills ( in terms of massive spin-2 ( i.e. we decompose the tensorrepresentations product of of Lorentz two massless group: vectorform representation field into and irreducible dual to a pseudo-scalar, i.e. axion. In terms of degrees of freedom we have In the double copy constructiontified the asymptotic with states the in the tensorindices. gravitational products For theory example, are of the iden- double gauge copy theory of pure asymptotic Yang-Mills states, theory gives ignoring the their following states: colour one parameter ( 4 Double copy of4.1 massive Yang-Mills Degrees of freedom where the coupling constant of Yang-Mills three-point colour-stripped amplitudeproducts if of we write two the spin-1for polarization polarization tensors double vectors, as copy toκ work we need to choose U where JHEP12(2020)030 ] µ 1 A ), is (4.8) (4.5) (4.6) (4.7) (4.2) (4.3) (4.4) 4.5 is required ) is chosen, . We define 2 from explicit 2.6 C corresponds to √ [] B.3 . It is conjectured [ = 0 , k ) n 2 + m j , , − n i σ  i ) . As we show in ˜ n 2 α + ν i A φ 2 ( p p , i n  polarization vector to be: µ n ,. . However instead of working with the m − ρ jk jk ) p 3 ( ) . p ) satisfy the colour-kinematics duality. ) respectively, are decomposed into polar- i A ) ˜ B h B A ( α φ ( µν ( µν ν + 3 ( µν    2.1 ˜   Q A µνρσ µν ε 2 κ ∝ → → η i i ] X  and m )) k ν – 11 – field, jk i 2 2 ˜  κ = 3 δ µ j ˜  − 1 [ µ  κ √ j 3 B n √  ˜ denotes the symmetric traceless part corresponding  , (( µ i j µ  M  =  ˜ = n 2 κ ) ) (())  B φ ( µν µν ( i , and the antisymmetric part denoted as   and ) h = i (  n n M ]. We follow the same procedure and conjecture that the following 2 . The dualization procedure is explained in appendix field polarization tensor and µν B B ) into irreducible representations. Schematically this is done as follows: ) to three-point amplitudes explicitly giving the following relation: field in this paper, we construct the action in terms of the vector field 4.2 are little group indices, SO(3) µν is the four-momentum of the external state and the factor of are the kinematic factors of the second massive Yang-Mills. The products of Yang- B σ i p j, k ˜ n In the usual double copy procedure, once the correct representation for ( transverse and normalized. 4.3 Double copyWe of apply three-point ( amplitudes the polarization tensor correspondingcalculation to in helicity the basis scalar, we find it to be which up to a sign could equally have been fixed by the requirement that it is a tracefull, where for the correct normalization. The trace part of the tensor product, given in ( to the graviton polarization, the spin-1 polarization inmassive terms of the which is dual to the map between where where Mills polarization tensors in ization tensors of the fieldsof in a the tensor gravitational product theory.in This of 4d corresponds two it to vector is decomposition representations of the little group (for massive particles the colour factors can be replaceda with gravitational kinematic theory factors [ in orderexpression to gives obtain an an amplitude amplitude of in a massive gravity theory: that it is always possiblefactors to satisfy choose this a by representationcase choosing for a that the amplitude gauge is for and not whichfour-point performing guaranteed kinematic amplitude field to calculated redefinitions. be directly true In from but the ( massive we have checked that the kinematic factors of kinematic factors must obey the same relation i.e. JHEP12(2020)030 ) . ). 7 24 1 4.2 − (4.9) = 4.11 (4.15) (4.10) (4.11) (4.12) (4.13) (4.14) = 4 ), ( ]). The κ 3  , κ ) 68 ! , 2.14 and 59 αβ 2 1  ), ( 2 −  αβ 3 m να =  2.13 3 − 3  µν 4 µ κ  u ), ( 2  , stripped off. By sub- µν α 1 1   abc 2.12 ν 2 f p + µ 1 p αβ 4 +  ]). The second term on the right 2 αβ µα 2 3 59 m  ) from the double copy prescription,  cyclic permutations of 1,2,3 α 2 − µν 3  +  t ν 1 AA β  at high energies. Since all of them have 2 µν ν 2 1 p 6 → p  amplitude calculated using dRGT massive α µ 1 E 1 p p + )), the lowest scale appearing in the resulting hφ  β + hh is a vector. Furthermore we find that none of ν 3 ν 2 – 12 –   ) we get the following three-point vertices in a 4.2 αβ 4 αβ  µ 1 A 3 →  µα 2 2  4.7  µ αβ µ 2 2 3 m   p µν hh µ 1 1 ν  −  1 µν 2 β 2  p s matches three graviton amplitude of massive gravity if we p +2 µν α 1 ) and ( using the parametrization of the theory as in [ 1 p +2 β 1  hhh p µ 4 4.6 − 2

α 1 /  M p amplitude which is calculated using ( 4 µν 3 matches the 3 point amplitude of massive gravity with µ 1 µν  using the parametrization of [ 3  ), ( m αβ  2 = 1 µν 3 2 2 µν ν 2 hh  2  , we find the following: 2 7 3 hhh κ  4.3 . This is expected, since we know theories containing massive spin-2 m ν 192  c µ 1 ν 2 2 2 µν 3  M p 15 p → 16 3 − κm 2 amplitudes are different from those obtained from vector and scalar ki- mGr µ 1 i  µ ) corresponds to a scalar exchange with three-point vertex given in (  4 m 3 1 (or p m = 3 κ p µν 1 − M hh √ 3 3 2 1  4 16 µν κ 5 √ κ φφh ( 4 3 ) and ( 4.15 d  − 8 √ 3 11 in front (can be seen from (  i i i i M 2 mGr κ 4 2.7 iκ = 2 Pl − i − − − iκ M − 3 ). By comparing it with and ======κ and /M 1 4 = = 4.3 4 hhh Having fixed the spin-2 interactions, we then construct the scattering amplitudes for all As mentioned before, φhh φφh φφφ AAh AAφ = = 4 are zero as one would expect since M AAh φφh M M M M 3 M M 2 c and make anfeatures ansatz emerge. for the WeA action find which that gives allthe 3 these amplitudes and amplitudes. scale 4 with point Aκ energy amplitudes couple more containing of that odd general numbers of with the free coefficients( in the massive gravity actionhand side chosen of to ( be other 2-2 scattering processes (for example and ( gravity action, usual equivalence principle requirementstioned for we a see massless that spin-2 particle. As4.4 already men- Double copy ofWe four-point start amplitudes with M netic terms minimally coupled to gravity (forM example a minimally coupled scalar wouldfield give do not have diffeomorphismthe symmetry, and reference we metric allow which couplings in between this our fields case and is the Minkowski metric. In this way we evade the choose stituting ( gravitational theory: where the 3 point amplitudes have their structure constants, JHEP12(2020)030 ) , − ! µ ν δ scat- µ 4.16 up to (4.16) A 2 = - h µ 2 ) and the µ ν A . The self 2 g K m 4.16 1 2 interactions are − ) φ α α µν  K ] F 2 µν µν g [Φ F quartic contact terms − 1 4 − + 2 − µν φ 2 to all orders, it is natural to scale), we have intentionally K [Φ] µν φ ( 3  2 3 F φ φ Λ Λ ν m 2 µν 1 2 κ ∇ is the reference metric, F m φ − 3 3 µ φ µν √ interactions to all orders. This process κ √ 1 ∇ µ η 3 1 4 16 ∇ 24 Λ φ − − + µ that breaks diffeomorphism invariances only φ  ) we see that the self interactions of να η ∇ µ ν ν F 1 2 A K – 13 – µ . The indices are raised/lowered with µ µ 4.15 − να ] A F , the well-known highest possible scale for a Lorentz A 2 K µ µ 3 [ −K / m is a ‘spurion’ field for the breaking of diffeomorphisms. K n 1 3
µν 1 8 U K 2 κ ) and ( n K √ terms. However changing the off-shell structure in this + κ m α µν − µ Pl 3 4 K =2 4.14 K 2 X F n φ  M ) 2 2 2 , contain galileon interactions (the cubic term in ( φ φ να and the crucial contact terms which fix the form of the κ φ 2 m F is the dynamical metric, ∇ term and two additional two and four derivative contact terms to = κm κ ( φ m 3 3 µν 3 ν ]+ 3 . The reference metric µν 3 φ Λ g K √ ∇ 2 [ − K 8 1 2 √ µ κh and R and vice versa without changing the on-shell vertex. A similar story )), can be described by dRGT massive gravity action. Anticipating that the ∇ − + − 2 2 + 2 3 , and in this sense κ . Although we have not calculated beyond four-point level, the implicit h -matrix is invariant under field redefinitions, the cubic φ = A.1 K

