Supergravitational Conformal Galileons JHEP08(2017)014 11 12 13 16 29 Bulk 5 Ads ]

JHEP08(2017)014 Springer July 11, 2017 June 23, 2017 August 4, 2017 : : : bulk space. We 5 AdS Accepted Received Published Published for SISSA by https://doi.org/10.1007/JHEP08(2017)014 -theory vacua. Here, not only are branes M [email protected] , . 3 = 1 supersymmetric. This motivates us to compute the worldvolume N 1705.06729 = 1 supergravity, opening the door to possible new cosmological scenarios The Authors. = 1 supergravitation. In this paper, for simplicity, we begin the process, not p-branes, Supergravity Models, Superstrings and Heterotic Strings, Super- = 1 supersymmetrize the associated worldvolume theory and then generalize N c

= 1 supersymmetric three-branes, first in flat superspace and then to general- The worldvolume actions of 3+1 dimensional bosonic branes embedded in a , N N [email protected] N Department of Physics andPhiladelphia, Astronomy, PA University 19104–6396, of U.S.A. Pennsylvania, E-mail: Open Access Article funded by SCOAP ArXiv ePrint: the results to Keywords: symmetric Effective Theories that they must be actions of ize them to within the context of aon superstring a vacuum but, co-dimension rather, one forproceed brane the to conformal embedded Galileons in arising a maximally symmetric “bouncing” cosmology, where apresently pre-Big observed Bang expanding contracting universe. phase Perhapsarise bounces the is smoothly most within natural to the arena the required for context for such of branes the superstring to consistency and ticle of the physics theory, occurs but, at in low many energy. cases, the However, such exact theories spectrum have of the par- additional constraint Abstract: five-dimensional bulk space can leadconformal to Galileons, important and effective may, when fieldtial the theories, role Null such in Energy as cosmological Condition the theories is of DBI violated, the play early an universe. essen- These include Galileon Genesis and Rehan Deen and Burt Ovrut Supergravitational conformal Galileons JHEP08(2017)014 11 12 13 16 29 bulk 5 AdS ]. To begin 1 22 31 SUSY 4 ¯ L 8 29 = 1 supersymmetry-both in superfields = 1 supergravity N 10 N = 1 supergravity. 16 – 1 – N 23 = 1 supergravity, thus allowing for curved space- N 33 5 ] that a probe three-brane embedded in an maximally 2 : conformal Galileons 5 = 1 supersymmetric extension of conformal Galileons in flat N AdS 1 -field terms in the Lagrangian F 2 3 4 1 ¯ ¯ ¯ ¯ L L L L 5.2 The low5.3 momentum, curved spacetime Low limit momentum, curved spacetime limit-including 4.3 4.4 4.5 The 5.1 The N=1 supergravity Galileons 4.1 4.2 The worldvolume action, and the associated dynamics, of a bosonic 3+1 brane embed- ded in a background five-dimensionalwith, bulk the space structure are of the oftopological worldvolume considerable and theory interest dynamical itself [ properties, has dependingexample, been on it shown the to was symmetries possess demonstrated of remarkable in the bulk [ space. For straightforward context of aspace. three-brane embedded This in 3+1 aknown brane maximally to bosonic symmetric produce worldvolume the theoryplicitly theory is, computing of as the conformal we Galileons.superspace will and discuss Hence, then below, in generalizing already this them to work we will be ex- method for extending theembedded bosonic in worldvolume non-dynamical theories bulk of spacesand 3+1 to in dimensional flat the probe associated branes couple component such fields, worldvolume and theories second,time to once as this well as has gravitational been dynamics. accomplished, to Here, we will carry this out within the relatively B Useful supergravity identities 1 Introduction This paper is intended as a preliminary step to accomplish the following; first, to present a A constructing higher-derivative SUGRA Lagrangians 5 Extension of conformal Galileons to 3 A flat 3-brane in 4 Supersymmetric conformal Galileons Contents 1 Introduction 2 Co-dimension 1 brane action JHEP08(2017)014 5 ]. ]. ]. 3 M 23 21 ] of an 20 -theory to – = 1 super- 18 M N ], under certain ] and found to be 26 24 -theory. Furthermore, M ]. In this theory, the particles and in- 17 – 2 – Hence, in this phase, the universe is contracting. The 1 ] to different maximally symmetric bulk spaces-including, ]. In addition to this natural setting for particle physics and 5 , 22 4 ] that the worldvolume theory of a three-brane moving relativistically 7 ]. In these, a contracting Friedman-Robinson-Walker (FRW) geometry , 6 space led to the theory of relativistic DBI conformal Galileons which, in 16 – 5 8 background-that is, the DBI conformal Galileons– can, for an appropriate choice AdS 5 . These bosonic brane worldvolume actions were shown to contain new effective 5 orbifold, whereas a “hidden sector” composed of unobserved particles occurs on a AdS -theory membranes. This potential was explicitly computed in [ 2 ]. dS Z Within this context, the lowest order kinetic energy for the 3+ 1 brane position modulus was presented Although these bosonic braneworld scenarios are interesting, the fact remains that 25 M 1 -theory led to the postulation of the “Ekpyrotic” theory of early universe cosmology [ / 1 a steep, negativescalar exponential. fluctuations of thequantum brane fluctuations modulus that evolving are down nearly this scale potential invariant. producein As two-point [ discussed in [ is, these vacua, priorsymmetric. to These possible realistic spontaneousM vacua symmetry of breaking, supersymmetric are three-branesIn all this embedded theory, in a heterotic relativistictracted three-brane toward embedded the in the observable five-dimensional wallof bulk via space a is potential at- energy, which arises from the exchange second orbifold wall-separated fromNaturally the embedded first within by a thiswrapped specific five-dimensional on five-dimensional bulk a geometry space [ holomorphicand, are curve), hence, 3+1 whose consistency branes existence [ (five-branes 3+1 is brane worldvolume required theories, for there anomaly is a cancellation second, very significant, new ingredient. That can arise as theous spectrum types of of specific branes.five-dimensions superstring known vacua One as that very Heterotic simultaneously concreteteractions M-Theory include example of [ vari- is the the StandardS compactification Model of arise on the so-called “observable” wall [ branes of varying dimensions embeddedurally in within higher-dimensional the bulk spaces contextwhereas arise of the most spectrum supersymmetric nat- and stringhoc interactions theory of manner and particle to physics bosonic must cosmological simply scenarios, be it added is in well-known an that ad the Standard Model begins as a non-singular flat geometryso-called and Galileon then “expands” Genesis. to the Thethe Universe appropriate fact that circumstances, we that observe- admit bosonic NECical violation, brane scenarios has worldvolume also [ theories led can, tocan under “bouncing” bounce cosmolog- smoothly through the “Big Bang” to the present expanding spacetime. ered in this manner, bosonic three-braneslead embedded to in new a higher and dimensionaldemonstrated exotic bulk in theories space [ can of cosmologyin and an the early universe.of For coupling example, parameters, it was admitNull Energy a Condition stable, (NEC). This Poincare allows invariant for background a that cosmological theory violates in the which the Universe the low momentum limit,This reproduced was generalized the in five [ and conformal Galileons firstfield discussed theory in generalizations [ ofimposed Galileons, on each them. reflecting the In background addition symmetry to groups the novel 3+1 dimensional effective field theories discov- symmetric JHEP08(2017)014 and ψ = 1 super- ], we review 5 N . These terms, , which, to this , 2 4 F ) F ], this was done by ∗ ]. This restricts the = 1 supersymmetric F 32 , 33 ], these turn out to be N , a Weyl fermion 31 ] that these fluctuations χ 5 , 27 4 , ] for computing the bosonic , being very complicated and and 2 such as ( 5 φ 5 L , F 4 , again following [ 3 ]. However, the complete 28 , these super-Lagrangians were expanded to all – 3 – φ was omitted). These four super-Lagrangians were 1 = 1 supersymmetry. This was previously discussed L N bulk space. These Lagrangians are then expanded in a 5 AdS . However, this expansion was incomplete. In order to study the F to flat space φ , we review the formalism presented in [ = 1 supergravity. Specifically, we will do the following. 2 is devoted to extending these four conformal Galileons from bosonic theories but only to quadratic order in all other component fields. There were two reasons N 4 φ ]. However, based on previous non-supersymmetric work [ 30 ], where the superfield Lagrangians for four of the five conformal Galileons were , 34 29 Section In section behaviour of the original realorders in scalar field for this. The first was tosecond allow reason a was discussion to of permit some a of simpleorder the analysis of dynamics of expansion, of the does the complex fermion auxiliary not field field. contain higher-order The terms in of a real scalarin field [ presented (the first conformal Galileon then expanded into their componenta fields, complex two auxiliary real field scalars the four conformal DBI Lagrangians specificallyin associated a with embedding maximally the three-brane symmetric derivative expansion and alltheir terms own sub-Lagrangians. with the Remarkably,the as same first pointed four number out conformal of in Galileons. derivatives [ assembled into of terms with two specialtensors properties; only 1) and they 2) arenumber they constructed of lead such from to Lagrangians worldvolume second to geometric four order five. out equations Using of of the the specific motion fiveunnecessary metric, [ such we for Lagrangians-the give the fifth the Lagrangian, purposes general of form for this paper. In section actions to worldvolume actions of 3+1 branestries. embedded in First, maximally symmetric thecoordinate bulk system. generic space We geome- form then present of the the general form five-dimensional of metric the worldvolume is action composed introduced in a specific cosmological theories of themalism early for universe, computing it supersymmetric wouldIn worldvolume seem brane this prudent actions paper, to in westarting try complete begin to with generality. the create the process a bosonicsuperspace for- of and actions then, calculating discussed finally, these coupling above, them actions then to in gravitation supersymmetrizing by a generalizing them systematic the in worldvolume fashion, flat A first attempt toin [ do this was“modelling” carried the three-brane out as within abulk solitonic the space. kink context of Although a of some chiral of heteroticcoefficients, superfield the were in string geometric found the terms, theory five-dimensional by and these particularlythree-brane methods, a worldvolume the computation theory of general was their theory far of from an complete. Given its potential importance in time that we presently observe.can pass Furthermore, it through was the shownobservational “bounce” data in with from [ almost the no CMB.the distortion An exponential and, effective potential hence, field was theory aresymmetric for constructed consistent worldvolume the with in action 3+1 [ of brane modulus the in three-brane has never been explicitly constructed. conditions the NEC can be violated and the universe “bounces” to the expanding space- JHEP08(2017)014 and 3, to , F 2 ]. This field to µ , ∂ F 39 , = 1 38 i with its aux- , ]. However, in i µν L 36 -such as g F ]. In this paper, we , and auxiliary field χ 34 , , solve for the φ χ = 1 supergravity. However, due . The final issue that arises when N 4 . Having written out the complete equation of motion. Related to this, M χ should be treated as an auxiliary field or = 1 supersymmetric conformal Galileons, F N – 4 – can be explicitly solved for and present the results. , when expanded to all orders in the component scalar M 3 L conformal Galileon to term, were previously discussed in a non-Galileon context equation of motion was solved and a discussion given of 4 and . To do this, we expand upon the formalism previously dis- F F L 5 µ b . This has two parts. First, one has to show that the potential χ and and complex auxiliary scalar φ , both in superfields and in component fields, and show that it leads to µ 1 b L ], we again expand into component fields-this time ignoring the fermion ] as well as, within the context of new minimal supergravity, [ 34 37 bosonic Galileon had been ignored in the analysis of [ , 1 35 L . Hence, it is no longer clear whether ]. The associated cubic ∂F 35 Having carefully discussed the flat space ∗ = 1 supergravitational Lagrangian for the first three conformal Galileons. We have also = 1 supergravity in section , as well as to all orders in the supergravity multiplet scalars; that is, N extended the supersymmetric to the complexity ofwe the will computation, use this severalcontext result non-trivial of will results low-energy, curved from be superspace this presented Lagrangians. work elsewhere. at However, the end of this paper within the to all orders inF the Galileon supermultipletiliary scalar vector components, field expansion into scalar fields,supergravity we auxiliary then fields show, inThese detail, solutions that are the then equations put back of into motion the for entire the Lagrangian, thus producing the complete cussed in [ is analytically a veryand tedious then process. again However, expand wepreviously, each we carry ignore such it both supergravity the out Lagrangian Weylwell completely fermion as into associated in the its with superfields gravitino the component of Galileon fields. supermultiplet supergravity. as However, As as above, we expand each such Lagrangian leading, first and, finally,results second are order. inserted back To into the simplify full the Lagrangian analysis, andwe only the then the associated extend the physics leading first discussed. order N three super Lagrangians, that is, supersymmetric It turns out that supersymmetric fields, contains terms proportional to∂F derivatives of the “auxiliary” field as a dynamical degree ofthe freedom. context In of this a paper, derivative we expansion carefully and discuss a this issue specific and, solution within for two real scalar fields energy so-derived, allows for stableone solutions must show of under what the conditions theBoth associated of kinetic these energy terms issues are are nonone ghost-like. discussed expands and to solved all in orders section in the scalar component fields is, perhaps, the most important. presented in [ entirely, but working to allnew orders issues in that the will scalar bethat fields. presented the This and opens solved up insupersymmetrize three this very paper. important The firsta specific arises potential due energy to in the the theory. fact The second issue has to do with the stability of the in [ the three different “branches” ofinto the possible Lagrangian violation that of now emerged. theinteresting physics NEC This arising and work from also other these looked “bouncing”the new properties present branches paper, in was also this we discussed context. do in something Some [ very different. Using the same superfield Lagrangian along with the usual quadratic JHEP08(2017)014 ) A 4 3. , X X ( 2 3 to ) are (2.1) (2.2) , , 1 2 X AB , , ( 1 G , µν g = 0 coordinates. µ = 0 . However, to be A , µ µ X X , ν = 0 is the time-like 3 as m, n, . . . =constant, where the , A dX 5 2 µ , B X 1 is the unit normal vector. n , dX , we use some of the results A ) A ∇ ], one chooses coordinates 5 n 5, where any given five-tuplet = 0 X ν , ( 5 ), where B , 3 e 3. The worldvolume coordinates A , 4 µν X µ , 2 ( g A 2 , 2 e , ) and the intrinsic metric ) 1 1 AB 5 5 , = , satisfying the above conditions and, in G X X ( ( is the transverse normal coordinate and µν = 0 A f f = 0 . It will be clear from the context when these refer 5 ] for constructing the worldvolume action X A µ + X 5 , 2 , ,K ) µ 4 ) 5 – 5 – σ µ, ν, . . . 