µν S φK A g η µν − Φ = √ , µ ν x µν ) 4 g η d 1 Since the As already stated, from ( − Z g = -point scattering amplitudes are controlled by the scale p S the Stückelberg fields/Goldstone modes givesthe us interacting an Lagrangian indication and of theand this best explains appendix way many to of structure nonlinearly our realized choices diffeomorphism of symmetry interactions presenta in in set the ( of Stückelberg interactions formulation at fixes all orders as is familiar in effective theories with broken symmetries. limit is a Galileon-likechosen to theory put (which the ismanifestly cubic implicit Galileon-like. interactions In in in other the a wordstheory form the gives us for desire guidance to which in have thebeyond writing a quartic what the Galileon-like is interactions nonlinear decoupling immediately off-shell limit are inferred structurethe also of from diffeomorphism the the symmetry theory on-shell is that broken scattering goes by amplitudes, the even mass though term. That is the decoupling limit for ambiguous since we may for examplefor use field a redefinitions potential to tradeholds the cubic for Galileon the term way also changes the form of the quartic interactions. Anticipating that the decoupling interactions of the scalar, quartic one in ( this order. The action hasinvariant in been terms intentionally of written inenters a through manner which is diffeomorphism where ( tering amplitude are given in appendix combinations which are naturaltheory, namely from those the that point automaticallyis lead of somewhat to view labourious, and of we the quote decoupling only our limit final effective form for the action which is invariant theory of massive gravity. quartic order in n write the interactions for all the fields in the dRGT form, taking particular care to choose theory to this order is JHEP12(2020)030 , 0 (5.5) → (5.4) (5.1) (5.2) (5.3) [Π]) , m µν  ] η 3 − φ 0 µν α α Π 0 (Π ] + 2[Π ) in as covariant ν ν 2 φ limit, it is sufficient Π ν 0 3 4.16 µ µ φ∂ Λ µ Φ 3[Π][Π ∂ β 0 3 3 α  − 0 ] 1 ν 2 3 0 4Λ in the decoupling limit, since , π . µ [Φ ε B π π − [Π] as the building block. Hence for ν ∂ 3 3  − ∂ K Λ φ 2 µ , AB µναβ A,V 6 3 ∂ ε η 1 L Λ as the helicity-0 modes of the spin-2 [Φ] 3 3 χ , AB 6 3 1  := + Λ η µ 3 π 7 φ ∂ 2 B we replace it by √ ) 3 3 − 3 3 48Λ 11 µν 1 , the decoupling limit is defined by Φ 1 m . The decoupling limit Lagrangian is found µ Λ Λ ν 24 A Π Pl ∂χ + A ∂ µν 3 ( + V η − A  = 1 2 √ ] µ 1 /M Pl Φ 3 φ – 14 – 0 A µ by − 12 µν 1 α α ∂ 2 is kept finite. The Lagrangian has been written in = 2 K Φ ) → + mM µν 0 , and our choice of = φ g κ ν ν Pl µ µ  µ − ν ν x Φ ] + 2[Φ fixed A ∂ 0 µν M 2 µ 3 ( Π A µ µ 2 f v Λ µ µ 1 2 x , Π lim m 0 + Π β c 0 = − → = α as the helicity-1 and − 0 + m A 3[Φ][Φ 3 3 ν µν 0 π ] for a review), after denoting the reference metric from which A Φ µν Λ µ − X ε V 57 Π 3 → µν and the metric µν π h [Φ] Π µναβ  + µν ε  φ 6 3 Π φ 6 3 µν 1 . Remembering that 3 3 Λ 1 h Λ 1 by Λ µν E 3 3 √ 11 4 µν µν 11 κh √ 864 8 h K + 1 2 − + + is the original Stückelberg scalar, the helicity-0 state of the spin-1. The normal- decoupling limit µν in such a way that χ = η 3 0 is constructed by is invariant under Crucially, we have Λ = DL ν → µ µν L µν to replace to be (keeping track only of those terms which contribute to quartic order) which explains the emergence ofΠ the Galileon symmetry for all terms in the Lagrangian for which the coefficients are finite in the a finite (and non-zero)g kinetic term in theκ decoupling limit. Thea metric judicious may way to be ensure denoted that no term diverges in this limit. particle. Further for the massive spin-1 state where izations, which are standard, are chosen so that all the additional Stückelberg fields have we further decompose so that we may identify theory. We haveform intentionally written as the possible, interacting sorecipe Lagrangian (see that ( for the example decouplingK [ limit is easily derived. Following the standard Having successfully constructed thetheory, at interaction least to Lagrangian quartic for order, itus the is insight useful into double to the understand copy interactions its that effective decouplingthe arise limit. effective beyond This theory, 2-2 will but scattering, give and itthe the will overall massive also structure of allow Yang-Mills us decoupling to limit understand better and the that connection for between the double copy massive gravity 5 JHEP12(2020)030 , ) n µναβ (5.8) (5.9) (5.6) (5.7) − D d 4 (5.10) (5.11) K Π are not Π 1 C c µ structure. − Φ n A ε B b ] coupled to Φ ) Φ terms, namely 69 6 3 εεδ and φ, π ( 1 ( n µ 24Λ hππ b V 4 + =1 X n and D d φ Π αβ # int bi-Galileon D d hπ C c )+ F D d h . Since L 3 Π 3 3 contains standard cubic and C Π Π . B b + Λ ∂ C c c , / 2 Π µναβ µν Π ) ∂ εεδ 6 3 K µν ( Π B b B b 1 A,V π Φ ∂χ π h Φ µν 0 ( 3 3 L 24Λ A 6 3 a b F 1 2 δ 1 int bi-Galileon . We have separated out the spin-2 1 4 + 4Λ + 3Λ L 1 and )+ − µν D d n − D d √ , and the kinetic term coefficients 2 η preserves spin-2 gauge invariance (linear 3 3 − ˜ ) − 8 µ Π Π Λ 3 ABCD αβ C c A µν / C c ε µν − ∂φ Π ν ˜ Π F ( h 1 Π . This is the decoupling limit remnant of full µν ∂ πδ D d 1 2 B b µ − B b abcd – 15 – 1 2 Π ξ n ˜ Φ δ − Π ε ν − 3 3 Φ µναβ C c A a 1 2 ν + ∂ 2 2 1 ˜ h Φ . The term K ) A − 4Λ + φ µν B b µ ) is invariant under two separate Galileon symmetries µν εεδ 3 ˜ h ∂ ν = ∂π Φ 1 ( − F ξ ( √ 6 3 n 5.5 µ and thus describes a bi-Galileon theory [ 4 1 = 3 4 = D d ABCD a µν ∂ − ε − h 3Λ µ Π − µν 1 3 =1 µν x E π + C c X √ n h = µ δ F ], and they are schematically abcd µν µν φ u = ε µν 72 B , b ˜ h h 62 6 3 δ µ h ˜ E π − 1 2 + A,V )+ 2 1 V 1 2 ν L  µν φ → ∂ Π 24Λ ˜ h 2 and ) is indeed identically conserved by virtue of the double 1 2 , which is characteristic of the massive gravity decoupling limit, needs + − µν → scattering amplitude, we cannot take seriously the inferred coefficients of the quintic ˜ h π ν µν εεδ 2 b 5.7 = φ ABCD ( V ∂ X ε − µ π , a 0 ∂ 2 µ ∂ DL a x abcd L are tensors constructed from µ = ε = = v bi-Galileon interactions: = ab µν + 8 ˜ F Π µναβ π aA K The tensor ( The tensor Strictly speaking there are also quintic interactions, however since we have only fixed the Lagrangian by X 8 → int bi-Galileon L reproducing the interactions. where quartic The resulting Lagrangian then takes the form performing a ‘demixing’ transformation that removes the mixed We may make use of the fact that up to total derivatives The full decoupling limit actionπ ( a massless spin-2 field. Indeed it may be put in a more manifest bi-Galileon form by diffeomorphisms) diffeomorphism invariance. Explicitly its form is and sourced, classically it isin consistent loop to processes. set them to zero. Theyto would be of identically course conserved contribute to ensure that and spin-1 helicity-1 contributionsparticularly which complicated even [ in the case of standard massive gravity is where where all indices are raised and lowered with JHEP12(2020)030 . . ) is π ˜ π . − ˜ =0 1.4 π ) = λ µν  4  = (5.13) (5.14) (5.15) (5.12) D d Pl , b φ D d 1 3 Π 1 3 and the M Π √ C c 2 , b C c / amplitudes 2 D Π + d 3 Π B b 2 π , b W B b p 1 − ± C c )Π Π that correspond , , b 3 0 A Z a 0 ∼ − φ κ , ) = b B b ( 0 φ π πδ + , 2 ) Y ν 1 0 3 ∂ √ 4 A a , and -point amplitudes will µ κ κ ∂ n + X and π 3 3 µν , + (4 1 π Λ 6 3 2 ) 4 X / µν 3 1 κ ∼ 3 µν 4Λ Π √ 1 which does not correspond to ABCD (4 h p π ε µν 24 ) + ). Indeed the combination − , 3 K + ( 1 8 6 D d κ 9 2 abcd . D d √ D.4 ) = 0 − ε Π ν 0 ABCD , h 3 ,  ε C c C 3 κ µ 0 D d 8 ) is  Π ∂ = √ Π c (2 + 3 B abcd b , ∂ C c 3 3 . The decoupling limit of dRGT massive δ ε 8 1 D.5 ) 1 B b 6 3 Π 3 µν − (2 + 3 4Λ h 1 B b κ h cannot be removed with a local field redefini- A 8Λ a Π + δ 3 A a + ˜ – 16 – Π ˜ h µν ˜ h D d ) = ( (2 + 3 XY ZW ˜ h 3 , the canonically normalized unitary gauge helicity-zero mode πη 3 3 ABCD ) C 1 2 2 ε 1 , a ) and explicit in ( ∂ h ABCD 4Λ 2 ( c + ε ∂ O abcd , a + D.3 ε B b 1 µν + abcd ˜ h 1 2 D d ˜ h , a ε µν is (ignoring helicity-1 contributions) A a Π ) 0 − h δ = 3 4 a C c ( Pl κ κ δ 1 = M µν B b . + h δ ) contact term in the decoupling limit, as described in equation ( = DL 4 ABCD 1 2 0 and κ ε L 11  864 µν 3 (4 , ΣΣ κ K 3 6 3 abcd scattering amplitude. Furthermore higher order ε , whence the combination arising in ( 1 √ π 1 2 4Λ 432 0 ABCD 2 197 , ε / − + 0 , 3 , 1 6 = 0 p , abcd , ε 3 − 2 √ DL = 144 ± 25 interaction which cannot itself be removed with a field redefinition since it contributes L µν As noted in the introduction, since the decoupling limit of massive Yang-Mills is a The quartic interactions of the form − To see this, note that at leading order in the decoupling limit , 9 X 5 96 is in effect receive contributions from intermediate gravitonquartic exchange Galileon which theory. do notstandard Hence massive arise gravity in the in the special any pure form. Galileon does not strictlySince in speaking unitary gauge arise in and so we onlyhπππ have a non-vanishing quarticto Galileon the term when there is also a non-zero there is no cubic Galileonthe term. value obtained This from requires double copy. Even with this choice, we then have Since the special Galileon inperforming four the dimensions demixing is a pure quartic Galileon, we need that after where nonlinear sigma model andhave the expected double copy the of massive theInterestingly gravity however, latter theory this is to was the be never specialgravity possible that never Galileon, since gives corresponding we rise the to to might decoupling a aGalileon limit special special interactions of Galileon. arise Galileon. dRGT This massive from isgravity easily mixing for seen with by general the manner in which the that arise from the and implicit in theexactly full the combination answer which ( identifiesto the diagonalized the parts spin-1 of helicity-0 polarization tensor squared coefficients are given( by tion, as is well knownit from the is standard precisely massive these gravity case. interactions This that is describe as it the should nonzero be helicity since where we have used the shorthand JHEP12(2020)030 (6.1)  ) ) ) etc. — the 24 14 24 s s  s ) αs + 24  + s ) + 17 + 17 13 s  s 24 n 23 23 s + 2 2 24 s + s s to 23 12 + 2 s s + 4 s + 6 ( + 4 n , 23 )( 14 24 2 14 s 24 ) s s s 13 s )( s -point amplitudes which are 13 23 s n 24 (13 (10 ) + s + s so that the amplitude remains 13 +  13 24 s 12 + s s i 2 24 s ( 12 ( s + 13 s 23 + ( 12 s 12 s 2 23 + ∆ s 23 24 2 ), satisfy the same colour/kinematics )) s ) + ) + 4 ( s s i ( + 24 23 n 24 2.6 ) + s + s s 14  + 4 s 2 24 ) + 2(2 + 14 + → + s 24 )) s + 24 s i 23 14 s + 23 14 13 s n 14 s s 2 14 s s – 17 – ( s s 23 + ] the conditions on the spectrum of the theory for + 6 + s  + 14 − in the denominator s ) 66 14 + 4 + 23 13 14 12 ij s s s s 14 s s 23 ( s 14 + s + s ) + 2 23 . For, example in massive Yang-Mills case these shifts (17 + ) + 14 s 13 ij + 23 13 s 14 s 24 s s s 13 s s ( s 13 ) + ( + s + 12 ( + 17 14 s + 4( 12 , calculated directly from Feynman rules do not satisfy Jacobi + 2 14 i s 2 14 s 2 12 s s s 2 14 n 12 ] which appear to be part of a large web of interconnected theories, ( 14 + s + s s 24 ( 12 (9 + 4 s 13 + 108 2 12 , 6 2 13 s + 4 ] that at 5 points the double copy of massive Yang-Mills amplitude + 13  s ’s are squared. In [ s 2 13 2 12  i m 11  + s 66 23 s 2 13 3 12 , n s s s 23 7 12 9 s , + +4 +2 s 117   5 + 36 +4 (  2 8 24 4 12 s m s m m + 2 + ( + 2 + 320 It is beyond the scope of this paper to consider higher which has some complicated expressions ofwhen zeros the which are shifted the locations of unphysical poles from a local Lagrangian.Mills The kinematic reason factors, foridentities, such so poles is they thatunchanged. need at to As 5 be point itpoles the turns shifted massive in out as Yang- kinematic for invariants, contain generic the theories following such polynomial shifts of are non local, i.e. they have are non-trivially lifted by the presence of a massneeded term. to check whether theit double was copy shown procedure ingives remains [ spurious intact at poles all and orders, therefore however that cannot be matched with an amplitude calculated leads to finite contributionsthe to decoupling the limit gravity andnot amplitudes. double have copy been One procedures anticipated consequencewell do known from of relations not the this between commute, the decoupling isGalileons a scattering etc that limit amplitudes result [ theories of which nonlinear alone. could sigma models, Hence special the by now scattering, we neverthelessambiguity find that arises that in it theso-called massless remains generalized case gauge intact (e.g. transformations) the is to ability fixedthat to in this colour the shift kinematics order. massive holds. case bykinematic Furthermore the the Interestingly factors requirement manner contain the a in term which which it is is singular fixed in is the such decoupling that limit, the but nevertheless In this papertheory, we i.e. massive explored gravity, as a the doublefor possibility copy of doing massive of this Yang-Mills theory. constructing is Ournormalizing to prescription an by demand interacting that the massive the massiveduality spin-2 kinematic scalar as factors propagator the for ( the massless massive case. theory, defined This by is a nontrivial requirement even at the level of 2-2 6 Discussion JHEP12(2020)030 , or similarly 3 ) only at three Λ 4.16 -point functions at a given n . Since as we have seen the decoupling 3 Λ -point functions do not precisely match those n ). That is we can infer the leading contributions – 18 – ) from the decoupling limit rather than double 4.2 4.16 or a parametrically lower scale on the Yang-Mills side, Λ effective theory whose interactions match with the double ) is consistent to all orders (from the low energy point of 3 Λ ) may only be true at leading order in an expansion in powers 4.16 4.2 ] are not satisfied. One reasonable conjecture is that this should be 66 , is a local ) strictly to all orders. Since nearly all cases in which the double copy ), we do not regard this as necessarily terminal for several reasons. It is 5 4.2 4.2 ), or some close relative to it, does have a role to play in an appropriately double ), or at least partially in a weaker form. The present work serves as a starting point , i.e. at leading order in the decoupling limit. What we have been able to construct, as It is also worth remembering that the de Rham-Gabadadze-Tolley model of massive Perhaps more importantly though, the rules of the double copy paradigm for massive 4.16 4.2 m one example of anbe effective interesting theory to for explore a extensionscopy spontaneously on paradigm both broken may sides gauge be to preserved symmetry. understandof either It to fully ( what would in extent the the sensefor double of such a an strict analysis relation and along for the establishing lines the relation between the decoupling limits on each organizing the structure of thein EFT. ( It remains possiblecopied that spontaneously the broken effective gauge theory theory. outlined gravity is really just one examplecontain of any large number family of of massive interacting effective spin-2 theories and which lower may states just as massive Yang-Mills is just implied by the double copyto conjecture ( the higher pointcopy. That interactions this in is possible ( role despite played having by only the computed nonlinearlyin up realized giving to symmetry, from 4-point mass, the order and underlying is how symmetry due it breaking to determines the the leading interactions of the low energy theory, outlined in section copy prescription with massive Yang-Millsleading up to terms 4-point are order. given Theview) in effective and ( theory we whose may useorder it in to compute an arbitrary EFT local expansion, (analytic) even if these theories, even if they apply, arethe not well conjecture established ( and it mayhas not be been meaningful well to established impose something are like for equation massless ( theories,of it is not unreasonable to suppose that the fact that ifspin-1, we then only the focus double copy on procedureGalileon those is for known amplitudes which all to that interactions work arise arise to atlimit all from the orders and scale helicity-0 since double modes it copy gives of procedures a the special do not commute, this does not constitute a proof. because both sides ofoperators the suppressed double by copy the are scale for effective the theories. sole purpose We of are ensuringspectral free the conditions colour/kinematics to of relation add [ remains intact irrelevant evenpossible when in the such a manner toat ensure a that the parametrically gravitational lower theory scale, remains a theory to all orders. Some support for this comes from satisfy them. Therefore,matched the with double amplitudes copy calculated ofand from four massive the points. Yang-Mills gravitational amplitudes action Whilethe can in conjecture this ( only ( result be isworth clearly noting negative that in in terms the present of context the there naive is more application freedom of that the conventional story avoiding such poles are derived and it was shown that massive Yang-Mills theory does not JHEP12(2020)030 was m Does this give 10 ) or new fields such that the double copy ] may be resolved by satisfying constraints 2.1 66 – 19 – were found even in 4pt amplitudes. It was suggested these poles 2 which has to be added to the spectrum. Therefore, an interesting ) m m 2 (2 ] which is superficially quite different to ghost-free massive − s 70 -point amplitudes, or is it only possible if the spectral conditions n -point functions in [ n ] and spurious poles 54 ) which could be applied to soft massive theory in which the mass for the spin-1 ] are satisfied? If no extra fields or higher order operators can fix the spurious There are many interesting future directions that these results suggest. For example, 4.2 66 The double copy of two spontaneously broken Yang-Mills theories where all fields have mass 10 studied in [ show the existence of afuture state direction of would mass beof to another understand state if as the well. 5pt spurious poles in massive Yang-Mills signal a presence We would like toMike thank Duff Mariana and Carrillo Laura JohnsonSTFC Gonzalez, for grant Clifford useful ST/P000762/1. Cheung, comments. Claudia JRRoyal The Society is de work for supported Rham, of support by AJT at an is ICL supported STFC through studentship. by a an Wolfson AJT Research thanks Merit the Award. dynamics of massive gravitythese theories various considerations exhibited to by future the work. Vainshtein mechanism. We leave Acknowledgments copy of Yang-Mills action coupledYang-Mills to considered here, a and Higgs what field, isus the which resulting any is gravitational insight the theory? into UV UV completionhold at completing of loop massive massive level gravity in theories? anyrelations? way? Does Are The the there procedure some latter simple outlined would extensions of be the highly classical double nontrivial copy given the known complicated nonlinear poles then maybe this suggestsgravitational that theory there elsewhere. is someYang-Mills For problem theory of example, with constructing as a suchsatisfied single an in a more adjoint massive anonymous than 10 fermion referee dimensions theand because pointed gravitational colour no theory out, with kinematics supersymmetry more for than duality cannot 8 cannot exist gravitini be in 4 dimensions. Also, can we construct the double conditions since the spectrum itself is continuous. is it possible toprocedure add works irrelevant for operators all toof ( [ and the special Galileon,our and present the discussion similar suggest emergenceto that of ( there the may bi-Galileonfield be effective on theory an the in analogous gauge doubleidentified side copy for emerges prescription from higher aon resonance. the In interactions this of manner the the continuum spurious modes pole without issue needing to satisfy specific spectrum ubiquitously in decouplingwas limits first of noted these toGabadadze-Porrati theories. arise model in [ Indeed a the diffeomorphismgravity. invariant original five There Galileon dimensional the model theory,be effective the viewed four as Dvali- dimensional a continuum graviton of emerges massive as states. a The resonance, intimate connection i.e. between may the NLSM side. What is interesting to note is that the class of Galileon effective theories emerge JHEP12(2020)030 φ φ σ ν ρσ K (A.8) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) (A.4) Φ µν 0 φ β β F σ σ K ∇ 0 ∇ φ α α 0 ρ β β φ µν K 0 ∇ µ K ν ν F 0 2 2 ρ ν ∇ F φ 0 κ m φ µ ν µ ∇ ∇ µ ρ 0 φ φ K 1 2 F α α φ φ 0 µ 0 ∇ α κ ν α 96 α ] − β m ρ K ν 0 ∇ µ ν φ K ∇ φφ α φ F + 0 3 0  K φ ] ρ ν ν ν ∇ 1 µν µν 0 β 3 µ √ µν [ K ∇ µ φ − K 0 F 0 K ε F ∇ σ ∇ ν 16 µ ρ µ α µσ φ φ φ 2 F 2 ] µν ∇ ∇ ∇ µ ν + σ κ F m φ K ρ µναβ φ ]+2[ ∇ µν K ∇ µ 2 ε κ µ 1 µσ β φ F ∇ [ ] ∇ 0 µν 48 3 µ ρ K 1 ν Φ − ∇ σ ) to reproduce the desired quartic β β β 0 ∇ ][ φ 16 0 F 19 − √ ∇ ∇ F β νρ ν 0 α ρ K + 2 1 ρ κ K 0 2 − σµ ρ α 2 K + 48 κ ν [ ∇ 0 ∇ 4.16 0 ∇ 3[ 3 2 κ F ν m φ µ 0 φφ ρσ ∇ 4 ρσ + 2 2 − 0 √ µ − ε µ ρσ 0 κ 21 F κ α α ε µρ 3 K m 128 β β m 1 ∇ ] F + 0 F K φ 128 ν µν 0 Φ νρ β β 0 + 0 3 K 0 1 νρ β β 1 [ µναβ 96 µ A α α  µν F α α K K ] ε + µναβ √  F 0 µ Φ − Φ – 20 – 3 ∇ 0 2 ε F Φ 8 α α 0 + φ 3 8 µσ 0 φ A α α κ µν ν ν 2 φ 2 ν ν 2 σ m K F 2 φ + ν 1 κ 0 K + F κ Φ Φ 0 σ m κ 0 ν ν φ m ∇ 3 0 2 µν ν ν φ φ 2 ∇ µ µ 1 µ µ 0 φ Φ ∇ 64 σ ν 3 νρ √ κ 0 1 F 1 K ρ m α φ α K 1 ] + 2[Φ 24 φ K 0 . ) µ µ ρ K 11 0 K √ 2 192 β ν 1 µ µ K ∇ 16 + φ 0 ρ β 0 Φ σ 4 + µρ 0 ∇ 3 − ν α 0 − K ν ν ∇ 0 12 0 + µν α 0 ∇ 256 β ∇ + ρσ A 0 σ ν β 0 ∇ ρ β µν K 0 0 µ ν ν + 0 F + α prefactor. 0 Φ µ 0 φ µ µσ + F 0 α K σ µ φφ µ µ ε 0 ∇ µα ν φ µ 2 3[Φ][Φ F g 0 0 ∇ νσ ρσ ε µ ν µν ) ν σ ∇ Φ 0 µν µ µ ∇ ρσ − ∇ − K µρ Φ µ − A β ε ε K K = 0 µν µ K µν ν F ε µ ν 3 µ ρ F √ µναβ α µρ µν F νρ 0 F F µναβ ν ε A A µν µν ν K F ρ ρ ε K 0 A µ 2 µν φ φ 2 [Φ] µναβ µναβ ∇ F Φ µ µ 4 ∇ ∇ µναβ κ ρσ µν ε ε µν 2  ρσ A ε F 2 2 m m κ ε ( 4 4 ∇ m 2 2 2 F R 2 2 2 κ κ F φ m ( 3 2 1 m m 1 1 κ 2 κ κ κ 4 1 κ m m 2 4 1 2 3 m m 2 µν κ √ 3 m 3 3 κ µναβ 3 κm 17 m κ 1 11 F 1 1 √ m 1 1 8 48 ε 11 = 1 1 √ √ √ 3 1 √ − 12 48 16 1 32 − 3 − − − − 8 192 192 4 − ρ 1 7 ] 16 − 1 √ 12 48 144 512 − + + + + + ν 11 384 8 + + + + + 3456 A = = = = = µ = = [ = ∇ (4) hhhφ (4) hhφφ (4) hφφφ (4) AAhh (4) AAAA L L L (4) AhAφ (4) AφAφ L φφφφ (4) L L L with where we have used Below are theinteractions. various contact All terms terms are neededenter written the in in action a ( with covariant a form, with the understanding thatL they A Contact terms JHEP12(2020)030 (B.7) (B.5) (B.6) (B.1) (B.2) (B.3) (B.4) . We define a ]. In terms of t 2 i.e 71 √ and three momenta = θ a the three-momentum. T p . . ]) as c ν ) , , . ν 0 71 A ) ) ) .  b µ θ θ θ ), µ 0 , !  A θ ν : sin sin 2 cos a p abc B.3 i, , + 2 µ f − T , m cos ) c p . 2 − ,E ν + i, T 1 ± ν  0 ab , p √  θ, − + δ µ − 0 0 µ θ, abc   , θ, + µν θ, if cos + η a µ ] = + sin b − cos A 0 ν ν − sin T = 0 = , ,   ν ] = a – 21 – b ∂ ∗ than the structure constants in [ µ λ ± µ + µ , (0 (0 T , p p, E )    [ ( is written as: ν λ ( ( s 2 2 2 ,T ν ± µ −  2 1 1  a 2 6 p 1 ( a ν √ √ √ √ m T r 1 1 T a µ ± µν µ λ [ √ √  A