2 ) for X ( ]. σ dX ( = 1 supersymmetric Galileons presented in section 40 A AB = ( N G X ν B B e µ have dimensions of length. We begin by defining a foliation dX A e A A ] spacetime indices are denoted by Latin letters 3 only. The function = X 40 , dX 2 ) µν , ¯ g 1 X , ( )) on the brane is a point in the bulk space written in AB 5 and the associated metric by = 0 geometry to be discussed in this paper, the origin of such a coordinate are the tangent vectors to the brane and σ , G ( 3 µ 5 = 1 supergravity extensions given in section A µ , , (5) 2 µ ∂σ N , ∂X ) is an arbitrary metric on the foliation and is a function of the four leaf 1 AdS X , X = ( µ ,...X µν A ) = 0 g e σ ( A , We note, however, that in [ Now consider a physical 3+1 brane embedded in the bulk space. Denote a set of intrinsic Finally, we point out that for the (0) 2 A X One expects the worldvolume action to becompatible composed with much entirely of of thewe will literature the denote on spacetime geometrical higher-derivative indices supersymmetryto tensors by and spacetime, Greek supergravity, letters as in opposed this to work spinorial, indices. by the five “embedding”( functions The induced metric and extrinsic curvature on the brane worldvolume are thenwhere given by such as the system is completely arbitrary and carries no intrinsic information. worldvolume coordinates of thealso brane have by dimensions of length. The location of the brane in the bulk space is specified coordinates dimensionless and will depend on theIt specific bulk is space important and foliation to geometriesparticular, notice of interest. that the the location coordinates ofcould be their physically origin, important have in not some contexts, been for uniquely any bulk specified. space of Although maximal this symmetry, and 2) the extrinsicinduced curvature on metric. each Under ofthe these the metric leaves circumstances, takes of the the form foliation is proportional to the where constant runs over a continuous rangethat which depends the on coordinates the choice onNote of an bulk that space. arbitrary we It leaf follows haveindicate of denoted that the the these foliation are four are therestrict coordinate given coordinates the indices by on foliation the so leaves that of it a is time-like 1) foliation. Gaussian Now, normal further with respect to the metric of a 3-braneX in a 4+1-dimensionaldirection. bulk The space. coordinates of Denote the bulk the space composed bulkso of that time-like space slices. the coordinates leaves Following of [ by the foliation are the surfaces associated with and for their and follow the notation presented in [ 2 Co-dimension 1 braneIn action this section, we review the formalism [ JHEP08(2017)014 ) ]. µν 5 , 2.7 K (2.9) (2.6) (2.7) (2.8) (2.3) (2.5) 4 and  , must be π . It follows µν , F ν g
µν ), however, is . Infinitesimal ∇ g µ σ π ( α σ µβν µ 5 are, in this gauge, ¯ R ∇ X , µ 0 . µ f σ f ν . ¯ ∇ , ν dx + 2 µ µν dx ) has dimensions of length. µ µν dx µ g ,K 0 ) dx x ). Therefore, the metric ( X ) µν ( . ( x ff respectively. It follows that the ¯ g x ( π 3 ( constructed from ¯ µν . ρ , π + g ) ) (2.4) 2 µν F 2 µ π , g σ ≡ ¯ g ν 2 ( 1 α βµν )) X ) 5 − , ) = ¯ x A ( ∇ R ρ ( x ( √ µ X π ( X π f σ ρ ( = 0 µ 4 ∂ f d + −∇ ) = µ 2 ξ +  σ Z ( 3 and ) can now be written as 2 , µ γ – 6 – , dρ π ). Although, naively, there would appear to be µ 2 = dρ ) = σ = 2.1 , ) to set σ = ≡ (
σ 1 of the foliation. The function µ ( = ) , B µν ξ 2.4 A σ µ α µβν B ) = ( ¯ 5 σ R dX = 0 X δX ( , A X dX µ µ and the curvature A , µ ¯ X ∇ µ dX µ , ) x π, K ¯ dX ∇ ν µν ) X ≡ ( ∇ X µ ,K ( π µ AB ) and define X µν AB G ∇ ¯ g ), one notes that the scalar field specifying the 3+1 brane location 2.5 G + L 2.6 µν σ 3 that defines the location of the 3+1 brane relative to the choice of origin g 4 ), as well as , discussed in this paper, the location of the coordinate origin is completely 2 d 2 ) 5 , π 2.2 1 Z ( coordinates. We reiterate that, although in some contexts the specific choice of )and ( , is a scalar function. Furthermore, the brane action, and, hence, f = A AdS 2.5 = F = 0 S X µ µν ¯ g , For clarity, let us relate our notation to that which often appears in the literature [ µ It then follows that the induced metric and the extrinsic curvature on the brane are given by Using ( relative to a chosenrestricted origin to can the be brane worldvolume expressed becomes as arbitrary and carries no intrinsic information. Note that With this in mind, we willnormal denote the coordinate four by foliation coordinates andgeneric the bulk transverse space Gaussian metric appearing in ( That is, there isX really only a singleof scalar the function of thethe transverse coordinate foliation origin coordinates could besuch physically as important, in a bulk space of maximal symmetry, fixed to becompletely the unconstrained bulk by coordinates thisgauge gauge specified by choice. ( Henceforth, we will always work in the five scalar degrees ofthat freedom one on can use the the 3-brane gauge worldvolume, freedom it ( is straightforward to show Inverting this expression, it is clear that the worldvolume coordinates invariant under arbitrary diffeomorphisms of the worldvolumediffeomorphisms coordinates are of the form for arbitrary local gauge parameters defined in ( that the worldvolume action must be of the form where associated with the embedding of the brane into the target space; that is, ¯ JHEP08(2017)014 ] ) 5 , 2.7  4 (2.13) (2.10) (2.11) (2.12) (2.14)
, ] ] if one 2 4 5 π , 4 , 2 ] + [ 3 π  .
] 3 µν [Π][ term is particularly π 5 K − L  2 1 µν f ¯ g 2 [Π] + 4[ ¯ R γ 2 ]. The 1 2 f . It has been shown [ 5 2 2 µν , − ] + 2 , leads to an equation of motion g γ , f i 2 4 4 , c µν ) . 2 0 ¯ [Π R 2 π ) (  , − i π f 2 2 0 4[Π] + L ∇ i − ( c GB ) for an arbitrary metric of the form ( dπ − [Π] 2 1 ¯ 1  π f R,  3 µν 0  =1 γ ] 5 i ¯ ¯ g, g ¯  g K Z K 2.12 3 is a Gauss-Bonnet boundary term given by ) – 7 – + ¯ g − g K, − 2 3 − 4 π 1 + [ γff γ π − − 2 √ √ √ = Σ − GB q ν γ 3 2 √ − √ − + L K ∇ K 2 = ) + 2 π = = = = = γ 2 µ 2 γ µν 3 4 5 1 2 2 γ and [Π] + ∇ K L L L L L 2 2 − ) µν f + π 1 + (1 ) will generically lead to equations of motion for the physical and α µαν 3 ∇ − 2 − ¯ γR ( ) are satisfied. R ) ( ) ) K 0 2 2.3 0 2 2 00 1 1 3 f µν π f γ γ f ( − ( g γ 2.12 3 2 − ∂/∂π 4 − ), for any choices of coefficients f f R f = ¯ 0 2 = = 1 + 6 ), ( (5 ¯ f 0 0 R dπ r + 6 − 2.11 1 γ f , GB 4 π 3 2.11  K ) that are higher than second order in derivatives and, hence, possibly µν f g gf Z x  K ( g − g − are constant real coefficients, π i µν − − √ √ c g √ − √ − ) that is only second order in derivatives. In this paper, we will assume that ) and ( µ = ¯ = = = = 2.3 X 4 2 3 1 K ( L L L L π Evaluating each of the Lagrangians in ( An action of the form ( is arduous and has beenlong carried and out not in necessary severalrest papers for of [ the this work paper. to The be remaining discussed four here. Lagrangians are Hence, found we to will be ignore it in the All indices are raisedthat and Lagrangian traces ( takenfor with respect toboth ¯ ( with propagate extra ghost degrees ofrestricts freedom. the Lagrangian Remarkably, this to can be be of avoided the [ form where the scalar field respectively, where JHEP08(2017)014 , 4 π 2) L , ν ) is ] = 2 (3.1) (3.2) (3.4) ∇ (4 π µ 3.3 and so This ge- 3 3 L ≡ ∇ , coordinates 2 1). A , [Π] (3.3) L µν , . For example,
X (3 1 2 p µν L γ g − ). More specifically, 4 2.8 . For example, [ γ π π , R 2 ν ∇ − 2 e dx − with isometry algebra µ n 2 
R 5 3 Π dx π π −  µν 3 and 2.  AdS , η 3 2 metric written in the ) 2 4 π 0), and denote the flat metric on the [Π]
) , ≡ ∇  5 5 . π ) become ∂π 3 − ] > ( ( π ) is of the form ( R . These are precisely the conformal DBI ( γ − n  f π ∂ 4 R − π 2 2 8 R AdS 1 R e 2.6 2.14 π + e γ 2  + π R – 8 –
π ) = 4 dρ 1 + R ∇ → 2 4 π − e γ ( = and so on. Similarly, we also define contractions e q 3 f γ + 1 1 and 2 respectively. Hence, the constant coefficients B 2 R π ]. = 2 ) , 2 ν ]. It can be shown that each of the terms in ( 0 41 γ γ − , ∇ dX and so on. Note that the Lagrangians 5 ∂π 3 , 1 µ ( A  : conformal Galileons π  − − [Π] + π 4 5 ∇ ν R 2 , 2 2 π π dX 2 e R Π ∇ 2 ν ) and definition (  π − 1 γ ∇ AB using the notation [ ν ) have mass dimensions 5 e AdS − µ 2.4 π 1 + G R 4 ∇ 2 π π − ∇ R radius of curvature by µ 4 q −  2.11 ∇ e − π 5 ∇ 3 [Π] R 4 2 e π π  6 ] = −  µ 4 R R 2 e γ 2 AdS ∇ − − γ − , one finds that the target space metric is given by + ] denotes the trace of n-powers of [Π] with respect to n , [Π µν = = = = ] are defined as above with ] = η π 1 2 3 4 3 n µ in action ( π π L L L L . We follow the notation common in the literature. Defining Π 4 ∇ ,[ µ µν ], [ , c π ) have mass dimensions g 3 n ∇ µ , c ∇ 2 The case of the 5-dimensional Poincare space, leading to the “Poincare” Galileons, and their extension 2.14 π 3 , c µ 1 and [Π Galileons, first presented in [ to supersymmetry has been discussed in [ respectively, where where It follows that the four Lagrangians given in ( subject to gauge choiceif ( we denote the foliations by imally symmetric” 5-dimensional anti-de Sitterand the space foliation leaves areometry “flat”-that is is, easily have shown Poincare to isometry satisfynormal algebra the above with two respect assumptions that to the theto foliations target the are Gaussian induced space metric. metric and It the then extrinsic follows that curvature is the proportional in ( c 3 A flat 3-braneHenceforth, in we will restrict our discussion to the case where the target space is the “max- metric the bracket [Π [Π] = of powers of∇ Π with In these expressions, all covariant derivatives and curvatures are with respect to the foliation JHEP08(2017)014 i c (3.8) (3.9) (3.5) (3.6) (3.7) (3.10) # ) admits 4  3.3 π . ˆ π ν ∂ M ∂ ν  x µ 3 20 x ˆ π . 2 − R ν ) that the coefficients ∂  ν − 3.8 ) that, in this expansion, ˆ π x π 2 ν µ µ ∂ x 3.6 ∂ µ M 2 ∂  − 2  x ˆ π , ˆ π ), one can, up to total derivatives, ,   µ 3 1 2 ) and ( ν R 4 c ∂ ˆ R π c ∂ 2 3.3 / ν µ M + M 3.5 x ∂ M ∂ i π R , 2 + mass scale by ¯ = 1 + 12 L 2 e i µ 5   4 ¯ c 3   + 16 x c R ˆ c π M M M ˆ 2 π µ =1  2 c ν 4 i M ∂ AdS = 2 ∂ + – 9 – + 6 + 6 µ M  ˆ π + 4 µ ∂ , 3 4 2 = Σ 1 5 µ x 3 4 2 ) to be dimensionless and, hence, each coefficient ¯ 1  c c c π L c R M M M M + 1 2 3.9 = 2 2 = ˆ as π = = = = + π i ˆ 1 2 3 4  π 4 M 2 µ ¯ ¯ ¯ ¯ c c c c 4 are arbitrary, it follows from ( ˆ L π, δ i ) are precisely the first four conformal Galileons. Since the 2 . Performing this expansion and combining terms with the   µ c 2 2 ˆ π  ∂ ˆ ) π ... µ 3.9 ∂ M ˆ =1 π ∂  4 i M x M ∂ M   = 1 −  1 4 π , δ 1 2 = Σ ) is invariant under the conformal Galileon symmetry µ 1 5 2 − ∂ − , i L − µ = 1 i  arising in different Lagrangians ( 3.9 2 " ˆ c π x ˆ π ˆ 2 2 π )  δ ∂ M M  ∂ 2  ˆ M π R −  ∂ M π π 4ˆ 2ˆ ˆ π =  − − 4 are also unconstrained. We find from ( ∂ M e e π 2ˆ 1 4 1 2  δπ ... − 1 2 − e − = = = = = 1 2 3 4 1 Defining the dimensionless field and the ¯ ¯ ¯ ¯ L L L L , i i ¯ has dimension 4. Note thatoriginal ( coefficients c each Lagrangian in ( We have chosen each Lagrangian in ( are real constants and where respectively, it is clearan that expansion each in powers of of thesame ( power four of conformal ( DBIre-express Lagrangians the in action ( invariant, up to a total divergence, under the transformations JHEP08(2017)014 ) p 2 ) 4.1 ∂ . In M (4.1) i (3.11) c ). This 3.9 , but with one 5 were extended 5 ¯ L ... given in ( field defined in ( 1 ¯ 4 L = 2 φ ] that all terms with ( i . Now perform the derivative , ) 42 i , ¯ 3 L have mass dimensions 0 and 2.11 ( i c ), the worldvolume Lagrangian for as in i M L i , 1. This is unique to the case of the conformal c ∂/ i ¯ ) in terms of the L  i =1 5 i ˆ ¯ π c 2 . It is well-known [ 3.9 e ) 2 and Lagrangians ) =1 ∂ 4 i = Σ M ≡ π ∂ 4 M ) L – 10 – φ 4 in ( = Σ ∂φ ( L ... 4 expansion is the following. Unlike the discussion in this section, 3 φ 2 ) 4 is given by ∂ = 1 M − in the sum 5 i )-since it is not necessary in this paper. However, it can 2 5 , ) 2 L i -field which occurs in the component field expansion of the ) M ¯ L AdS and each constant coefficient ¯ F ∂φ 5 to all orders in ( ( ∂/ i ∂φ φ ( ] ¯ ... L 4 4  40 1 1 φ φ ], the real scalar field ˆ ] and, hence, do not contribute to the theory. 3 4 2 = 1 1 φ 34 i 2 − − 42 , 3 for = = = i = 1 supersymmetry. To do this, it was convenient to define a dimensionless 3 1 2 L ¯ ¯ ¯ L L L N = 1. Here, we will review this analysis, again neglecting set to unity. The result is M 4 form a total divergence and, hence, can be ignored in the action. Therefore, this expansion is M A more complete discussion of the ( We begin by presenting p > 4 let us here includeexpansion the of Lagrangian the for exact and does notGalileons require that that we one are demand discussing. that ( this section, as wellnotation as presented in in the [ next section on supergravity, wewith use results and follow the adds a potential energy termsuperpotential to the to scalar appear Lagrangian in and,discussion the hence, requires of superfield a the action. non-vanishing auxiliary Insuper-Lagrangian. turn, In this particular, necessitates weits a give a more equation careful subtle of analysis motion of how and it what can constraints, be eliminated if via any, that puts on the coefficients ¯ real scalar field and set important new ingredient. That is, we now include the Lagrangian 4 Supersymmetric conformal Galileons In a previous paper [ to flat space 4 respectively. As discussedwhich previously, is we eighth-order are, ineasily for ( be simplicity, included ignoring without the changingthan any fifth of 8 Galileon our in results. the Notetotal derivative that divergence expansion all [ terms of of the order DBI greater conformal Galileons can be shown to be a a flat 3-brane embedded in where each Lagrangian We conclude that, expanded up to sixth-order in ( JHEP08(2017)014 2 φ / we is a 5 (4.4) (4.5) (4.2) (4.3) , ψ φ so as to , ) χ ], both in range of the 2, then the 4.1 / 34 ( ∞ and 1 + − φ φ ,  are dimensionless, ˆ π < 4 φ , since this will be ) < χ  F 0 surface is equivalent to defined in 4 ∂φ ) ( have dimensions 0, 1 φ → −∞ 3 ∂φ and F. 1 φ φ ( F ∗ 4 3 F 4 φ − 4 5 φ, χ 1 φ + ,µν and, hence, , + were constructed in [ φ 2 2 φ has mass dimension ,ν ,µν ) ) 4 6 † φ φ ¯ θ ) L . ∂χ ,µ ) ( . The results are the following. ,µν 2 ∂φ 4 φ . Of course, the ( F φφ in terms of the field 2 1 φ φ 4 and )  2 φ ) 3 3 2 3(Φ + Φ ¯ θ 1 ¯ ∂φ φ − ¯ L 20 θ ( 3.9 A, ψ, F 2 ( , θθ 2 – 11 – ) + −