 F = = = µ ∂ = = = = 1 3 =1 X 2 1 λ − =0 =1 µ = µ λ µ λ =0 ± ± = p   µν λ µ λ = = as a   µν µν λ µν λ   F abc f . The polarization tensors for the spin-2 field with different helici- λ − µ  λ 1) is defined as: − 2 m = ( 4 ∗ the field strength tensor ) − is the scattering angle in the centre of mass frame and µ λ s  abc θ ( f √ 1 2 = where ties are constructed fromcoefficients the as polarization (we vectors review with the appropriate construction Clebsch-Gordan in (CG) detail in where The polarizations clearly satisfy the transverse and completeness relations, p We define the polarization vectors in the helicity basis as follows: B.2 Polarizations The four momenta in the centre of mass frame with scattering angle These are related tothe the structure usual constants, generators (for example in [ which again are largerthese by a factor of B.1 Lie algebraWe generators use of the the following conventions gauge for group the generators, B Conventions JHEP12(2020)030 . > > 0 2 (B.8) (B.9) , λ (B.13) (B.10) (B.11) (B.12) , λ 1 | . 1 2 = ⊗| ≤ > λ ), we can see 1 , λ > ≤ 0 | , λ B.5 2 1 , | − ,  i.e αβ G , with is given as, µν  > λ G + ν 00  2 3 − µ , λ  − 1 . , − , .  ⊗| να 00 00 00 λ ν − ν ν , λ ν λ ν G 2   p > 00 0   ). By substituting ( 0 0 µ 0 + µ λ ν . λ µ +1 ν m µβ λ µ λ µ    p  , 0   G , λ 00 +1 ν is, λ µ with helicity 00 00 − λ  +2 +1 µ 1  + B.15 λ λ , 0 |  0 ,λ 0 + 00 0 ν 2 λ J J,λ 2 λ λ = 0  λ +1 µ 0 µν , 0 = C  0 µ C C 0 νβ λ η +2  , 00 µ µλ 00 00 = +2 2  C × +1+1 G λ  λ λ – 22 – 0 0 0 X 00 λ λ C 3 3 X X λ λ , λ > µα λ 1 1 0 2 √ X √ λ | = = 1 = G = =  ,  = = = ,λ +2 2 1 , ) µν 2 J,λ µν 0 2 µν ,   φ . This polarization state is obtained by considering the  0 µν ( µν = = 0 00   λ ∗ ) which follows from the CG coefficient. ) ≡ µν λ  + ) 4.7 αβ λ µ 0 φ  ( µν p λ (  . in ( µν λ ν  2 3 p 1 2 µ m √ p − , from the tensor product of two massive spin-1 states we get a massive 2 = 0 = = X . We start from the spin-0 state which is obtained from + λ 00 λ 4.1 λ µν η + are the CG coefficients given in ( 0 = 00 λ ) can be expressed as: λ , with 0 0 = λ µα > C G λ 00 B.10 The spin-2 state is obtained from , λ 1 To give an explicit example, the helicity that ( Hence, the factor of where where ⊗| following: spin-2, a massive spin-1 andhow a the massive gravity spin-0 on-shell on the statesThe gravity are polarization side. constructed tensor In from of this such the section product, particle we of review spin where B.3 Construction of gravityAs states mentioned from in mYM The polarization tensors satisfy the transverse, traceless and completeness relations JHEP12(2020)030 (C.5) (C.1) (C.2) (C.3) (C.4) (B.14) (B.15) , 2 3 3 1 √ r = = 0 = 0 0 0 , , , 2 1 0 00 00 00 , , ) ) dλ mB − − 2 G 1 mB ( √ , ( d 2 1 = ∧ √ ∧ ∗ ). By integrating the last term by ]. The Stückelberg action of free 1 ) A − 0 , = 72 1 , , C.3 − + dλ 1 . , C − Λ G , − − Λ 2 = 0 = 0 − 0 d to be m dA. : C ∧ ∗ = + + G G mB dH 1 = ( G − ∗ B λ − mdB 1 2 1 1 2 , − . ,C,C,C – 23 – 1 field in 4d 0 − = 0 , , 6 2 3 1 − → → ,λ − − 2 C − 1 1 2 µν − G √ √ √  λ B C B = λ = 1 dG dB dB = = = α 1 = 2 , 2 1 0+ 0 − 1 0 1 + + , , , , , ∧ ∗ ∧ ∗ − 2 0 2 2 1 C − −− +0 − ++ +2 = X C λ C C C − C dB dB 1 2 ======1 2 . To do that we need to impose Bianchi identities, ) 1 1 2 1 0 0 − − − , − , , , , , h 2 1 µν ( 0+ +0 0 2 2 1 + ++ + ++ dλ  Z Z C C C C C C , is − = = B S S mB = : : : G is a 1-form Lagrange multiplier imposing ( and is a 1-form Stückelberg field which is needed to restore the gauge symmetry which Spin-0 Spin-1 Spin-2 A λ dB = where parts we can find the equation of motion for with Lagrange multipliers. We first do it for The first step in the dualizationH procedure is to rewrite the action in terms of field strengths, where acts on the fields as follows: C Dualization of theWe massive follow the dualizationmassive 2-form procedure field, explained in [ we do not focus on specific choices, for example for the graviton polarisation we have, In this paper we use the polarization states to be a superposition of different helicities and JHEP12(2020)030 , ) , , 1 ) ) 2 p 3 ( p (C.6) (C.7) (C.8) (C.9) (D.1) (D.2) p 0 ( . This (C.11) (C.10) , ( 0 0 µν χ , 0 to be ,  0 µν 0 µν  S  H tensor and S α S β γ ε , and the po- ) )+ ) )+ 1 A )+ ). 4 2 ( p 3  p p ( p ( ( 0 , , ( , 4.6 0 , 0 , , 2 µν 0 ) , A 2 µν 2 µν  2 µν   5  5 5 dχ T 5 fixed T T T α χdH. − 3 σ β which in coordinate basis γ / 1 + mdB. )+ ) )+ )+ 1 )+ 4 pl ∧ 2 3 p mA p mdA p ( p ( ( ( mH M A 1 ( to be normalised (i.e. consistent 1 1 2 1 − ∧ ) , . − − + − ∗ , − , , 2 µν m ∧ ∗ , A 2 µν 2 µν ) we can see that in unitary gauge,  . ( µ 2 µν A ρσ )   ρσ ) = 4   ) 4 4 dA 4 B T = ( + T B T dχ C.8 ν T dφ ( α 3 σ β  γ dB ∧ ∗ ∇ − ν Λ − dA p )+ )+ )+ )+ 1 ) with a scalar Lagrange multiplier, 4 2 dA 3 p ∧ ∗ µνρσ p p 1 2 p ( mA ε ( ( ( mdA ( µνρσ C.2 ( ε +1 − dA m – 24 – +1 1 , +1 fields is 1 2 +1 , , i , 2 µν 2 keeping 1 2 2 µν 2 µν m 2 µν  − ∗   A  3 − − dB 3 3 3 − T ∝ , = T T T = ) α σ β γ dA ∧ ∗ H A and H µ ) gives the Stueckelberg action for massive spin-1 field, ( µ  )+ A )+ )+ )+ ∧ ∗ ∧ ∗ B → ∞ 1 dB 4 2 3 p C.7 p p 1 2 p ( pl and impose ( ( ( H ( 2 dA 2 2 − 2 1 2 − 2 1 − − , H − , , , will be of the form: , 2 µν fields defined as: (setting the vectors to zero for simplicity) − Z 2 µν 2 µν ) and imposing normalisation condition gives ( µν 2 −     B ,M 2 φ 2 2 by 2 ( 2 Z = 0 T Z  T T T m α , σ S β = γ = → dB − and B S . )+ S )+ )+ )+ ) = m h 1 4 2 3 4 p 2 decoupling limit, p p p p ( ( ( ( p ( 3 0 , +2 +2 +2 Λ +2 , 0 µν , , , µν 2  2 µν 2 µν 2 µν     S 1 1 )). This relation can be inverted by multiplying both sides by the 1 1 σ is now the Stückelberg scalar field. From ( T T T T β γ σ + α χ B.6 , the relation between the limit = = = = µν µν µν µν = 0 , known as Proca action: 2 3 4 1 , which using     of the full scatteringsuperpositions amplitude of obtained from double copy with external states arbitrary D Double copy of the 4-pointWe take scattering the amplitude in the decoupling where the overall constant canwith be ( found by requiring p can be written as: This means that thelarization relationship tensor between of the polarization vector of A where χ Now again we integrate last term by parts and find the equation ofSubstituting motion for this back in the ( Now we can replace gives the following Substituting this back to the action and integrating by parts the last term we get JHEP12(2020)030 4 T S σ S 3 σ 3 4 4 S S σ 5 5 5 5 2 T 5 T T S T 2 σ σ 4 5 3 T T T T σ T 2 S σ σ γ σ √ 3 6 T T T 5 σ σ σ σ 5 5 5 √ σ σ 5 √ σ σ σ 3 4 5 T S 3 √ √ T T 6 S T T 2 3 5 5 5 γ γ T T T 5 T γ γ + 6 S γ γ 3 4 γ S T 4 2 T T T √ γ γ γ √ T 5 4 γ 4 γ S S T T 4 5 σ S γ γ γ σ 5 4 3 T T 5 2 γ S T T T β β 4 σ σ 5 5 5 T T σ S T T T β β 1 2 T S S T σ 3 4 β β β T S S T T T 5 σ γ S γ 2 6 β β β T 5 5 3 β σ σ 4 β T T 3 √ γ γ γ 4 β β β T 3 5 3 6 S 2 σ 4 3 3 5 β T T T 5 T 5 √ √ 4 5 5 α α 4 5 4 T γ T T T T 5 β S T T γ T T 2 T √ α α 6 2 T 2 2 √ α T T T 5 γ T T T 2 β γ T γ α α α γ T 5 α 4 2 2 α σ 6 σ β β β T 5 4 T α α α 3 3 √ √ α 3 T σ − − 5 T 5 3 4 β T T √ √ T 12 12 6 6 T α T S σ + 6 T − σ T T T 3 4 12 28 + 11 2 6 β α β β + 6 1 + 2 γ 3 γ γ 10 T T 3 α α α 4 5 S − − + 12 − − + 4 4 5 5 T T  S − − + 12 2 2 2 S σ σ T − − T + 2 T σ 5 5 5 3 4 T 5 T γ T γ − S 3 5 σ σ + 6 σ 4 5 3 2 + 6 α 5 α 5 T T T T T T β √ √ √ S β S β T T 2 σ 4 6 T T T S 2 T 2 6 6 6 T  6 σ σ σ σ σ σ σ S σ √ 3 γ γ T σ σ σ σ 3 4 5 4 4 5 σ √ 3 2 β 5 √ 6 5 3 σ √ γ √ √ 5 5 5 + 6 5 3 − − − √ T T T T T S T 2 T S T T 6 + 2 6 6 6 √ 2 √ T T T 2 T √ T γ γ γ γ γ γ γ 3 4 γ γ S T β β 2 + 2 S 2  γ γ γ γ 4 3 5 5 T 4 3 √ γ 3 1 S γ T T σ 2 2 − − − 5 σ − 5 4 5 11 2 T 2 4 γ T T T T T T √ 5 T T β 4 σ σ S 2 + 6 T 3 T T T 5 T γ 5 1 T T 5 S √ √ 5 β β β β β β T β β γ T S S − σ T β β β T S β T 5 5 5 5 T 3 4 β σ 6 6 T σ √ 3 5 T 4 β γ γ γ 5 3 4 3 5 5 6 2 3 3 σ β β T T T T σ T T σ σ T T 6 3 4 T α S T 3 T T T 5 5 2 T 5 T − − T 4 T T α α α α α α 2 α α T T √ β γ β T T T √ γ T γ T α α α T σ σ α 3 4 2 2 3 2 4 β β 2 γ 2 γ 5 √ γ 3 γ S α α 12 T T 5 5 T T 5 6 5 √ √ 4 2 12 T β + 6 + 2 − σ σ T T − α √ − T 6 α 6  T T 4 γ + 6 + 12 + 12 − + 12 4 S 6 2 α α 6 S S 3 5 S + 12 − + 12 − S β β + 4 + 2 β T 6 + 2 S T 5 5 3 4 γ S 2 2 − − 3 T T σ σ σ 3 β 4 5 5 3 α 4 γ σ T T T T − S S − σ − 4 √ 2 3 β σ 5 5 T T T T 1 √ √ S T √ σ σ σ σ √ 2 σ σ 1 1 T − S S T T 6 6 S T σ σ σ σ 3 4 4 3 σ β 4 √ √ – 25 – 4 3 6 T γ T γ σ 5 4 5 5 + 6 5 5 σ √ γ β 2 + 6 4 5 σ 6 T T T T 4 σ σ 2 5 − − 4 √ √ S T T T T 2 T S T S − T T γ γ γ γ − 5 S T 3 √ 1 1 γ T γ γ γ γ T 3 5 γ γ γ 2 + 6 − σ 3 4 γ γ S S 5 √ 1 T 5 σ β √ 3 4 5 4 4 T T 3 5 5 5 γ β S S T T T T √ 3 β β 3 T γ T 6 T √ γ γ T T T T T T 6 T S T T 5 4 β β β β 2 σ σ − T T 3 σ σ 5 σ S β β β β β 5 5 5 5 β γ β β β T T S S 1 γ + 6 − 3 σ T √ S T 4 5 5 4 3 S 4 2 3 2 3 β T T T T − γ γ T α α S 2 T S σ γ 4 5 β T T T T T σ T T S T T T α α α α 2 2 S S γ 4 5 β γ γ 5 S 3 + 2 5 T T β α σ α α α α α α α α 5 3 3 3 − β β T T T σ T 4 σ 3 6 √ √ 2 α 12 12 √ T 4 4 T T T γ β γ α S T S 12 6 12 6 12 5 S T T β β β √ 6 17 √ α 2 − γ γ σ − − + 6 + 10 + 11 + 6 + 4 3 T σ 4 α 6 6 α 4 3 − − − − − + 12 + 2 S √ 2 √ 4 5 S β − 6 S S S S S + 4 T T √ γ 5 5 3 3 4 5 2 T − + 6 T T S σ − − σ σ σ σ σ 5 β β 4  T T T T T T − S γ − + 6 3 4 5 σ σ σ S 2 3 5 2 2 + 6 5 6 T S − S σ σ σ σ σ σ + 6 3 3 σ 6 γ T T T T T S 4 3 − T β γ 4 5 3 5 5 3 σ √ √ 3 3 √ T T S γ γ γ − + 6  T T σ σ σ σ 2 4 5 √ 5 T T T T T T S 4 5 5 5 γ γ T 4 5 σ 6 S σ √ σ − S 4 T T S T γ γ γ γ γ γ γ T S S S T T T β 3 + 2 3 √ T T γ S S S 5 3 5 3 3 5 5 γ T γ γ γ 5 σ 3 √ σ σ β β β β β γ γ 3 2 5 γ γ γ T T T T T S T T 5 √ 5 3 √ σ T S 5 3 3 4 5 T 11 3 5 − S T S T T S β β β β β β β β S T T T S S γ σ T T T T T T T α 3 4 5 5 5 5 4 5 β β β β β β 5 5 γ  γ γ γ σ σ − 6 β β α α α α α 3 5 2 5 2 3 T T T T T T T T 4 1 4 2 3 1 2 T T  2 3 2 2 2 2 2  T T T T T T T T T T T T T γ α α β α α α α α α S γ γ β β β β √ β √ α √ √ √ √ √ α α α α α 6 12 α 6 2 4 6 10 12 6 6 6 6 2 6 stu 2304 + 2 + 2 + 6 + 2 − − − + 10 − − − − − + 12 + 12 − + 6 − + 12 + 10 + 2 − + 12 − − − + 6 + 2 + 4 + 10 − i → 4 M This gives the following amplitude: JHEP12(2020)030 ) ), 2 2 3 3 3 6 2.6 √ √ √ √ √ √ (E.2) (E.1) (D.5) (D.3) (D.4) 4 5 1 5 5 5  T T T T T T XXX 2 σ σ σ σ σ σ . 2 S S S S S √ 6 ) T γ γ γ γ γ 5 (+2 3 3 5 1 γ 4 √ ν + S T .  1 T T T T S β σ M T 2 β β β β β µ − 5  5 3 5 5 5  σ T T S T T T T T S γ α σ + γ α α α α α 2 S 1 ν − 6 + 2 √  β 5 5 µ + √ 6 + 2 T +  5 , T 2 + 6 2 + 6 2 + 4 3 + 4 3 + 4 √ α T 5 σ )(  2 √ √ √ √ √ , σ T S , , S 3 4 5 2 5 2 T , ) 5 ) ) γ σ ) T T T T T β 1 u σ 4 2 T 3 m 2 p 5 σ σ σ σ σ n p p γ p 3 + 2 ( 5 5 4 5 1 ( T ( 1 u ( + 2 √   − 0 µ T T T T T 0 µ γ c µ 0 √ T 0 µ  S   5 γ γ γ γ γ S u  we recognize that 3 − β γ 3 3 S S S S S T 3 σ 5  α 5 σ 2 β γ 2 β β β β β β 5 ν + + T 1 T 4 3 3 5 5 5  T √ α √ 2 β T T T T T T µ + σ ( ) + S t ) + ) +  ) + 5 α α α α α α m 1 + γ 4 2 6 . n 3 T p t 1 p p 5  p ( γ c √ − ( ( + ( 6 + 4 6 T 1 2 1 1 1 t γ S − µ √ √ − µ µ − + 2 + 6 2 + 6 2 + 6 3 + 4 3 + 4 6 + 4 − µ  −  σ  5 5  + 2 ν 0 ). By plugging the polarization vectors which 2 5 2 √ √ √ √ √ √ 2 T T 5    3 4 5 2 5 2 α 2 T σ β γ σ σ T µ 0 S T T T T T T γ s 2 5 – 26 –  σ m β 2 σ σ σ σ σ σ n 2.14 T T S 3 4 3  s 2 ) +  S S S ) + ) + γ γ ) + γ − 1 c 1 5 T T T S 4 2 5 5 γ γ γ 3 can only be a scalar mode and this amplitude then 2 √ p 5 5 γ γ γ s T p p p β S T T ( ( ( ( T T S S S β √ 2 β β β  + X 1 5 5 2 β β  β β β +1 µ +1 µ +1 µ +1 µ 2 √ 5 2 1 5 5 4 2 ) and ( 5 − T T T     T T T T T T T 1 1 1 Λ α α α massive Yang-Mills amplitude and comes specifically from 1 m S + β α α α α α α α β γ σ ν 0 σ is precisely the combination of polarizations that picks out 5  2.13  5 = µ 0 T = = = = S T 6  β β γ 6 + 2 6 + 4 6 + 4 µ µ µ µ ), ( 3 + 4 3 + 4 3 + 2 2 + 6 2 + 6 2 + 6 2 2 √ ).  1 2 3 4 1 √ √ √ +1000     √ istu 4 mYM 1 5 5 √ √ √ √ √ √ √ 6 96 1 5 5 3 4 5 2 1.4 T T T 2.12 A √ + T T T T T T − mode has polarization tensor σ σ σ 5 1 5 σ σ σ σ σ σ 5 S 5 5 ) = = S T T T S S S T σ T T γ γ γ γ γ γ γ +2 0 β S , 2 5 5 2 5 5 4 γ γ µν 0 γ S T S T T T T T T  β β β β β β β β β S − 5 5 2 5 5 5 4 3 4 XXX β T T T T T T T T T  S α α α α α α α α α + ’s given by ( β (+2 0 . We may easily see that n , contact term ( 4 µν 2 + 4 + 4 − + 4 + 4 + 6 + 6 + 4 + 4 + 6 0  M 5 T This amplitude simplifies considerable if we focus on scattering processes of the form ΣΣ β XXX the 4-point amplitudes of massive Yang-Mills is expressed as: with the are arbitrary superpositions of all helicities given as: E Decoupling limit ofIn massive this Yang-Mills section amplitude weis derive expected the to decoupling be limit theto of amplitude show the of that massive NLSM, we Yang-Mills derivetaking amplitude recover the the which the kinematic decoupling factors 4 limit and point and double performing amplitude copy the of it double a copy special do Galileon. not commute. We From also ( show that Since the helicity is the double copy ofthe the The combination the helicity-0 squared term +2 takes the form JHEP12(2020)030 , u ˆ n (E.7) (E.8) (E.9) (E.3) (E.4) (E.5) (E.6) 2 ’s and (E.10) 1 , m ˆ n  )+ 2 , 0 u ) m ˆ n u u c − s,t,u ˆ n . . + u 3 ) Σ( t t γ c 2 + 3 − σ 3 2 m + 3 0 t s decoupling limit is the s m β − ( m ˆ n c 3 3 t u − ˆ α iu n 12 Λ )( t  = ) − u t + stu, = − 2 3 2 s, t, u u 0 s σ s , n m ( 6 3 2 ˆ t 3 n Σ( u s γ ˆ n 2 − c 2 ˆ 3 n 2 Λ 1 s β 1 16Λ , n ). The amplitude can be written as, stu, 2 3 m ) m  ) + 2 s 3 α s 6 3 σ i , + )+ 1 − 6 3 2 3 . The explicit expressions for the E.15 Λ −  γ =  u Σ 2 3 2 + ( u 2 )( i 0 i 0 ( β u 16Λ m s u t s,t,u m ˆ 2 3 n and can see that the colour-kinematics duality it m 4 12 n c i 6 3 ˆ n − , α u ˆ − u ΣΣ n – 27 – Σ( Λ m − i − c E.14 c = 2 u − = 0 u ) + ) since they cancel in the full amplitude. However = = 3 3 u )( =1 m u mYM i X u t + = n + m − A − 2 E.3 2 pl + t DC 2  + ˆ 2 i t t t t E.13 1 2 t m ( A M Λ m n = ˆ n s 0 u t n + , n t t m c c − n ) c − s = terms could have contributed to the double copy amplitude,  u t + ˆ u t − 2 fixed fixed, is zero by virtue of Jacobi identity. Focusing on the n ˆ n s ’s, they can be rearranged in the following form (as mentioned 2 Λ u , n − + ˆ 1 n ˆ ) and ( + DC n n s , Λ t 0 2 terms in ( ˆ ( n 12Λ A 2 + , s i 2 → s 3 m 0 2 m is n 1 12 , we would have obtained a theory whose ˆ and n E.12 m − 0 t s m ) s m − c n − − /m c t → = lim = s 1 s − n )+ = = 0  t  m s DL 4 s, t, u u 2 2 2 n + A 1 n Λ m Λ s,t,u Σ( 2 + 0 s = = Σ( t m n 2 n s are given in ( − 3 n m + s ) m mYM 4 − s )):  s n A It seems that we could have defined the kinematic factors of the full massive Yang-Mills Using the kinematic factors of this amplitude we double copy it and obtain the follow- We see that only helicity-0 polarization states remain interacting in this decoupling 2 Pl 1.1 = 1 s, t, u s M n without using special galileon because only i.e. we would have obtained without them themassless colour-kinematics double duality copy is wherethis not at duality. four-points satisfied. If any we representation tried This of to is kinematic double factors in copy, i.e. satisfy contrast to the which is equal to the scattering amplitude of atheory galileon without theory. the Note that in this limitis we satisfied. have ing: limit. The kinematic factors of this amplitude are, and as mentioned in thedecoupling introduction, limit the lastnon-zero term term, which the seems amplitude at in first the ill decoupling defined limit in is the as follows: Σ( in ( with and four momenta into the JHEP12(2020)030 3 2 2 σ 1 )
σ 2 σ σ 3 3