2 ) = 1 φ ) ¯ 2 † . In this paper, we present the same superfield -field in higher-derivative Lagrangians without a L 2 † ) ∂φ and the complex scalar φ Φ ,µν F ( ]. In the present paper, however, we will carefully Φ Φ = ( , + , φ 4 ∂φ ψ action is given by 34 2 ( 1 φ ,ν is a complex “auxiliary” field that can, for Lagrangians (Φ ) regime of field (Φ µ 2 2 φ φ ¯ ∂ F K follows from integration by parts. L − K ,µ 2 ∞  ) φ 4 ( + 2 = = ¯ 2 L ) ) ∂φ ( µ ∂φ ∂φ ( ∂ ( SUSY 2 3 2 2 < φ < ¯ L 4 φ = 1 chiral superfield 1 1 φ φ 5 4 2 can only approach zero, but never achieve it. Hence, nowhere in the range of ) is a complex field composed of two real scalar fields ]. However, unlike that paper, we will not display any component field N − − φ − -field, this time in the presence of a non-vanishing potential. Ignoring iχ 34 F = = + 4 ¯ in an φ L ( φ 2 1 √ field, but the supersymmetrization, althought completely equivalent to that presented in this and, hence, π = 2 ¯ is changed to the 0 L A π We have written the conformal Galileons in Having presented the Galileon Lagrangians associated with the real scalar field 5 , the supersymmetric extensions of → −∞ 1 ¯ ˆ greatly simplify the extensionfield to ˆ the supersymmetric case.π By doing so,do the the Lagrangians, or anywith other the quantity in ˆ ourpaper, derivation, is diverge. far It more is complicated-both equally to possible construct to and work as directly mathematical expressions. the complete supersymmetrized 4.1 Defining order in allexpressions component as fields in [ except terms containing the fermion-since thiswe is give not the of full interestimportant in component for this our field work. discussion expansion On of for the the other equation all hand, for scalars and 1 respectively.potential The energy role was discussed ofre-examined in the the [ L superfields and in their component field expansion-working, however, only to quadratic two-component Weyl spinor and with at most two derivativesusing on the its scalar equation fields, of beand motion. eliminated since from the Note the anticommuting super-Lagrangian that superspacecomplex since coordinate scalar scalars A, the Weyl spinor now embed Here where the second versions of JHEP08(2017)014 ¯ θ 2 ¯ θ ) (4.8) (4.9) (4.6) (4.7) φ θθ (4.10)

 ∂χ · ) # 2 † ) = 0. For ∂φ ∂χ Φ F . · ( 2 F ∗ ¯ ) ) ∗ D 4 F ∗ † F 4 F ∂φ φ F ( Φ F ∗ (  = µ ) in terms of the + 2 ¯ 3 4 F D 8 ) 2 ( φ 24 φ ) Φ χ 4.5 4 F ∂ + 4 ∂χ D φ . ( ∂χ )+ − φ ). This dimensionless ( ∗ Φ respectively, where we 4 2 + F F 2  12 φ F ∗ ) D µ 2 ∗ F 4.5 µ F ) 4 M M ∗ ∂ F + ) ∗ ∂φ 2 ) when † 4 F ( ) 4 1 ∂χ F ∂ FF F φ ) ( 4 (  ∗ 1 2 4.2 − 2 and into ( 3 φ ∂φ ) ,µ + 4 ( ∂χ 2 φ K F ( ) χ − 2 2 4 ∂ µ R (Φ+Φ M 4 3 ∂χ 4 − ∂ 12 φ  / ∂ i ) ( " φ ∂χ 3 φ ∗ φ ∂A∂A · 4 4 + 3 4 4 F  + M ∂χ ∂χ φ φ + ( = 1 ( 2 − − ∂φ F  ) ∗ ¯ (  4 θ + ,µ φ ∗ 1 ¯ θ 4 2 3 φ 4 χ 8 M )  3 F ∂φ 1 φ φ θθ φ ∂A 3 i ( 2

+ 2 · 8 )  φ  4 3 ∂φ − # 2 4 ) ( ) stand for φ 2 ) −  − + conformal Galileon Lagrangian ) ∂χ . 4 ∂A 2 ( 4 c – 12 – ∂φ = φ 3 4.5 . ∗ ∂χ ( 3  ¯ ¯ ∂χ ( θ L  ¯ ( 4 θ 1 φ K 4 1 2 +h 2 φ action is given as a specific sum of two superfield 2 in ( − ∂φ M θθ 3 φ ) † ) and the lowest component of the Kahler potential

∂ 2 − 3  = Φ # ∂A∂A F 4 ¯ iχ ∂φ 2 ¯ θ L + 4 )  ( ¯ θ . − ¯ 2 1 φ D + c 4 ) θθ . 2 ∂φ Φ

3 φ = and ( φ 2 − ∂φ ( # D 4 = 1. This will be the case for the remainder of this paper, ( +h † ∂ 2 3 + φ 1 † Φ = 4 Φ √ 4 φ Φ 3 φ D ¯ SUSY 2 M D 2 2 ¯  −  † L = ¯ ) D φ 3 + Φ SUSY 2 ) Φ A  φ ¯ † ¯ ∂χ D L 2 · D  ) Φ 1 3 Φ 1 ∂φ D φ ∂φ ( D ( Φ (Φ+Φ − ). We find that 3  3 1 D φ "  3 1 φ 4 ) 4.4 1 8 2 ) † + + † = = 1 3 1 ¯ L (Φ+Φ (Φ+Φ SUSY 3 " ¯ " L fermion but have included allgive a of supersymmetric the extension scalar of fields the to all orders. These can be combined to respectively. Note that, as discussed above, we have dropped all terms containing the and Lagrangians. These are 4.2 The complete supersymmetrized It follows that thehave, symbols for simplicity, set unless otherwise specified. Finally, wecomplex find scalar it field convenient todefined re-express in ( ( specificity, it is usefulLagrangian to then reintroduce becomes the mass Note that this matches the corresponding expression in ( JHEP08(2017)014 ), 4.2 2
) ∗ (4.12) (4.11) ¯ θ ¯ θ F in ( ∂χ µ ( θθ 4

µ ¯ L ∂ F ∂  .  2 ) c ) ) + ∗ . ∗ A F A ∂χ µ ( + h + ∂ µ − ∗ † ∂ A Φ A F 2 ( . ( 2 1 φ  
∂ ¯ 4 D ∗ · F ¯ θ ∗ ,µν ∗ ¯ Φ) θ )+ χ ∗ FF F
θθ D ) 2 ,µ

∗ ,ν ∂χ χ ) . The result is Φ χ · FF F ∂F = 0. The second term can be φ † − ∂F  2 D · K + )  ∂A Φ ) in terms of the complex scalar ( ∂φ A ∗ F ,µν
· ¯ ( } − D χ ,µν  ∂χ ,µ µ † ¯ ) 2 D = · φ ∂F ∂ ,µ ∗ χ 4.10 = 0, as it should. Furthermore, it is Φ
∂A ) ,µ ∗ χ A χ ¯ + D D, F 3 φ ∂φ F ( 2 { ∗ ∂χ ( − φ · } ) ∂A ,µ + 6( µ 2 † F = has cancelled between the first and second 1 ¯ φ ∂ A D 2 φ + 2 and used integration by parts. Note that ( 2 ∂φ ) 4 χ ,µ φ ) (
∗ ∂ −  + 2 2 φ D, µ · ) when F ∗ F ) + ,µ { 2 ∗ ∂ ∗ 1 ∂A φ 2 – 13 – φ A ( (Φ + Φ 4.2 F ∂χ Φ) 2 1 φ 2 ∂F FF ∂F } ( 1  ) · 2 ∗ φ
+ and D ¯ 2 ∗ ) when D −
F φ in ( + Φ φ + χ ,µν ∂A 2 2  4 4.2  ) D, ) χ ∂F D , 2 ∂A ¯ 2( 2 ( 2 + 2
{ L ) φ ,µν ) )  2 }  ∂φ ,µν ∂χ 2 ) χ 4 in ( ¯ ( ( D  χ 2 ) ∂χ ∂φ ∂φ µ ) ( ∗ 3 3 ( ( ,µν ∂χ µ † + ) ∂ µ ¯ . It is convenient to use the second expression for µ µ A L D, ∂ χ ∗ 2 ∂ 4 ∂ 6 ∂ { ) 2 +( ¯ ,µ ,µν A 2 L 2 + ) respectively. This will play an important role in our discussion ,µ 2 1 ,µ 2 φ ) 1 φ φ ) φ ) ∂φ φ † + − 3 A ( 2 1 2 4.9 ( − ∂φ φ ,µν µ 1 φ ∂φ 1 A φ ( ∂ φ ( ( 2 1  2 µ − 2 ∂ 128(Φ + Φ 2 2 − 1 φ + + = 2 1 1 φ φ 2 1 φ ) and ( = = 64(Φ+Φ 4 ) reduces to − + − 4.8 = = SUSY 3 ¯ 4.10 L equation. Again, it is useful to re-express ( 2nd term 4 , and the lowest component of the Kahler potential SUSY 4 F 1stterm ¯ L , ¯ L SUSY 4 A ¯ L where, in component fields,to we all have dropped orders all inthis terms reduces the containing to scalar the the fields supersymmetrized fermion, first as but term work for terms. For the first term, consider 4.3 We now supersymmetrize obtained using integrating byterms. parts. We This proceed expression by is first simpler, constructing consisting the of supersymmetric only extension three for each of these important to note that theexpressions, quartic ( term ( of the field Note that ( JHEP08(2017)014 4 ), ¯ L . 4.14 (4.14) F F µ ∂

and ∗ ¯ θ ,ν ,ν ¯ θ ) and ( F χ µ φ iφ iφ θθ ,

 ) φ − + 2 † 4.13 F ∂ ) ∗ ,ν ,ν χ χ F ∂χ ¯ θ ) ) (4.13) 2 ¯ θ · 2 χ χ φ θθ ,µν ,µν (Φ + Φ

+ + ∂φ } χ χ
+ . ( 2 2 ¯ c φ φ ν D 3 . φ ¯ ( ( θ 1 1 φ φ 1 F ¯ φ θ . 2 2 ). Finally, consider the third    ∗ ) ,µ 4 D, +h θθ )

+ ,µ ,µ † + + F ) 4.2 }{ φ φ
∂φ † Φ φ ∂φ ¯ θ ,µ ,µ 2 2 2 · ¯ ( ,µ ¯ θ D

 ¯ 1 1 φ φ ∗ 3 ∗ D in ( φ 2 θθ 2 2 1 φ )

∗ 4 ,ν D, ∂χ ∂χ ) 4 Φ) F ¯ ). Note that in both ( (Φ+Φ † ∂F { χ · · L F FF µ ( † ∂χ } D + Φ µ ) + ) + ∂ ( 2  ¯ − φ Φ Φ 4.2 D ¯ ∗ 2 D φ F ) ¯ †  i∂φ i∂φ D ,ν ∂χ ∂χ D F F ∂ ∗ 2 (  D, † Φ · · φ ) in (   } 2 ∂φ F ¯ Φ ) (

D }{ ,µ ¯ 4 + 2 + 2 3 D ( ∂φ 1 ¯ 3 ¯ ¯ ∂φ ∂φ χ D φ 2 2 L ( D } ( ( 1 φ ∂χ ,µν ,µν ) ) µ – 14 – ¯ D, Φ 2 D 2 ∂ ,µ ,µ { D, iχ iχ i ) φ D ∂χ ∂χ φ φ { ) + ,µ + ( † − ( ( D, † 2 3 3 φ Φ − + i i { φ . φ Φ φ φ 2 φ ∂φ − − D + ¯  1 φ  · D Φ)  2 2 3 φ 2 ,µν ,µν 2 2 ∗ † − − ν 4 ) ) ) ) ) D φ φ (Φ+Φ ) †  Φ 2 2 ∗ F − } ) ) 4 Φ ∂F ¯ 2 ∂φ ∂φ D ) ,µ ∂φ ( ¯ ) ,ν ,ν ∂χ D ( ( D ( 2 Φ F ( ( φ φ FF ∂χ ∂χ 1 )  ( ∂φ } ( ( 3 ∂φ 2 2 2 2 ∗ ,ν D D, ( ( 3 3 ¯ 1 φ { 1 1 1 1 φ φ φ φ ,µ ,µ φ D 1 1 Φ µ F 3 4 φ φ 2 4 2 4 ( φ φ ∂ ,µ 1 φ ], there are two inequivalent ways of supersymmetrizing this term. D 2 2 3 3 D, 64(Φ + Φ 4 F φ ) 3 ) 1 1 + − + + + − − { † φ φ 34 ) 3 2 2 † 1 1 = = φ φ ) ∂φ   ( † 1 µ 1 + + + + − ∂ 1 2 1 φ 4 128(Φ+Φ 64(Φ+Φ = 0, this is simply the second term for 3rd term , 64(Φ+Φ = + + SUSY 4 F = 0, this gives the third term for ¯ L = = F χ ). This is given by = SUSY 4 ¯ χ L 4.2 Putting these three terms together, we get a complete supersymmetrization of fermion, but, as in the first term, working toin all ( orders in the scalar fields For the component field expressions have been obtained by dropping all terms containing the When term. As discussed in [ For simplicity, we will focus on the easiest such supersymmetrization. This is given by JHEP08(2017)014 ) ∗ ) ∗ F A (4.15) (4.16) ν A + ∂ ∗ + A
. ( F ) ) A ∗ 2 µ µ ( ) χ χ ∂ F ∂ ∗ µ 2
) + + µ ∂ ) ∗ ,ν φ φ ,µ ∂  φ ( ( FF ∗ A φ ∂χ ( (   F A ,ν φ − ,µν µ
2 ν ,µ ,µ χ ) | χ  ∂ A φ φ ∗ 2 3 ( − ,µ F ) 1 2 2 | ν F ∂ φ 2 2 ,ν χ 1 1 φ φ |
| ν ∂ ∂F 3 φ 2 2 ∂χ · ∂ 2

F F (  φ )+ | ,µ ∗ ∗ µ ∂F ), this can be re-written as ,ν ,ν +
F ∗ χ | )+ )+ + ∂ ∗ ∗ ∂φ F ) 2 
iφ iφ · 2 iχ | ∂A ∗ ) µ 2 2 A F ∗ ∂χ ∂χ ( 1 φ ∗ ) i − + FF · · ∂ φ ∗ F ν F + 4 µ |  2 2 φ A ∂ µ ,ν ,ν A ∂
φ ∂F  ∂χ ∂φ ∂φ µ ∂ χ χ (  ( µ − + ( ) + φ ( ( )+
∂ + ∗ 2 2 2 ∗ ∗ ∗ ∂ ∗ ) ∗ 1  − ∗ +12 ,µ ,µ A ν F A √ 2 F A 2 ( ∂χ 2 φ φ F ,µν ,µν ∗ A ) ν A · | ) ν ∂χ 3 3 µ ν χ χ  FF + · i i ∂ F =  F ∂ φ φ 2 2 ∂ µ ∂χ ∂φ 3 ∂φ F ∗  A F ∂ ν ∂F + 1 ( 1 1 µ ( φ φ ∂ ∗ 4 | ∂φ ( φ A χ − − µ ∂ ∗ | 2 2 F µ ( 2 ∂ F 2 µ +( 2 2 ∂ | µ F AF ∂  µ ∗ ∂ 3 ) ) F F ∂ φ ) + + 1 ∂ ) µ − + ,ν µ )+ | ∂ φ ∗ + F 2  ∂ φ  ∂ ,µ 2 ,µ ,µ
∂χ ∂χ ∂A A ) + µ A ∂χ 2 ∗ + ∂χ |

ν ( ( ∂φ φ ,µ ν A · ∗ ) · 2 · ∂ µ ∗ F 4 φ | ∂ 3 A F ∂ ,µ ,µ ) φ + – 15 – 1 µ ∂  ∗ F ∂χ ∂χ ν µ 2 φ µ 2  φ φ − ∂φ ∂φ F ∗ 2 ∂φ · · ν 1 ) ∂A µ | ∂F ∂ 2 ( ( ∂ ( A 3 3 φ ) 2 ( ∗ ( ∂ ) ) + ∗ 1 1 F ∗ µ ) ∗ ν µ A∂ φ φ 2 ∗ µ A∂
∗ 3 + ∂ ∂ ν ) F F ∂ ∂ i∂φ i∂φ A 1 ) ν ∂ ∂χ ∗ φ + + ,µν +(   2 ∂A ∂ ,µ ν ( ∂A ∂ +