2 γ


3 σ γ 3 3 2 γ 3 1
γ 1 2 1 1 1 σ σ β 3 stu, 3 σ σ β γ β
2 2 σ 2 σ 3 s, t, u 1 σ 2 (E.11) (E.12) β 1 2 +3 γ 1 1 σ 2 3 γ γ γ 2 α 14 − 1 1 γ 3 γ )) − γ 2 2 +5 σ Σ( − 3 σ 2 σ γ 3 3 2 γ 1 3
6 3 σ − − 3 )) u 3 1

− σ β 3 β β σ γ 2 1 1 σ 3 2 σ 3 4 1 γ γ − 2 2 5 1 ) 2 σ γ 1
γ + 1 σ β 1 +9 β γ 3 σ 2 σ 3 γ 3 β α γ
3 ) 1 4 γ 2 1 1 − 3 β 2 1 σ 2 γ α 2 σ 6 1 γ β γ β σ 1 γ α + +9 + β 3 − + 2 α σ 2 α 3 11 2
γ
1 2 − − γ 6 2 + 3 β

γ 2 3 +5 α 1 2 σ 1 3 −
σ +8 − − 3 +3 − 7 2 σ σ 1 m +4 − α σ 3 3 2 α 2 1 2 14 +
+5 1 σ 3 − σ γ 3 3 1 γ σ − 3
α 2 σ 3 − σ σ

σ 2 + γ 2 7 γ − σ 3 σ 3 2 3 3 3 1 σ 1 γ σ
γ 3 − σ 1 γ 1 3 σ γ γ γ 3 3 γ σ 3 σ 2 3 1 )+( 3 γ 3 2 3 2 β 1 γ − − 2 γ 13 +5 σ β 2 )+( γ γ 3 σ − α 1 4 1 γ 3 1 2 β β γ 6 1 1 1 1 − 3 β m β 1 − 1 − γ 2 β α σ σ 1 γ σ 2 β + 2 α γ m 2 2 σ − β 2 1 2 11 3 β +3 3 − β σ
1 +4 + α 1 σ σ γ γ − 1 + 1 β

3 γ 1 2 2 − 2
γ 1 − − 2 16



σ 3 2 3
σ 6 β γ α β 3 3 β γ + 3 3 3 γ σ +



3 1 β σ ( 2 +4 3 − σ 3 1 3 2 σ γ 2 σ σ − ( 3 + − σ γ − 3 + 2 3 β 2
γ 3 1 α 5 σ β σ 3 σ 3
1 γ 1 1 1 +5 2 σ γ 2 α σ 1 2 2 3 γ γ 3 3 3 +12 γ 8 α σ ( 3 β + 1 σ γ 2 3 1 β γ γ β γ
β ( α σ 1 − 1
γ 3 γ σ β 3 4 3 9 11 − β 3 1
3 2 3 γ
− +7 3 3 γ 3 1 1 1 are given below: σ 2 1 β σ 3 α σ α 3 2 3 α σ 3 β − − − 3 γ β β +12 − α γ 3 σ 3 3 σ γ 3 u 2 σ 2 2 2 1 5 − γ t
2 α 2 + 3 2 γ 1 γ − β 1 6 3 γ ˆ + n 3 2 11 σ σ α σ α
σ γ 3 β γ 3 2 3 γ + s +
2 2 3 − 2 3 σ σ 2 + , t β − 9 3 β
β t α + 3 2 γ γ − γ 2 2 − 3 8Λ γ σ 14 3 2 3 −
1 + σ ˆ 1 γ +12 σ γ γ n
3 + 7 σ − +2 3 2 16 σ σ
2 3 α β 2 ) σ
γ ) − , 2 3 +4 s +4 +4 3 2 3 γ σ − 1 1 2 1 β 2 σ s 3 σ 2
− + σ 1 1 + +5 3 γ γ 6 σ 1 γ 2 γ 2 γ σ ˆ m n σ 2 3 3
3 1 1 1 − 3 + 3 γ σ σ − – 28 – γ σ γ 3 s 1 γ σ − , σ
1 1 1 1 − β β γ − σ 2 − σ 1 7 γ )
σ 1 3 3 3 − γ γ 1 1 5


2 2 σ β γ σ + 2 9 − +5 3 γ σ σ 3 3 4 4 γ 1 − γ γ 2 γ 9 3 σ +7 γ 3 2 2 +4 − 1 3 2 ( ( 1 6 γ σ σ γ 3 σ 1 − 3 3 3 1 β +2 γ γ 3 σ σ − 2 3 1 3 β 3 σ − 3 β 2 σ − 2 2
β β + 5 σ σ γ β γ γ 1 β β 1 s, t, u γ 3 σ 3 3 2 3 2 3 γ γ 1 4 11 4 3 3 3 γ 2 3 γ γ +5 − + − γ γ 3 σ β σ σ 12 α β 9 α 3 β α α Σ( + 9 2 2 − − − 3 3 2 2 2 3 1 3 γ 1 − 3 2 3 2 σ σ − + γ 1 + − β γ β σ γ α − s 1 α 3 2 σ
σ σ σ 2 3
)+ α 3 α 2 )+ respectively. However, in our case, when we square 3 1 3 2 3 2 γ γ 2 2 σ 2 + 14 γ σ + 2 1 β s +4 +5 +9 1 γ γ γ γ 6 3 3
σ 2 + β σ σ 2 β σ polarizations and the decoupling limit of the resulting theory is − − 2 3 2 1 1 1 γ 3 2


16 γ β 2
− − α 1 1 − − 3 3 3 3 β β β γ σ σ σ 2 σ 1 − +4 3 1 1 α 1 1 3 3 3 1 2 σ σ
σ σ α α 2 − β σ σ +9 3 3 ± γ γ γ 3 3 σ σ s s, t, u γ β m 3 σ +9 +9 +9 3 1 ( 3 1 1 + γ γ contribution to the double copy scattering amplitude 6 − γ γ ( stu 3 − 3 2 3 + 3 γ γ 3 α
9 β β
σ 3 2 stu 4 3 β 3 − 2 3 2 +9 5 γ γ


σ σ σ 3 1 2 5 √ i 1 3 − γ 3 3 3 3 1 2 1 − − β σ σ γ σ s √ 3 m 3 1 2 3 β − 1 2 2 3 2 γ γ γ 3 β α β
σ σ 7 2 2 /m 4 1 2 β β β β β 1 2 1 labels 3 2 3 1 γ γ σ σ α 1 3 1 3 1 2 3 2 2  1 1 2 2 β √ γ γ γ α u β β √ stu α √ 3 4 α 3 α 3 α 7 12 β α α β α ( α 2 γ α γ s 2 , ( √ 2 i − + − + − − + + − × − + +9 − + +2 +2 + + + + + +5 +9 +2 i , = = )= ) s = 1 ˆ n i s,t,u ( s,t,u The explicit expressions for 2 DC Σ Σ( A not the double copyof of taking the decoupling decoupling limit limit and of performing the double massive copy Yang-Mills, do i.e. not the commute. operations , they sum to a which contains helicity where JHEP12(2020)030 2 2 1 σ
1 2 1 σ 2 1 3 γ 3 2 2
2 σ σ 1 σ 1 σ γ σ γ σ σ 3 1 1 γ 1 σ 3 3 β 2
1 1 σ γ γ 2 γ 2 +
1 γ γ 3 − 3 2 3 β 2
2 1 σ 1 5 3 β 1 2 1 γ σ 3 σ 1 1 (E.13) 2 γ 3 σ 1 σ 2 γ σ β 2 1 σ 2 σ σ σ β 3 + β 2 σ σ
4 3 3 γ σ 1 4 σ
1
3 + 24 γ 3 2 γ β 1 1 1 3 β γ 3 2 γ 2 3 β 3 β
γ γ − γ 5 γ
γ m σ γ γ − 1 +7 γ − β γ 2 σ σ 4 +5

1 2 2 4 21 1 + 1 3 + 2


+ 1 2 2 +3 β
2 3 2 2 σ +4 γ β σ 1 γ + β 3
3 β
σ − 2 +5 1 σ β − − σ γ 2 σ 2 σ 2 α +5 1 2 3 σ 3 3 σ 1 2 γ 1



α 2 1 − σ 3

1 β 3 γ 1 γ 19
1 β 3 σ +4 3 σ − γ σ 3 2 γ 2 2 γ 5 3 1 + σ γ σ 3 γ 2 3  σ γ 5 1 α 2 1 γ − 3 3 2 β 1 + 2 β σ γ
2 γ 1 1 σ β β 2 +52 σ − γ − 8 2 γ σ + σ 1 3 3 β γ β 3 γ 3 β + 2 5
2 α γ − + 1 2 2 + 1
3 σ γ 9 σ σ + γ 5
3 +15 − β 1 γ
α γ − 2 3 +3 γ 2 γ 1 3 8 β − 1 5 2 + 3 γ 2 − 1 3 2 +4 4 σ 3 1