∗ ∗ χ ∂A 1 φ + F 2 ∂ φ ( ∂F ∂A − +2 +2 µ (( ∗ · ( ) 2 · ∂F ( 2 µ 2 µ F
2 2 F µ ∂ · 1 ∗ ) ,µν ,µν ,µν ,µν µ ) ) ) F ,µν ∂ F φ ν ∂ + ∂ ,µ ) ∂φ χ χ ∂ χ 3 + ∂A ∂ 2 F iχ iχ ( 1 − χ + − 2 ) 2 ∂F ∗ ∂χ ∂χ φ − ,µ ,µ F ∂A ∂A µ 3 ∗ ) 3 ( ( ∗ + 2(
+ − φ ( ( ∗ φ + µ χ χ 1 A φ 2 + µ µ ∂χ ∂ − − A  ) 2 ν − ,µ + + − ( ( A∂ ∂ 2 2 2 ∂ ,µν ,µν ∂A 2 ∂ FF φ µ  ) ∗ ) ) AF ν ( − FF ) φ φ ∂F ∂χ µ 2 µ + ∂ µ · µ ∗ ,µν ,µν ,µν ∂ µ φ A ) 2 ∗  ∂φ ∂φ ∂χ ∂ ∂ φ φ φ ∂ µ ) A ,ν ,ν  ( (  ( +( + ) 2 ν A∂ A∂ ,µ ∂ φ φ 4 ,µ ,µ 3 2 µ ∂A 2  ∂A  ν ∂ ) ν ) ,µν ) φ ν ( ∂φ 2 2 2 2 ) φ φ ∂ ( ∗ 2 ( ∂ ∗ 2 2 F φ 1 1 1 1 φ φ φ φ F ∂F µ AF ) 1 ,µ ) µ µ ∂χ µ 2 2 2 ∂φ 4 2 φ 4 2 ∗ F A ∗ A∂ µ ∂ 1 1 1 A∂ ( ∂ φ ( ∂A 2 ∂ ν φ φ φ ∂ ν ν F ∂ 3 2 2 − + + − + A 6 + 2 ∂ − ∂A (( 1 ∂ ν 2 2 ∂ 1 1 1 φ φ φ − − − ( 1 6 µ µ + − A φ φ 4 2 2 ∂    ( (   ∂ 3 ∂  A + + + + + + − + ( + − + + + − − − + = SUSY 4 Expressed in terms of the complex scalar field ¯ L JHEP08(2017)014 4 ). ), W M iχ ) 4.2 ) re- χ + (4.17) (4.18) + ) φ ∗ in ( ( ,ν φ 4.15 2 ( φ F 1 4 supersym- √ µ ¯  L 6 ,µν ,µ = χ F ∂ φ ,µ 2 A − ) and ( and to the superfield to be dimension- χ 1 φ 3 2 F , is given by 3 2 A µ ¯ φ W L 5-space with an 4.10 that contain at least ∂
)+ , + ∗ 5 ,ν ) with 2 ∗
SUSY 1 F ¯ . 2 ), ( ∂χ L ¯ A ( iφ L ) · ∗ ( SUSY ,µ AdS FF − F 4.5 ∂χ L  W ∂φ χ ∗ ,ν ( ∗ (
3 χ i φ = ,µ − φ ), ( 2 ∂A 3 φ  ∂W ) c 2 3 W ,µν i ) , φ + 4.17 +¯ ). How does one supersymmetrize χ and scalar field ∂φ 2 ( − F ∂χ F , and is given by 1 φ ∗ 4.2 2 µ 2 ) F +( SUSY i ∂ ¯ ∂A SUSY 1 + φ ∂W L ,µ in (  ∂χ ¯ i SUSY L ( 2 ,µ φ  ¯ c 1 )
= 2 3 ,µ ¯ Galileon. 1 ) L ¯ =1 φ θ φ F 4 i 5 ∂χ ¯ θ ⊂ L ∂χ 3 | ( µ L · ∂φ 1 + † φ ∂ ( – 16 – 4 F ∗ ∗ 3 φ = Σ are all real, each has dimension 4 but are other- + ,µν W 3 i∂φ 2 F 1
F i χ SUSY F φ  + c 4 + L
1 +2 ,µν φ − 2 ,µν θθ 2 )
2 SUSY χ | χ ,µν ) ) has dimension 0, it follows that the superpotential c L ,µ 2 ,µ 3 ∂φ W F iχ +¯ φ χ M χ ∂χ ( ∗ ( − 4 + = + − F − 3 φ ,µ ∗ 2 ∗ 2 ,µν ,µν ,µν φ ) 4 are given by expressions ( φ φ φ + , ∂A SUSY 1 3 ∂W ∂φ ,µ  φ ,ν ¯ ( , ,µν 1 L φ ,µ φ  2 c φ , φ 2 2 3 2 2 1 +¯ 2 1 1 1 1 φ φ φ φ φ 1 φ 4 2 F (recall, really − = 1 − − is a holomorphic function of the chiral superfield Φ introduced above. It − + +    i F 4 3 4 ∂A ∂W c c c -field terms in the Lagrangian , W 1 c +¯ +¯ +¯ F ) must also have mass dimension 0. = ¯ 3 SUSY i W 1 ¯ M L ¯ L -field. This will be denoted by SUSY F F Recall that we are ignoring, for simplicity, the Having introduced the superpotential term, we can now write the entire L 6 4.5 The In this section, we willone isolate and discuss only those terms in foliation. It is given by where spectively. The constant coefficientswise arbitrary. ¯ as defined above. Note thatless, since and we since are taking (really metric Lagrangian for the worldvolume action of a 3-brane in follows that its supersymmetric Lagrangian, which we denote by In the component field expression on the right-hand side, In the preceding subsections,expressions-ignoring we the have presented fermion-for both theHowever, the supersymmetrization there superfield is of and also componentit? a field pure As potential is well term Lagrangian. known, this is accomplished by adding a superpotential 4.4 JHEP08(2017)014 ) , as ) ∗ 4.10 4 are χ (4.19) (4.20) + ∂F φ ( p > or  , each paired ,µ φ F ∂F ) 2 where 1. Be that as it 1 φ ∂φ p 2 · and 2 ∗ ) 
) by doing an expan- , there are also terms A )+ ∂ 2 M ,ν Lagrangians in ( ∂F ) F ( 3.9 ∗ ∂χ iφ ∂ 2 · M F + F is zero and, hence, one can ∗ ∗ ∂φ ,ν ( SUSY 4 -where we have momentarily and its derivatives. For ex- F F χ ¯ 2 and ν ,µ L W 3 ) 1 φ ∗ F φ ∂ 3 . Assuming, for a moment, that F ∂ 4 M ,µν i F c µ φ χ , and , such Lagrangians would lead to a ∂ 2 ∂F ∗ + F ,ν )+¯ ∗ 1 φ 2 F φ 2 ) . ∂φ ,µ ∂F 1 · + or ∂χ SUSY 3 φ ( ,µ ¯ 2 L 
1 F ,µ ∂F φ ( ∗ 2 φ 4 2 ]. Perhaps more intriguing is the second case ∂χ c 3 F )  F · 1 ∗ µ φ ν – 17 – +¯ ∂ ) in powers of ( ∂ 36 F M + , ∗ ( in our calculations and, furthermore, simplify the i∂φ F ∂ , and its elimination from the Lagrangian, in the
∂F is inserted back into the Lagrangian. For example,  2 F  3.3 F 35 · 5 F
µ 3 ∗ ¯ − ,µν 1 L F ∂ φ 2 χ
,µν ) 4 ∂F terms with one or more derivatives acting on them. This ,µ c ,µ
iχ χ φ ∂χ 2 +¯ F ( ) ]. In the present paper, however, we will deal directly with + ,ν + . Since terms in this expansion with ( − χ 12 ∂F 2 ∂χ 2 ,µν · ,µν ) R ) − φ ∗ ∗ / φ ,ν +( -field is dynamical and no longer a true auxiliary field. There ∂φ ,µ 2 φ ,ν ( F ) φ FF φ = 1 ,µ ( F ∂F 2 2 . This would have three inequivalent solutions and lead, for example, χ 2 ∂φ ∗ φ 1 1 1 φ φ ( . By definition, this Lagrangian contains pure scalar terms with four φ -field, we will henceforth assume F M F  2 4 2 3 F -field is no longer a simple auxiliary field. There are two reasons for this. 2 2 3 ¯ i φ − L 1 6 1 + − . Secondly, there are terms with both a single derivative φ 2 φ φ  F 2 4 4 4 4 4 ) c c c c c F +¯ +¯ +¯ +¯ +¯ ∗ F ) respectively contain term involving the field = 0 as the solution [ 4.15 F First, recall that we have obtained the conformal Galileons in ( Clearly, the Additionally, note thatand the ( supersymmetric ample, consider sion of the DBIrestored conformal the Galileons mass ( a total divergence, itmay, is since we not would strictlydiscussion like necessary of to to the ignore assume that ( been discussed in thetake trivial case where thethe superpotential issue of derivativesnon-trivial on case where the the field superpotential does not vanish. We do this as follows. when the Lagrangian contains would imply that the is nothing wrong withirreducible supermultiplet this containing from two dynamical thewith complex point scalars a Weyl of spinor, view does of exist. supersymmetry The dynamics representations-an of such theories, to our knowledge, has only well as terms containingthere two are derivatives no such terms as cubic with equation derivatives for acting on to three different expressionsthese when Lagrangians would havebeen different explored potential in energy several contexts functions. in [ This has previously The first is that, inof addition order to terms ( proportional to JHEP08(2017)014 2 ) ∂ M (4.25) (4.22) (4.23) (4.28) (4.24) (4.26) (4.27) (4.21) ) becomes ), it follows iχ 4.19 + φ . ( 2 1 (0) √ F ∗ = as an expansion in ( (0) A F = 0, the potential becomes F χ 4 1 φ 2 . . c , as required. It is also of interest 2 4

) + ¯ . . . . ∗ ∗ iχ ∗ 4 4 0 3 2 ∂A ) SUSY 1 ∂W

φ 2 ¯ − F A

, 1 L ∂A 4 ∗ χ ∂W ∗ 1 φ 4 12 φ 4 ( ∂A φ ∂W + 2 1 2 2 1 φ 2 1

¯ ¯ ∂A  c c ¯ ¯ c c ∂W 2 ¯ ¯ 1 2 c c 4 ¯ ¯ c c 1 2 φ − φ

c ( r r 2 1 2 − – 18 – . However, the same Lagrangian contains terms, 1 2 1 ¯ ¯ c c 2 1 is a constant. = 4 + ¯ F ¯ c φ M =

− ) yields =  = ). This is easily satisfied if one chooses (0) (0) = so that, when one sets 2 ) = (0) F V F W V 4.2 F 4.22 W ∂φ then implies that (0) SUSY(0) F ), it follows that one must also choose ( ∂A 3 ∂W ∗ F L in ( 1 φ 1 W 2 c (0) ) that 4.20 F = ¯ SUSY 1 , which have two powers of derivatives acting on scalars multi- 4.25 ¯ L F ∗ = 0 the field Lagrangians be of the same order of magnitude. Since the terms F χ 2 SUSY(0) F ) L ∂φ SUSY i ( ¯ L 4 3 φ . It is natural, and greatly simplifies our analysis, if we demand that all terms = 0 this reproduces the potential in 2 F χ ∗ is a holomorphic function of the complex scalar field F , as required by 4 1 W φ ¯ c 4 With this in mind, it is reasonable to solve the equation for = to note that for this choice of It follows that when It follows from this and ( Hence, for One now must choose theV form of Since that the above expression is simply minus the potential energy Putting this expression back into ( The equation of motion for which we assume henceforth. as well. To zeroth order in this expansion, the relevant part of Lagrangian ( for example plied by in each of the involving derivatives satisfy ( powers of derivatives, such as JHEP08(2017)014 ) = 0, 4.28 (4.33) (4.30) (4.32) (4.31) (4.29) χ = 0 to the χ space. . Therefore, if the χ - 1 φ . Furthermore, at , the potential is everywhere 1 . c | is non-ghost like, which we will 1 . → ∞ ¯ c χ 0 χ . Hence, the solution >  | φ | 2 . 1 ¯ c 6 . ¯ | c | φ 1 2 and 2 ¯ ¯ c c | , = 0 = 0 are indeed possible. Note from ( ⇒ = 0 s plotted over a region of χ i 2 χ space is shown in figure < µ =0 µ V ∂ ), then χ 1 → ∞ ∂ – 19 – = | ) then becomes everywhere negative-which, for χ ¯ c 1 = - 2 = χ V φ = (0) 2 4.21 χ  4.27 ∂χ ∂ χ F , it is minimized as 0 for any value of φ (0) = F 2 χ < (0) 1 m 6 c ∗ φ 2¯ F to satisfy ( − ). For positive values of coefficient ¯ in a range of = (0) F V 4.27 . The potential energy =0 χ | 2 V 2 ∂χ ∂ = Figure 1 2 χ the potential grows larger as m φ Henceforth, in this paper, we would like to restrict ourselves to the subset of solution Note that in order for The potential energy coefficients are chosen so thatimpose the kinetic below, energy then term for solutionsthat, where in this case, example, is required invalue bouncing of universe cosmological scenarios. It follows that for any equations of motion would be fine-tunedcan and easily highly be unstable. corrected This by unsatisfactory simply situation imposing the condition that which we do henceforth. Potential ( First, of course, onethe must form show of that potential such ( solutionspositive are and, possible. for any We fixedthe begin value mass by of considering space where JHEP08(2017)014 ) = 0 field 4.35 ), we χ (4.35) (4.36) (4.37) (4.34) φ µ ∂ terms in 4.19 should be 2 = ) i c χ ∂χ ( 4 . Depending on 1 i φ , is minimized at c (0) φ kinetic energy term F and would vanish in ∗
and φ 2 χ (0)
2 , must be appropriately ) F . (0) (0) (0) ∂φ  0 F F ( 4 F = 0. Furthermore, we have ∗ ∗ ) 4 1 < φ (0) χ (0) 2 ∂φ µ
F ( F and ∂ ) back into Lagrangian ( 4 and, hence, its variation does not i (0) 4 .  3 ) that the coefficients ¯ c φ c 2 = 0 F χ ) 6¯ . ∗ . − 4.32 χ ) to the > are quadratic in − 2 4.35 ∂χ
) (0) (  ,µ F 4.5 (0) (0) (0) 4 equation of motion for which χ (0) 3 φ F 4 F F χ ∗ SUSY F c 2 F ∗ ∗ + -field ( in ( ∗ 6¯ L (0) (0) (0) + ,µ (0) only where a) it would could contribute to the (0) − – 20 – φ F 2 in F F F ( ) 3 F 3 3 χ c SUSY 2 c c (0) χ   ¯ 3¯ 3¯ ∂φ L 2 F ( ,µ ) ∗ − − + 3¯ φ 4 2 kinetic energy to be positive. 2 2 2 (0) 3 ∂φ φ ¯ c 1 ¯ c ¯ c ( 2 φ φ F 4 ] that Galileon Lagrangians can, for appropriate choices of 3 2 2 − + 1 ) ) c φ 4   4 1 34 c φ , 3 4 ∂φ ∂χ ¯ c + 3¯ ( ( c c 4 7 4 4 2 − ¯ + ¯ + ¯ + 3¯ c 1 1 φ φ ) ensuring that the potential energy, for fixed , which generically depend on ¯ 2 2 = χ − − 4.30 kinetic energy be ghost free. It follows from the second term in ( equations of motion and, hence, can be dropped from the Lagrangian. and χ SUSY F simplifies to equation of motion when one sets χ φ L kinetic energy term or b) is linear in ) yields χ χ ), and putting the constant and SUSY F 4.34 L φ 4.30 = 0. All other terms containing in ( , have kept the terms containing (0) (0) The constraint ( F F = 0 is not the only constraint that one might put on the coefficients ¯ SUSY F µ µ chosen so that However, it is well-known [ coefficients, violate the NECcan without choose the developing coefficient a of ghost the condensate. In such cases, one On the other hand, thedepends sign on and the magnitude type ofto of the develop physics coefficient one a of is “ghost the be interested condensate”-one violated-then in. way it For in follows example, which from if the the one null wants first energy the term condition in (NEC) ( can As discussed above, to obtainrequires solutions that of the the that one must therefore impose chosen. Adding the first twoL terms of both the χ the physical problem beingterms analyzed, for the both coefficients of the two-derivative kinetic energy ∂ lowest order vanish in the simplified the remaining∂ expressions using integrating by parts, which we can do since In deriving this expression, we have dropped all terms containing at least one power of find that Taking ( JHEP08(2017)014 ) = 0, SUSY F 4.30 -term (4.38) (4.40) (4.41) (4.42) (4.39) . Dif- ) and, . This L  (0) F χ F µ . expansion 4.32 µ ∂  2  φ∂ ) ) µ . χ ∂ ∂ ((0)+(1)) term, since we µ 4  F 1 ((0)+(1)) ((0)+(1)+(2)) ) ((0)+(1)+(2)) φ φ∂ (0) χ F 3 F µ F ((0)+(1)) µ ∗ ∂ F − 3 we use integration by F φ∂ ). As discussed above, χ µ , in the ( ∗ µ − ) back into the ((0)+(1)) ∂  ∂ ∗ F (2) χ 3 3 ((0)+(1)) µ 4.23 1 F ∗ (which are of second order φ F  ∂ 4 ), the relevant part of 4.41 , − 1 F φ  φ ( (1) χ i 2  i (2) replaced by c ) respectively, the relevant part 2 ((0)+(1)) F  4.22 ) + F + φ , F + ¯ F 2 ). ( (1) 2 + ) ∗ ∂χ i ) 4.39 − is given by the constant ( (1) ( F + ∂χ (1) 4 F 4.34 ∂χ ( is given by ( ( 2 3 φ (0) F to first order in the derivative expansion. ) + 4 2 3 2 F ) with 3 φ ((0)+(1)) + (0) F ∗ ) and ( 2 ∂χ + (0) + -field, up to and including terms arising in ( ((0)+(1)) F F 2 (0) F 2 F 4.19 ) + 3 2 ∗ F ) ∗ F 4.22 µ 2 = ) – 21 – ∂φ + ∂ ∂φ 1. One can insert ( ( = ∂A ∂W ( 2 ) ∂φ 4 1 3 2 ( c 3 φ  4 2 ∂φ + + ¯ ( 2 3 φ , where in terms involving ) ((0)+(1)) ∗ 2 ((0)+(1)) + φ 3 2 ∂ ∗ F F  φ + F + φ   φ φ 3 − ((0)+(1)+(2)) ). Recalling that we are always choosing constraints ( 1 = 0, it follows that ,µ  φ ((0)+(1))   F ), as we did in the zeroth order case. However, as far as the 3 χ φ χ F 1 2 3 − φ 3 µ ¯ ¯ c c i − 4.39  φ ∂ -term Lagrangian given in ( − 4.19  3 3 ∂A − ∂W F c c  | = are determined from ( 1 1 1 2 ¯ c ¯ c c + ¯ + ¯ in ( |  χ (0) term in this expansion is small compared to the (1) as follows. First recall that = ¯ F ) into ( s (0) 4 F 3 2 φ (1) F (2) ¯ ¯ = 0. Dropping terms proportional to c c 3 2 SUSY F F i 4.38 ¯ ¯ F c c 2 = L and (0) is computed from the zeroth order Lagrangian ( − − is given by the entire expression ( F ) so that expansion) and integrating the last term by parts using the fact that (0) = = µ (0) ∂ F F F 4.36 (1) F SUSY((0)+(1)) . Denoting SUSY F ((0)+(1)) F L It is clear, however, that one can consistently do a higher order expansion to completely With this in mind, we now calculate the field L F . Here, we demonstrate this by computing the next order, F 5 ¯ where of ferentiating this with respectparts to to remove a derivative,is one solved arrives for at the equation of motion for determine the perturbative solutionL for the of are working in theLagrangian limit where ( analysis in this papermore is yet concerned, higher there derivativewith is terms. the nothing zeroth to Hence, order be we gained will from not doing do this-simply that here-contenting ourselves It follows that Clearly, the Now insert ( and ( hence, in the we find that Denoting where then becomes JHEP08(2017)014 4 c and = 1 b , 7 (4.43) N and ¯ a ,ν ]. 3 φ c 40 ) into the ,µν actually, in . We will not χ ν 4.40 , ,µ (2) (Φ) of multiple µ 4. Finally, recall F F φχ 2 = 0. Second, recall − ) and ( . Putting everything 4 = 0. Hence, it follows , n >
∂ 2 (0) n ) χ ∂ 4.32 F µ = 0, it follows that these µ ∂ ∂χ ∂ alphabet–e.g. ( ), ( χ = µ − (0) ∂ χ 2 4.23 ) field. Using this, it follows that F Greek =  F ∂φ φ χ ( ) in flat superspace. By definition, ¯  = 1 supergravity θ µ
= 1 supergravity. This is accomplished ] in the following way. Tangent space bosonic ). The expression for 2 ∂ times a total derivative. When inserted ) ) so that N ,µ 40 4 N ). Insert ( x, θ, φ is the flat superspace differential operator. ∂χ 4.43 equation involving φφ 4.36 ¯ D F 4.40 − . + ( 2 – 22 – (1) ) (0) ,µν ) and, therefore, that F F χ ∂φ µ have the appropriate dimension whereas terms like ) and ( 2 ( | ,µν 4.32 χ∂ (0) φ (0) χ µ 4.30 2 F F + | φ 2 φ∂ |