2 +5 1 γ 3 σ σ 1 γ 1 β 3 γ +5 3 σ + 2 2 β 3 + σ σ σ 3 α − 1 1 1 β − σ 2 1 σ α +19 β 1 α 1 3 + − σ
σ β 2 2 σ 2 γ σ 2 2 σ 1
γ 3 β σ 1 β 1
3 8
β 5 γ 3 + σ 3 γ 3 γ − 7 3 2 σ 1 + + 3 γ σ γ γ 3 2 1 γ σ 1 3 γ σ γ +4 γ 1 3 γ

2 − γ β σ 5 4 2 3 m 3 1 +5 γ +
1 σ +13 β γ 3 σ 1 3 7 +5 3 γ t 3 3 3 σ γ 5 3 2 3 γ + 3 β +5 α 2 2 3 β β +5 2 1 3 σ σ σ β σ 2 γ γ β 1 1 − γ 21 − σ + − γ 1 1 3 +5 −


3 2 σ β 2 1 α − +4 2 2 + β + 1 1
1 σ 1 σ 2 s 3 +5 γ 2 1 γ 1 γ σ γ γ +2 1 2 −
2
β β 3 α 2 γ β σ + 2 σ 2 t 1 1 σ σ γ 3 σ σ +4
2 3 σ

σ 21 + γ β 4 3 γ 3 +
2 3 2 +13 3 γ β σ β 2 2 2 β 3 2 σ 2 σ 2 2 3 γ 3 γ 1 γ 1 γ 1 4 γ β 2 3 +5 +3 4 γ 2 σ − β +5 3 γ σ 3 σ 16 σ α γ σ 1 γ σ + 2 3 2 3 γ stu − 3 2
+6 1 γ 3 + α 2 − 2 +5 − 1 − 2 β 2 1 γ 3 γ γ β 1 γ − 3 β σ + 1 + +5 β γ 1 γ 3 +5 γ 2 3 √ γ 5 1 1 1 2 + σ γ 2 3 α 8 2 1 σ 2 2 β +3 σ 2 σ 2 σ + 2 σ β + 2 +4
β β β β + 1 − γ β − 1 1 β β σ 3 − σ 2 σ 1 2 1 β + 3 1 2 m 4 15 β 1 1 +5 3 γ 2 1 2 γ 2 √ γ σ 1 γ γ γ 3
+ σ α σ
σ 3 5 σ σ 2 α γ σ 1 2 γ 8 3 α γ 1 1 3

+4 − +5 3 1
+5 3 3 2 α 3 2 1 3 4 α γ 1 3 β β α 2 1 − 3 2 +5 β β σ γ 1 2 γ γ σ + σ − − γ β γ γ +22 β 8
2 σ 3 3 − γ γ γ 2 σ 3 γ 3 2 β + 2 1 6 2 − 3 + u 3 1 2 3 1 1 3
2 γ 1 β

σ + γ β + γ
+3 1 β σ − +5 γ 3 γ σ 1 − 3 3 − γ α β β β − m 2 − 3 4 β 1

1 1 2 σ 1 2 2 3 3 2
σ γ β
σ + 3 σ σ σ γ 2 − σ +22 σ + γ 3 σ 2 − β 2 +4 3 +5 3 3 β σ α
3
+2 13 3 σ 1 1 1 γ − 2


1 2 +5 3 γ 1 2 – 29 – 1 σ 3 3 γ β β 2 σ γ 3 γ 3 γ 2 3 γ − γ
σ 2 + +15 +3 γ γ β 3 2 3 γ 2 8 σ 4 σ γ +5 σ γ 5 1 2 3 β 3 − 3 3 σ σ 1 1 2 − β


− γ 1 2 3 − 2 γ 1 α α 5 5 3 3 3 γ β σ σ 3 − − β γ γ β σ 1 4 γ γ + γ 2 β 2 σ t 3 + +5 3 1 γ γ 3 3 +5 σ 2 3 3 4 3 3 σ + −
σ 1 − β + 1 2 3 1 − 1 β γ γ 1 − 3 − − γ β β σ 2 σ +3 3
β β + γ 1 +9 1 1 3 2 β 5 β γ 2 σ σ 1

3 3
2 γ +5 2 σ 3 γ 4 + σ 2 4 1 σ + 1 1 2 σ 2 β 2 2 β 2 3 σ 2 β γ σ σ σ
σ σ s 1 − +5 σ 1 2 γ γ 2 4 γ 3 1 2 − γ σ 1 1 17 3 γ − 3 5 α 2 1 γ +4 − − 3 +2 1 2 γ γ 2 2 γ 2 1 γ

β γ γ γ α σ 3 2 2 3 2 +4 − γ 3 2 σ σ +4 +5 − β β 3 − γ β 3 − 1 β 3 σ 3 2 2 β 2 2 +5 σ σ σ 1 σ 2 2 + + β β 1 + σ β γ β 2 3 1 3 γ γ β 1 +5 σ σ 2 2 +4 3 3 σ σ σ
σ 1 1 1 2 +5 +5 2 3 +5 +16 +3 γ γ γ γ 1


1 1 2 + 3 +5 15 σ γ σ σ 64 α 3 3 3 2 1 1 α 2 3 3 2 2 β β γ γ 1
3 1 3 2 γ γ γ γ σ +8 − +2 3 3 3
3 − β β 2 β γ 3 σ σ 2 1 γ σ σ σ σ
γ γ γ α γ 1 1 s 2 1 3 1 1 3 3 1 β β 8 1 1 γ σ β  2 β t +5 − σ σ +4 − + + +6 γ γ β 2 γ γ γ γ β 4 16 σ β 2 1 2 +5 +4 + +21 + − 3 2 1 1 1 2 3 1 1 − + 3 + γ 5 8 7 ) 4 − 1 1 1 2 2 2 γ γ − − σ σ 1 1 − 2 β β σ σ σ σ σ α stu 1 2 1 3 − +3 +8 + − − 2

σ σ σ σ
σ σ 2 3 2 1 1 − γ 5 2 σ 2 2 2 1 3 3 2 2 1 m 3 2 3 3 3 1 1 3 γ γ β β 3 1 3 γ γ γ γ σ √ 3 2 3 − − 4 i γ γ γ γ γ γ γ γ β 7 β σ σ σ σ σ σ − β γ 4 γ 4 5 2 3 β 3 2 3 1 3 2 3 2 2 2 1 2 3 2 1 2 β β 1 4 − 2 1 1 3 2 2 1 1 1 3 β γ β s β β γ β β β β β γ γ γ γ √ α γ β s α α β β β 4 α 5 α 24 4 5 2 α 4 7 α β 5 4( − +5 + +5 + + + +4 +3 + − + +5 +5 +2 − +3 + +5 − +13 − − − + +2 − +4 +5 − − + − = t ˆ n JHEP12(2020)030 2
1 1
3 σ 3 γ

1 γ 3 1 2 2 1 σ
σ 1 3 σ σ 3 γ 3 σ σ 1 σ 1 1 β σ 2 1 3 1 2 2 γ 2 1 6 σ γ σ 2 3 α γ 1 γ σ 1 σ σ γ γ γ 3 2 2 2

8 γ 1 +2 1 5 1 3 γ 2 m +
β 8 m (E.14) γ γ 3 σ 1 + 2 1 γ 3 2 σ γ γ 2 β 1 − 2 m
σ 1 3 σ +7 + σ 24 2 − − 5 σ β σ
7 σ σ 3 1 γ 2 1 1 2 2 γ β 3 1 γ 3 2 26 +2 3

1 1



γ − γ
σ 1 γ − γ



1 σ σ − γ +24 σ γ 3 1 γ σ γ 3 3 1 2 2 β + 2 3 3 1 3 2 2 β − +17 1 2 2 β 1 6 2 2 1 σ σ
γ σ 8 11 σ 1 σ +7 β σ σ 1 γ γ β σ − γ γ σ β 3 1 − σ β β 3 3 +5
3 2 2 2 2 + 2 4 2 3 6 2 1 3 σ 1 α 2 2 + γ γ − − 3 1 γ + γ σ γ
γ γ γ β 3 σ 11 β + γ β β σ 1 2 γ 2 3 γ 3 − 2 8
σ 3 − σ 2 + 3 2 5 +4 γ 1 3 2 4 1 9 + β σ 3 3 α σ 2 γ 2 1 − 2 σ 11 β + γ 2 9 − 2 γ β α σ − 2 3 2 1 α γ − 5 γ σ 2 1 3 σ γ 1 6 − α − − σ 1 3 + 2 α
γ +17 γ σ − 2 1 α 1 3 − γ 3 σ 2 3 γ + β 2 2 3
2 β 2 − 1 σ 2 3 − γ − 2 γ σ 1
β 1 t β σ β 2 γ − 3 σ σ t 2 3 2 +5 + β γ 3 1 σ 1 σ 3 1 2 γ  2 β 3 1 1 1 +7 σ α β 1 γ 3

σ 1 1 +7 + γ σ β σ + γ − α γ β 3 γ 2 σ 2 +7 3 1 − 11 σ 3 γ σ 1 γ 2 9 s +4 2 1 + 2 1 3 γ 3 +4 3 σ 1 γ 1 1 − σ 5 γ β
+19 + σ γ
s − β 1 +5 σ 1 − β γ



3 3 2 γ 3 σ 2 γ 1 3
2 α 3 +2 1 +5 2 1 + 3 1 β −


γ σ +7 1 + γ β 5 − 3 β

σ β γ 1 σ 1

+ σ σ 2 3 β 2 12 σ 1 σ β
2 1 γ + 8 17 3 3 3 2 2 2 6 +4 σ 1 γ 3 + 2 σ 2 3 2 3 − γ σ + σ 11 σ σ γ 3 α σ σ 3 γ 1 β 2 σ γ 2 γ 3 σ 3 − − γ γ 2 2


4 2 σ 3 − 11 1 3 1 γ 3 2 β + 2 σ γ − 2 − σ β 3 3 3 γ 2 γ + γ σ 2 γ γ β γ 2 3 3 1 β β 2 γ 3 1 1 − 4 11 3
γ σ σ σ 5 +4 2 +2 σ − γ + 2 +6 − γ σ α σ 2 β β σ 3 1 2 2 γ 3 3 11 2 1 +5 β 3
3
1 − − 1 3 β 2 3 − γ γ γ σ +2 +3 σ +19 γ σ 1 +4 σ 3 σ 1 β + + 1 2 β σ 1 β σ γ − 3
2 − 2 1 2 1 2 +13 1 + σ γ