3 2 Φ = 0, where 2 ) c ,µ ˙ (0) + 3 ¯ c α 2¯ χ is presented in ( ¯ F ) on the magnitude of the ]. Throughout this section we use results from, and follow the D ∂φ | ,µν − + 39 φ . φ (1) 2 – ,µ ) 4.21  F + 3( (2) φ ,µν (1) indices are taken from the middle of the φ 37 F φ , 2  F  ( φ = 1 superspace and, hence, to ,µ 35 φ 1  φ = 1 supersymmetry. We now further generalize these results by extending − ) for 2 4 2 4 2 N do not, even though this last term is proportional to ¯ ¯ ¯ ¯ c c c c bosonic φ . Third, drop all terms in the (0) N F F − + − (1) 4.43 . However, inserting it into these terms, integrating by parts and recalling that 2 F = is given by the constant ( c 2 | (2) (0) (0) F and ¯ F F | . Spacetime 1 However, our index labelling convention differs from [ We begin with a purely chiral superfield Φ( c 7 α φ , ˙ supersymmetry by integrating this chiral superfield, or any chiral function and spinor indices are chosenα from the start ofdeal the with Latin spacetime and spinor Greek indices alphabet in respectively this —- section. e.g. first presented in [ notation of, the book “Supersymmetry and Supergravity” by Baggerthis and satisfies Wess the [ constraint Note that one can generically construct a Lagrangian that is invariant under global 5 Extension of conformal GalileonsIn to the previous section,scalar field we to extended thethem to flat curved space conformalby Galileons using, of and then a greatly single expanding real upon, results on higher-derivative supergravitation we will always solvecontributions the exactly equations vanish. of Hence, motionexpression we ( so have dropped that this total derivative term from the There is one importantaddition caveat to in the arriving above, contains atback a into ( term the which Lagrangian, is thisand, would hence, be can higher be order ignored. into However, all it ¯ would terms have proportional to be to included ¯ in the terms proportional  together, we find that that that the expression for equation for that we have imposedterms ( of the form, say, we will always choose constraints ( JHEP08(2017)014 is a (5.7) (5.4) (5.5) (5.6) (5.1) (5.2) (5.3) E by the 2 ¯ D ), one can † ], and . A function Φ a , 40 µ e (Φ ., O c . ) by using the chiral † + h , which will not appear Φ .. ,  R c , which contains the Ricci 2 P . (Φ R M 3 O / .. ) c ., † ) + h . c † Φ . , .. Φ c , (Φ . ) + h K (Φ † − )) + h e Φ O ¯ θ ) , ) R R (Φ (Φ) + h 8 -over half of superspace. That is, 8 ) has as its supergravity analogue x, θ, F O ( − − W 2 i E 5.2 2 2 ¯ ., D ¯ – 23 – (Φ D c θ D Θ2 . ( ( 2 2 8 3 d E d θF 2 − d Z Z , Θ2 2  1 4 = 1 supergravity. To do this, one replaces the measure II = 1 supergravity extension of the first three conformal , d = Z E (Φ) + h 3 − ¯ N L L = are the covariant theta variables defined in [ Z N W Θ2 = 3 4 2 E 1 4 α L d L radius of curvature, also denoted by − − ), and equation ( Θ2 I 5 , Z 2 R = 3 d 2 8 P ¯ L L . However, it turn out that this, by itself, is insufficient. One must 1 8 M Z − AdS = 1 supersymmetric invariant Lagrangian by first applying the chiral 2 2 ¯ = = = D ¯ N D 1 2 3 ( to the function-hence turning it into a chiral superfield-before integrating ¯ ¯ ¯ 1 4 L L L , where the Θ 2 − E ¯ D 1 4 Θ2 2 − d in the highest component of its Θ expansion. Note that this quantity should not R with We now use similar methods to construct a Lagrangian that is invariant under local = 1 supersymmetry; that is, θ 2 5.1 The N=1 supergravityWe Galileons are now readyGalileons. to They give are the explicitly in our supergravitythen given expressions. by The chiral projector in curved superspace is projector. To do this,covariant one operator must first replacealso the introduce flat-space a differential new operator termscalar proportional to the chiral superfield be confused with the Once again, we can integrate over a more general function N d chiral density whose lowest componentof is purely the chiral superfields determinant of then yields the the vierbein invariant Lagrangian continue to get an projector over half of superspace, where we have madeFor the a Lagrangian more manifestly general real function by of adding both the chiral hermitian and conjugate. anti-chiral superfields, chiral superfields-such as the superpotential JHEP08(2017)014 F -not (5.8) (5.9) scale a , (5.11) (5.12) (5.10) µ ) 5 e ∗  ∗ , one can ) (5.13) P AdS FF ∗ FF M
− .. 2 ∂A c ) . · ∗ ∂A ∗ ., · c . + h ∂A ∂F  ]. Finally, recall that ( † ∂A + ( + h ( F 43 Φ 2  has mass dimension 1,  2 ) P † D ∗ K ∗  † M A Φ 1 3 2 Φ ∂A ) + 4 M ( ∗ − D D FM e F Φ Φ ∂A 2 2 3 ∂A ∂A + 2 3 · ∂ D D also have their mass dimensions ∗ 2 − P ] - see also [ Φ Φ − ∂F A M ∗ D D M 12 ( FF 3 4 µ , ∗ ) A   ∂ − − ∗ µ F ) ) µ + 3   2 ∂ P † † and K ib µ M 2 ∂A ∗ P  , K 4 3 µ  · 1 3 µ ib + 4 M ∗ ) b ∗ b 4 3 − 1 3 ∗ ∗ ∂A µ − F b − has dimension 3. In addition to the A ∂A ∂A ∗ ∂e + (Φ + Φ (Φ + Φ ∗ A µ 1 3 – 24 – = 1 gravity supermultiplet, that is, the vielbein ∗   ∗ ν ∂e ∂ FM ) ) W ∂ ∗ 2 A + A bulk space. Specifically, ∂A which will always appear raised to some exponent. ∂W ∗ µ − N R R µ + 8 ( ν ∗ F 5 8 8 ∂ e ∂ F ∗ 2 D A + ) µ 2 µ − − ∗ − µν M D F ∂ 2 2 − g AdS ( MM 2 P 2 + ∂A µν µ K D D 1 3 M ( b ( ( g  M ∂A , 2 2 ∂W ∗ 3 1 P 2 2 ∗ E E ) K ) 2 ∂A M −  ∗ F . These are of dimension 0, 3/2, 1 and 1 respectively. At the ) + of the 1 3 2 2 † R − Θ2 Θ2 ∂A ) ∂e ∂A iF F 2 2 4 − 1 2 M F R ∂A M d d 2( 1 3 2 A / ∗ − and is dimensionless. ∂e ∂A  M µ ( 4 Z Z + + ( −  ∂ and the auxiliary fields W  ) νa = 1 4 1 8 1 ∗ (Φ + Φ 2 3 P   e ). All fields in the α µ ) K − MF a µ 2 3 A M µ − − ∗ ψ N ∗ 6 e ) have previously been evaluated in [ b 1 3 M   A + 2 P = = − 1 2 = P πG e I ) = + A II , † 2 5.10 ( P , iM M 3 (8 WM µν 3 A Φ ¯ / L g ( ¯ − M , − + + L = = = on the left-hand side of these expressions is the determinant of the vierbein (Φ = 1 3 2 1 e ) and ( K 2 ¯ ¯ ¯ P L L L e e e 1 1 1 5.9 M In component fields, we find , we have introduced the gravitational reduced Planck mass in four dimensions, denoted , the gravitino a µ where to be confused with Euler’sAs in constant the preceding sections, we have continued to omit any terms containing the fermion in ( the metric by e specified with respect to end of this section, wereturn will demonstrate to how, the for conventions momenta of much smaller the than previous section. We also note that the expression given Note that we have now restoredto the canonical mass dimensions scale to thehas chiral dimension superfields 2, with and respect theM superpotential where JHEP08(2017)014 ) F . , a µ 5.12 b  (5.16) (5.17) (5.19) (5.15) (5.18) (5.14) ∗ ) trans- ), ( FF ∗ 5.13 5.11 A µ ∂ 2 P K M . 1 3 A.  e ν ∂ − hc. 2 P , 2 K ) M + µ ∗ 1 6
h e µ A µ ∂A b λ ( 3 ∂ ∂ c 2 P A K , term in Lagrangian ( µν µ M + ¯ g 3 ∂ as the sum of the individual terms 1 2 µ ¯ 4 , A L j − 8 + ν 3 . µ e 2 P − c ∂ b K A . These are 2 P M µ ∗ 2 ) P K µ + ¯ 1 3 M µ + 2 b D 2 e 1 6 ∂ M ¯ 2 ν A e µν L , but continue to write the rescaled metric λ ( µν c 0 ν 2 g δ AF F a g ¯ c ∂ L µ µ + ¯ µ e ∂ + ¯ – 25 – µ → D 2 P b 1 ) are given in the appendices. Note that each of ∂A ∂K K → , µ ¯ M µν L µν  b a 3 1 g 1 µ 2  P c 2 e P 2 e 5.10 P ∗ K 1 M M = ¯ + , g A M 2 1 3 µ 6 2 ¯ c ¯ 2 P L ) ∂ − K 1 3 3 only. M e ∗ ) to ( + , 2 3 2 ∂A = respectively. The Weyl rescaling restores the canonical Ricci , ( ∂K λ µν → 5.5 ∂A b ∗ 0 e e Γ a ), we perform the following Weyl rescaling µ ¯ A = 1 − L ), we find that the A e ν i µ e 1 → → ∂ . Details on how one arrives at the expressions in ( 5.6 , A ∂ i µ e , but also introduces a total derivative term which depends on the µ 4 R ¯ L 5.16 λ µν ∂ D and R  Γ , a complex scalar. These supergravity auxiliary fields arise in the Θ 2 P 3 µν ) µν ∂A g ∂K M and ∗ g eM  A E 1 2 2 i P + ’s are now dimensionless constants. In order to restore the non-linear sigma − i i M c A 3 ) from equations ( 3 ( − 5.13 = = . However, we carefully analyze all terms containing the auxiliary fields; that is, This time limited to µ µ We now integrate out the auxiliary fields of supergravity. We begin by first isolating We can once again write out the total Lagrangian 8 j α of the chiral superfield. In addition, we also omit all interactions involving the gravitino h µ where scalar term rescaling factor. However, sincewe this will total drop divergence it is henceforth. inside of an integral inthe the terms action, in the rescaled Lagrangian containing For example, using ( forms as We will denote theand rescaled vierbein Lagrangian as as model kinetic term in ( This induces the following transformations: the above Lagrangians has mass dimension 4. given above, where the ¯ ψ which arises from thefour-vector, chiral and superfield Φexpansions and of two newand auxiliary ( fields. These are ψ JHEP08(2017)014 (5.27) (5.22) (5.23) (5.24) (5.26) (5.25) (5.20) (5.21) , ∗ (collected  2 ∗ 2 ¯ FF ) L . This allows . ∗ ∗ . . Solving the 2 F 3 )   MM ∂K ¯ ∗ ∂A ∗ L FF ∂A ∂K F 2 P ( . K 2 2 M P ) FF 1 ∂A ∗ ∂K 1 3  ( FM M µ ∂A e 2 ∂K 2 F P ∗ ) h 2 ( K M P + ∗ 2 M P µ ∗ 2 K P ∗ 1 3 ∂K M h M F ∂A K F M e ( 2 2 3 4 2 P 2 M P ¯ 2 3 c 2 2 e P P 1 K F e K M 1 3 M M 1 = 2 3 2 M P 2 3 2 3 2 2 K ∗ + 2 − e 3 . M ¯ ¯ c c e ) arising from + 2 2 ∗ 2 3 ∗ µ N c ∗  e + h + 2 µ + 2 F , ∂K + ¯ by first re-writing the Lagrangian in µ ∂A ∗ FF 2 h X. F ) ∗ h 2 3 2 ∗ + 2  2 µ N ) P ¯ ¯ c c ∗ 0 F ∗ K ∗ j FF only. We find that M ∗ FM ¯ M ∂A 2 P L F 2 2 2 ( 1 3 P 1 ∗ M ∗ ) ) 2 ), we find M ∂A 2 ∗ P M − + ∗ ( M K e 2 X, P W M F and K 2 3 ∗ ∂A N ∗ + 2 2 ∗ 1 3 M P 0 ¯ ¯ F,F c c ( ∂A ) 5.18 3 ¯ e 1 3 ( µ N L M – 26 – ∂K M j ∂A e 3 2 ∂K 2 + ∂A P