σ 1 − − 1 +8 +12 σ 2 σ 7 1 γ σ 3 1 σ σ γ β + σ γ + 2 1 3 3 − 2 3 σ 3 3 3 3 +4 2 γ 1 3 γ 1 γ σ 2 3 2 1 σ 8 1
1 γ 1 α γ σ 23 σ σ σ 3 γ 2 γ 5 γ σ σ 3 γ σ σ 3 1 γ γ 2 γ β 3 σ γ 2 2 1 σ 2 +7 1 1 8 β 3 1 σ 9 − γ 3 γ + + 8 − 2 σ 3 6 γ 17 2 3 γ γ γ − 1 3 γ γ γ γ − 2 +13 7 3
β 1 2 2 − 3 γ 1 + +4 3 β +12 γ 1 β 1 γ σ 1 4 − 1 1 − σ − β 2 γ σ 2
+ + 1 β 2 α − β γ β 3 1 β + 1 3 σ 1 σ 3 4 σ 3 2 +9 +7 3 + β 2 2 − 6 2 β − σ σ γ 1 β 3 2 3 γ 3 σ 2 + σ 3 3 −
γ + 4 σ α 1 + 1 α 3 σ σ σ − γ 2 σ γ 3 β 3 + − β 1 +
8 3 σ σ 3
3 3 γ + 3 γ σ 11 2 − 3 2 3 γ 2
1 γ + 3 2 − 1 1 1 σ 3 σ γ 3
γ γ β 4 γ +5 3 2 + γ 2 3 2 σ β 3 3 1 – 30 – γ γ σ σ α σ 3 α − σ γ 23 2 1 2 2 3 σ + σ α β β 3 2 − − γ 1 1 3 σ + 2 α σ 5 +5 β σ γ s β σ 2 + + 13 3 3 7 1 1 γ γ − 1 2 γ 17 11 γ β 3 α 1 2 t 2 3 3 + 3
3 2 3 σ γ 3 1 1 σ σ β σ γ γ 3 3 − σ − 3 γ β α − − α
1 − + 3 3 σ − β γ β 3 + β σ +2 +2 1 3 1 1 +12 2 σ 2 2 σ 3
3 α γ 3 3 γ γ 3 γ − 3 1 + γ 3 3 2 3 +12 β 3 σ σ −

β + σ σ σ σ β γ α 2 +2 σ + γ 1 γ 2 3 2 +3 γ 1 2 γ 3 3



σ
1 +32 2 23 m
1 3 3
1 σ 13 +4 3 1 3 γ 3 3 2 2 2 − γ γ σ σ 1
+ β β γ α β 3 3 1 γ γ 2 1 β β 2 3 γ β 1 γ β − σ σ σ α β − 3 +12 σ γ σ + 2 2 s + β

σ 3 3 1 γ γ σ 3 1
1 2 2 − β +9 +7



2 3 2 +5 +5 + 2 1 2 11 γ 8 8 26 γ γ β 9 2 3 α − β 3 3 γ +4 β 1 1

σ
γ 1 σ σ β β γ β 3 t 1 3 +6 − 3 3 1 + − − − 2 + 3 σ σ 3 3 − 4 σ σ 1 192 β t t
2 1 2 2 2 2 2 3 2 +5 2 β γ α γ 3 α 3 3 σ σ 2 σ γ γ 7  2 3 α s +4 2 1 2 2 3 2 + γ γ γ σ σ 3 2 γ γ σ σ σ σ + σ α 2 β 1 + α − 3 1 1 2 2 1 2 1 − γ γ β β β γ σ + ) 3

σ 1 +9 γ γ

1 2 + γ γ γ γ 24 β β 17 11 2 − 3 +13 + 1 + 1 1 2 σ 3 1 σ +4 3 γ γ 2 m 1
1 3 − + +8 1 stu γ − −

σ 3 m β β σ σ 3 α β stu 3 2 2 3 1 1 3 − − +5 +5 +5 + +24 +17 − γ +36


σ σ 1 γ 4 i 2 4 σ √ γ 3 1 2 2 1 2 1 3 2 3 2 2 3 1 3 3
γ β σ σ σ σ s γ √ 2 2 − 1 3 3 − − 2 2 2 3 γ γ γ γ β 4 β β σ σ σ σ σ

σ σ σ σ + 2 8 2 β 3 3 2 1 s 1 2 1 2 2 3 2 1 1 2 β γ β γ σ γ γ
√ 2 3 2 3 2 1 2 2 3 2 1 γ γ γ γ √ β γ β β β γ γ γ γ γ 4( β 16 6 α γ α 9 9 α α α α 12 8 2 2 9 α β 9 st 7 β 23 11 − − − − − + − + − − +17 + + − + +5 +24 + +2 +5 +6 +5 − +3 − − − + − − + − − − = u ˆ n JHEP12(2020)030
2



1 1 3 2
3 3 σ σ σ 3 3 2 σ σ 1 2 σ σ 3 2 1 3 2 1 σ 1 σ σ 3 2 γ σ γ 1 σ γ 1 γ 2 3 γ
σ σ 1 1 γ γ 7 3 2 1 σ 2 γ 3 + 2 γ γ 8 3 3
γ σ γ 3 1 σ σ 17 1 − 1 3 γ σ γ 3 σ 3 2 9 β γ 3 3 γ 1 3 3 3 2 1 25 3 2 2 10 σ σ σ σ γ 4 β − +10 γ γ +9
1 β +19 γ 1 β σ σ 1 1 γ 12 σ 1 β σ
3 3 3 2 − 7 6 3 3 −

1 1 2 1 + 1 1 3 σ γ γ γ β γ +25 2 β − β 2 3 1 1 − σ 1 σ σ 3 γ γ
+5 γ 3 γ σ 8 σ 1 α − 3 β 2 2 +3 3 1 2 σ 3 σ 1 + 3 σ β 2 γ 1 α 5 3 β 1 σ 2 β 2 1 3 2 γ + 3
γ γ 3 3 +26 σ +8 β β 22 s σ γ − γ σ 3 +4 3 − σ + 3 + − γ γ α 2 1 γ σ σ 1 3 2 β β 1 8 1
+
γ 3 α
− 1 2 2

17 γ σ σ γ σ 3 σ γ β 3
t γ 3




2 +5 + 3 2 − γ 10 γ 3 37 σ +59 1 2 32 t 3 σ 12 σ 3 3 +50 − +11 +
σ 2 +17 2 1 γ σ γ σ 2 γ γ + 3 25
σ 2 +12 11 − 2 3 2 3 3 σ σ 2 +9 3 3 − 3 γ 3 σ σ 2 − σ 2 γ 3 1 3 3 3 − +3 − γ σ 2 σ σ σ σ 1 3 β γ 1 γ − σ +11 3 s β 21 γ γ 3 3 1 1 +25 γ γ 1 3 1 3 2 σ σ 2 stu γ σ γ 8 12 1 3 σ γ + β + 1 3 1 6 γ γ γ γ γ σ σ 1 2

2 σ − 2 γ − +8 3 11 β σ 1
√ 1 − 1 5 72 γ 3 γ 1 3 7 2 σ 2 γ β β γ 1 2 2 β + − 3 − − σ 2 +41 2 +5 2 γ γ γ 2 σ β 3 1 8 β 5 γ σ +10 γ σ
2 − − 1 1 2 +9 − σ 1 1 3 25 γ 5 3 σ 1 β 3 2 3 2 32 α α √ 1 2 + 1 2 2 1 γ σ + σ σ 1 − 2 5 σ γ σ γ 32 α γ β σ σ 2 2 − +37 3 3 3
γ 8 σ σ + σ σ 3 β 3 1 γ β − + + 19 5 3 2 3 2 2 2 − γ 2 3 3
γ γ σ β − γ γ s β
2 +2 γ γ 2 2 2 3 1 γ γ − γ γ σ σ 3 +5

− 3 7 − α 4 σ 2 2 +5 β β
2 +5 3 + γ 5 σ 3 σ 1 1 2 13 σ 32 +26 σ 2 +5 +15 3 +2
− β β 2 1 + γ γ 17 1 σ 3 m σ +9 +8 1 σ σ γ − + 2 − − 3 3 −
− σ γ − σ 3 2 1 γ 1 γ 1 2 2 σ
3


1 1 3 − +23 1 1 − +11

σ 3 σ σ 1 3 21 1 γ 2 γ γ γ 3 3 σ σ β 3 +5 +5 3 3 2 1 3 σ 1 1 3 σ σ σ σ β γ γ 12 1 σ β β 1 1 γ

σ 2 1



σ 3 1 2 − γ γ γ σ σ σ σ β 2 β 3 γ γ 3 3 2 3 3 + 3 3 γ 1 3 1 1 γ γ γ β 3 3 +5 +5 γ +3 32 3 8 5 β γ 1 2 γ σ σ + γ σ +3 γ γ 8 γ γ 21 σ +33 β α 1 2 1 2 3 2 2 2 3
β 1 σ 3 22
7 − +11 1 σ − β 2 σ σ
+ +22 − 3 1 14 +24 − γ γ + + 1 β 24 γ 1 α 19 γ σ 1 2 3 2 1 1 2 β α 2 1 1 − 2 2 −
β 8 σ γ 2 σ γ γ σ 14 − σ +37 σ γ − β +5

3 3 + − 2 σ σ σ σ σ γ 15 5 + 3 5 1 2 1 1 3 2 3 + − 2 1 2 3 2 1 2 3 γ σ σ −
γ − γ γ +13 +17


β 2 σ +24 +4 σ σ σ 2 1 γ γ γ β σ 3 − γ γ σ − 3 +5 3 1 3 1 3 3 5 2 3 3 1
2 3 1 3 σ 2 +3 γ 5 γ 2 9 β +4 1 σ σ β β 3 − − β γ 3 +8 γ 19 σ σ 2 γ γ σ σ γ 1 σ γ 1 σ 5 3 1 − σ 2 3 2 1 1 − 3 1 – 31 – − 1 1 σ 1 + 2 6 γ α σ β 1 1 γ γ − 3 1 +6 γ γ 2 β +37 γ 10 γ
σ σ β σ γ 1 β m 1 3 1 γ σ − 3 2 3 1 + γ 4 1 1 9 8 σ + σ 4 γ 5 γ + +59 2 +26 1 β − 2 α − +17 +5 3 +13 24 γ γ 1 γ
σ σ σ 2 5 1 2 21 γ − 2 − −





2 2 4 α 2 − σ 3 2 3 3 γ 3 γ σ 3 3 3 σ − +41 14 σ 3 2 2 2 σ +5 σ σ 23 γ γ 1 β 13 σ γ σ − 1 1 2 + 3 3 2 3 +24 50 2 2 3 σ σ σ 1 1 β γ σ σ 2 2 1 7 β t γ −
γ γ 2 3 2 2 3 1 +5 − 3 3 γ σ σ α β γ γ γ α γ σ 10 2 3 2 2 2 3 2 1 − γ γ γ 1 2 γ γ σ + β 5 2 + α 5 β σ σ +5 β β β 2 1 + + +23 γ 7 γ −
σ σ γ s +4 3 3 s 1 +33 + +5 +3 − 3 3 1 − − γ 19 7 11 2 3 σ 2 1 2 − γ γ 2 +15




1 1 59 1 14 σ 1 3 9 γ γ σ σ σ 12 σ α 1 1 3 3 3 3 3 − − − 1 +17 +13




α 2 σ σ 2 1 1 γ β σ 2 − 3 − 2 2 3 3 3 − +17 β β 1 1 σ σ σ σ σ 2 2 γ σ − 1 γ γ γ − γ 3 t 2 2 2 3 3 2 1 2 3 +5 +8 3 σ σ σ σ σ γ α γ σ σ 41 γ 4 2 3 2 1 3 − − γ γ γ γ γ σ 3 1 1 3 3 γ σ σ 5

α 12 8 + m + 1 3 2 1 1 γ γ γ γ γ 2 γ γ β σ σ 2 2 1 +17 +26
− +3 +22 γ + 33 γ γ 21 7 1 2 1 3 9 8 21 σ σ β s σ β 3 3 2 1 1 1 1 + 1 3 5 38 γ γ β β +33 − +25 +41 +6 − − +7 − − −






σ σ σ + β σ σ σ − γ γ − 2 2 3 2 1 3 3 3 2 3 3 2 3 2 +5 +13 − −



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