2 K 2 ∂A +   = ∂A 2 1 M P ∂K ¯ µ c 2 ( ¯ c P K ∗ , whose Lagrangian, after Weyl rescaling, is found j 1 3 3 3 2 M 1 M 1 3 2 2 P P ) µ µ M ∗ e K K 1 3 M 0 = ∗ b j M M − M ¯ 2 3 e − µ L 1 3 1 3 A W  NX + b 3 2 N 1 = e e + 2 2 P P + + 2 3 2 3 + + µ ∂A ∂K b ∗  is straightforward to solve. The solution is ∗ M M  A and its complex conjugate WF 2 2 ( 2 3 P ¯ ¯ c c 3 ∗ N ) 1 ¯ 1 3 3 4 c ∗ MF NN M F ∂A gives us WM ∂K 2 P A − − ∗ 2 3 P − K 2 1 µ ) K P M 2 M P b ∗ ∂K 1 + ∂A = = K 1 3 1 3 W M M A b e 2 P e A 0 3 2 1 2 1 P 2  ( 2 ¯ K P e M ¯ c + L M 3 1 2 1 3 P e 1 ¯ c 2 3 ¯ c M K A + M e 2 ( − − − ¯ 1 c 2 3 3 ¯ c M c 1 3 e = 1 − contains terms that depend on − c = + ¯ M F = 0 = = ¯ N ¯ N L 0 F X ¯ N e 1 L 0 N ¯ 0 ) from the higher derivative terms (denoted by L ¯ µ L e 1 j e 1 The equation of motion for where As in ordinary supergravity, weterms eliminate of one to express where We now turn toto the be auxiliary field in equation of motion for Inserting this result back into Lagrangian ( Here, we have taken care to distinguish the “usual” two-derivative terms from JHEP08(2017)014  3 ) , F (5.28) (5.30) (5.29) ∗  2  F ) ( i , ∗ F 2 P  ∗ K  ∗ M 3 ∗  F e ) ∗ ( FW ∗ 2 F  FF ) +4 ∗ 2 ∗ ∗ FF 2 WW ) 2 FF ) F F ) ∗  ) 2 P ( ∂K ∗ 2 K ∂A F ∗ M ) 2 P ∗ +( ∗ K e ∂A + M F ( ∂A ∂A 2 ( P e W · ∗ 3 ∂A i )( ∗ 2 ∂A M ∂K 2 ) ∗   F ) ∗ 2 ∂K 2  +( 2 ∂A  1 ∂A + 4   2 ∗ ) ¯ c ∗ , we arrive at the complete ( ¯ c 2 ∗ 0 ∗ 2 ∂A ) 2 F 2 P P · ) ¯ ∂A K K h ∂A + L ∂A M M W ( · F ∂A ∂A ∗ 2 2 2 3 ∂A  ∂K P ∗ 1 3 ( FW ∗ K ) ( ∗ ∂A +( e M e 2 F 2 P (  2 ) F F e 2 K ∂A ) ∗ + ( M ∗ ) ∗ 2 P ∂A ∂A +2 ∂K  ∗ K 3 1 ( M F 2 +2 ∂A F W ∂K e ∂A + ) ∂A ∂K 1 3 i ( ∂A ∗ ∗ ∗ ∗ ∗ h e  − FF + ∂A F + ( ∂K ) F ) ∗ 2 2 P + ∗ ∂K +4 ∗ W + ( ∂A 4 K ∂A  ) ∂A ∗ M 2 ∗ · 2 W F 2 ) P − ) 2 3 ∂A ∗ ∂A F ∗ 2 FF 1 · ∗ 2 ∂A · e ) + ( 2 2 ∂A FF ( M ) ∗ ∗ F ∗ ) 4 3 ) FW ∂A ) 2 2 FF ( 2 ∂A ∗ + ) + ( ) 2 ∗ ∂A ∂A ∂A ∂F F ∂A ∂K ∂A ∂K – 27 – ∂A ) ∗ ∂A ( ( ∗  F ( ∗ ( ∂A )(  ∂A K  ∗ ∗ 2 · F 2 P ∗ +( ( WF 2 ) K ∂K 2 ∂A 2 ∂A P  F P 2 P M ∂ ( ∂A  K ∂K 2 ∂A 1 ∂K ∂A ∂A 1 2 M ∂A∂A e ( P  µ ∂A + ) M M ( ∂A K ∂A ∂K · 1 3 ∗ ∗ FW 2 3 M h 2 2 3 2 2 P P P ( P ) 2 e P K K K K 2 2 µ P 2 3 K M M M M 2 ), we find ) P + 2 + + 1 M h ∂K e ∂A K ∂A ∗ ∂A 1 3 1 3 1 3 2 3 ∗ M M ( 1 3 ∂A with the remaining terms in ( A 4 4 3 P e 4 e e e ( A 1 3 )+4 2 e ν 1 2 + ) ∂A 2 3 ν F ∂ c 5.24 e ∗ M − + − ∗ ( + ∂ µ ∗ ∂A N 4 9 ∂K h ∂A 2 2 +8  3 2 +¯ ∂A 0 µ ∂K F · ∂A ¯ ¯ 2 c c D ∗ 4 3 ¯ ( 2 P ∗  ) L D ∗ )  K  NN ∗ 6 + M µν ∗ 2 P ∂F ) 2 2 ∂A P µν K 2 P g 2 3 µ P ( WW M · K ∂A ∗ A K g ∂W 2 ∗ M e h ∂A 1 M 1 1 3 2 P 2 and ( M 1 3 µ A 2 3  K F 1 3 e + + M 1 j 9 2 ∂A  e 2 ) 4 9 2 P  − ) N ∗ 2 e + F 2 3 2 3 P K 0 A ) e ∗ M ) 2 2 2  2 ( 1 P ¯  ∗ 2 L K 1 3 A ∂A ¯ P M c 3 ¯ c 3 2 2 ( P ∂A e , M ∂A 2 ∂A ∂W A 3 2 ¯ P c 2  3 ∂ 1 2( ( M 2 ¯ ¯ c c 1 2 ¯ c 1 3 3 b M 2 ∂A∂A 2  ¯ + P c 0 ) ¯ ¯ c c M   +4 +( ¯  c 3 6 2 ∗ ¯ 2 L 4 3 3 P ) A ¯ c M 1 3 K 2 ) ) ) P A ( + + ∗ M 1 K ∗ ∗ ∗ M 2 3 A + 1 = = A A A 2 3 = e 1 6 1 R− + A e 1 µ + + + ( 2 P N c 2 A P 0 h 2 3 A A A ( ¯ ¯ µ ( ( c ( M 2 = ¯ L M 2 2 ¯ c 1 3 0 j 3 3 3 2 e 1 ¯ c ¯ ¯ c c 3 4 ¯ ¯ ¯ c c c c L e 1 +¯ − + + +¯ − − where Combining Lagrangian. It is given by Substituting this result into ( JHEP08(2017)014
1 , , ∗ ∗ 2   ∗ A (5.35) (5.36) (5.34) (5.31) (5.32) (5.33) M  M , ∂A ,F 2 ) yields, ∂F ∂A  2  ∗ 2  ∂ F, ) · .  F ∗ and restore ∗ 4  )  5.29 M ) ∂A ), let us take ∗ ∗ F + ∗
W  FF ∗ M 3 ∂A 1. We find that 2 ( ∂A ∂A FF ∂A 2 5.29 · 3 ∗  (
) ) ∗ ∂A → 2 4 2 ∗ ∗ · ∗ ) ) ) e in ( ∗ ∗ A → M ∂F 1 , 0 ∂A ( ∂A F ∂F A + ∂A FF ( ¯ 4 L 2 ∂A µν F + 2 + W P A ) ∗ η K )+ 2 ( , M A ) ∗ +( 3 have dimension 4. Contin- ( 2 3 ∗ F c 2 ∂A +16( → A )+4 FF 2 e ) ( 3 2 ¯ c ∗ 2 ¯ c ) = ¯ + ∂A + ) . , µν ∗ − ∂A ∗ ∗ 2 ∂A 1 A g · K 2 ) ( ( c A 2 +( ) → M = ∗ ∂A ∗ 2 ∂  ∂A ∗  ( ) ∂F 4 ( + . For example, F ∂A∂A 2 2 A ( and F ) 4 ∗ ) 2 2 , FF ∗ P ∗ ∂A K ( c M + ∂A · M A F ∗ F ( ) and expressed in the canonical mass M ∂A ∂A +¯ 1 3 FF A / ( ( ∗ ) e i ( 3 4 iχ 3 ∗ ∂A +4 ∂A ∗ ¯ ) is exact, and has not used the → ∞ c ) ) 2 · 2 3 ∗ ∗ 16 A + ∂A 2 P M → M ∂A · A A → ∂A − M · 2  2 6 φ ∂A · 5.29 ∗ i 2 M 4 ( 3 A + + ) = ) 2 = 1, the flat superspace limit of ( c ) P ) 2 ∂A ∗ K ∗ – 28 – ∗ 1 ∗ ∂A A A M ∂A ∗ √ ( ( FF A 2 3 A 4 32 2 M has dimension 4. Similarly, one finds ∂A e − K + F ) + + = M A, A 2 2 1 − 2 to set +8( ( M = c 4 ∂ A A 2 4 − has dimension 1, and ¯ A ) ( ( ∂A )
∂A∂A +2( K 2 4 3 ∗ ∗ ∂A M ∗ 2 ∂W ) )( F ¯ c ∗ 1 ∂A = A M M ( c ∂A ∂A − 2 ∂A  ( · 4 4 · ) K 2 2 2 = ¯ 3 ∗ FF  ∗ ∗ ) ) ¯ ¯ c c ( ∗ M M 2 F ) F ∂A ∂A ∂A ∂A ∗ ( ∗ → M F ∂F 2( 2( ∂A 2 ∗ P → − 16(   ( + K ∂A M A ∗  ∂W 3 4 2 P 1 3 ∂W ) ) ∂A 6 K  3 M ∗ ∗ ∂A ) + e 2 · ∂A ∗ 3 2 ) · A A M F 1 6 e M A 1 + + ∂F 4 + +16 ∂A + ∂A − 1 ∗ ) is the sum of the flat superspace conformal Galileons of the previous section- A A ∂A ¯ c 4 ∂W ( ( 2( A F M ) 3 3 3 ∗  4 ¯ c c 9( ) 3 5.33 has mass dimension 0, 1 A ∗ → ) is now dimensionless while ¯ c − +¯ M ∗ = A + F 1 4 = ¯ W A µ A + 2 A 0 h ( ¯ + L µ A 2 ∂A ( ∂W As an important check on our supergravitational expression for h A ¯ c 1 3 ( ¯ ¯ c c − where uing this procedure and again setting where scale conventions defined near theization beginning conventions of used this in section. theprocedure. To previous return We section to use the the of mass field thethe scale normal- paper, dimensions in we the implement the constants by following letting ¯ equation ( now, however, written in terms of After the following integration by parts, the flat superspace limit. To do this, we let and We want to emphasize thatlimit the employed final in result the ( previous section. JHEP08(2017)014 µν g and  2 P (5.38) (5.39) (5.37) ∗ M FF ∗ . Be this as  2 P ∗ ) K R  ∗ 2 M )  ∂ A ∗ FF ) ∂A∂A ∗ A ∗  ). +  ) + A + ∗ K iχ ∗ A ) becomes 2 . ( ,F A − A A . 2 ( ∂ +   2 SUSY 4 ∂ −   A ∂A∂A φ ( ∗  ¯ ·∇ F 5.29 ∇ L ( A ∗ ∗ ∂ + ( 2 A F · 1 F ∗ √ ∗ FF ∇ ∗ 2 ·∇ . In this “cosmological” limit, ) FF ∗ F 2 ∗ ) ∗ = ) contribute to the low-energy ∂A 2 − ∗ F · K − A M A 2 ∂A A F ∂F ∇ · ∂ A 2 4.16  ·∇ ∂A F 2 ∂A∂A 2 2 ∗ −
A ∇ − ∂A ∗ ) 0 cosmo 2 − ) is in a cosmological context where 1) ∇ K ∗ ¯ 
) L 2 ∗ ∗ 2 A ∂A ∂ c + 6( A A ∂A∂A 5.29 +6( − +¯ 2 ∇ 2 − ) − detg )  A ∗ + 2 A ∗ ∗ (  − , we have also carried out a complete calculation ( ∗ ∂ A – 29 – 2 F +2 ∂A A · c p ∗ ∗ ( . This is a complicated and detailed analysis and ·∇ ∇ ∗ 2  ( of chiral matter also satisfy to be smaller than A + ¯ ) 2 F 2 Z 2 ) ∂F
∂A ) is exact and is valid for any momentum or magnitude F ∂W F  SUGRA 3 ∇ ∗ , the action associated with ( ∇ = ) written in terms of A ∂A ∗ ∗ ¯ 2 L F + P ∂A or
∇ F F S SUGRA 4 2( 5.29 F  ¯ ∗ 2 M A 2 ∗ L 2( 4.11 4 + 2  ∂  +2 ) ∇ 3 4 ∗ ∂A ) ∂A ) ∂W ∂W 2 ∗ A ∗  6  A + A 3 + 6 1 1 ) c + ), and ( + ∗ F A +¯ ( A A A 1 ( 4.7 ( R + ∂A ) greatly simplifies. Neglecting all terms-with the notable exception of − ∂W  − 2 P A 3 ), (  2 ( c M 1  5.29 c 2 3 + ¯ ¯ c c 4.17 = ¯ − +¯ , in this limit all terms in the expression ( 0 ¯ = L , 2) although spacetime can be curved and dynamical, its curvature P is the associated covariant derivative. M -that depend explicitly on SUGRA 3 0 cosmo ∇ . However, one important application of ( ¯ ¯ L R L F 2 P 2 M of the supergravity Lagrangian will be presented elsewhere.energy However, the “cosmological” results limit of defined thatfor above calculation are that relatively concern straightforward. thecurved We low- find superspce that, Lagrangian as –however, with the modification that one replaces the flat All indices in this expressionand are contracted with respect to a curved spacetime5.3 metric Low momentum, curvedSimilarly to spacetime the above limit-including discussion of where 3) the momentum and auxiliary field it may, 4) it is not necessarythe for expression ( 5.2 The low momentum,As mentioned curved above, expression spacetime ( limit of M  This is precisely the sumin of equations the ( dimensionless flat superspace conformal Galileons defined as expected, JHEP08(2017)014 ) in ) ) ∗ ∗ 4.40 (2) (5.40) (5.41) in the A A F + ), ( + 
2 (0) + | A A ∗ ( ( F F F µ µ 4.32 | (1) µ ∇  ∇ F ) ) ∇  ∗ ∗ ∗ + F given as the sum A A A
ν ) + − (0) ∗ ∇ A 2 A F | 2 F ( ( F | ν 0 cosmo ν ν F 2 = ¯ | F | ·∇ L ∇ | are given in ( ∇
∇ terms and by ∗ )
F ∗  ∗ F . The origin of these terms + 3 ). A F A (2) c ∗ A F |∇ ∗ ν ∇ ) 2 F F ( | +
∗ F  ∇ ∗ µ 4.13 ∗ ) µ F A µ F ∗ | A ( F ∇ µ  SUGRA 4 ∇ occur in µ ∇ and µ A µ ) ¯ 2 in ( ∗ in the ¯ L + ∇ ∗ ) one must add the term ∇ ∇ + ∇ ∗ ∗ − A +12 ν A (1) ∗ ν A A 2 ∂F A ) are with respect to the curved metric µν (1) | ∇ F 2 ( + and each derivative with the covariant ν F ∇ AF 5.39 2 F R , F ∗ F F A ∇ µ µ ∇ 9 FF ( ∗ ν ∗ ∇ µν 5.41 F + µ 2nd term (0) and ∇ ∇ + µ |∇ − g , F  µ ∇ F ν 2 ∗ ) SUSY 4 F ∇ 4 | µ + ∇ µ ∗ | ¯ F (0) ∇ 2 F ∇ L A )  A ∇ + ∇ ∗ A ∗ F can be replaced by F 2 − ∗ | ν
– 30 – 2 A ) A A 2 + |∇ ) and ( 2 ∗ A ) + ∇ F ν µ F = 4 ν A , there are several other terms involving the curva- ∗ 2 µ µ + A ∇ | ( ∇ ∇ − ∗ 2 A ∇ µ F ∇ ∇ ν ∇ ) 2 F 5.40 ∗ + | F ) ∗ ∇ A A ∇ ∗ ∇ ) ((
∗ ( A ) ν ν ∗ ) A µ ν µ ∗ F A SUGRA 3 ) A ∇ ∇ +( , by F ∇ ∇ A ∇ are the spacetime curvature scalar and Ricci tensor respec- ¯ ∗ 2 ( ∗ µ ∇ L F µ c ( −∇ µ + ·∇ + ·∇ F . It follows that to ( µ F ∇ µ ∗ , ¯ µν 2 ) ν A A 1 A ) ∇ µν + ( ∇ c ). The field R F −∇ ( ∇ g ∗ −∇ 2 µ A ∇ ∇ ∗ ∗ µ FF ) ∗ ( + A µ ∇ A 2( . The expressions for µ ∇ A ν A ( 4 µ 5.41 2 and ∇ R∇ − µ FF c A ∇ ∇ ∇ ) AF 2 ∇ ν µ ( ∇ 2 µ ∇ ) + 34 µ ν R µ 2 ) ∗ ∇ ∇ ) ∇  A ∇ ∇ A ∇ A ). A 2 + 2 A ν A ), one needs to add the terms 2 A  ν ) ∇ ν ∇  ) and ( ∇ ν ∗ ∇ ∇ 3 ( ∇ 2 ( (( ) ∗ ∇ 2 AF F A ) ∇ 5.39 ∗ µ µ 5.40 ) µ A µ µ ∗ F with the curved spacetime metric ν A 5.40 ν + 3 A 2 1 ∇ A ∇ ∇ 1 in ( 6 ∇ + ) respectively. ν ∇ 2 −∇ −∇ − ∇ A + µν µ ( ), ( − A ∇ η A ( (    ( 4.43 + − + −∇ + + + 128(  0 cosmo 5.39 4 . Here, we will simply point out that these new terms arise from commuting certain ¯ Finally, we note that various powers of However, unlike the case of  L ¯ c 4 ¯ c µν and ( Acknowledgments R. Deen and B.A.Ovrut Ovrut would are like supported to in thank part Anna by Ijjas DOE , contract Paul No. Steinhardt DE-SC0007901. and B. other members of the PCTS derivatives in the supergravity extension of of ( the terms proportional toterms ¯ proportional to ¯ to tively. Note that all contractionsg in ( ture tensor that also enteris the less “cosmological” straightforward and, limit in of addition this to paper, ( we will simply state the results. We find that in, derivative with respect to metric JHEP08(2017)014 , it O (A.4) (A.3) (A.1) (A.2) ) R 8

− , O ), which are 2 )

. However, we D R 8 O 5.10 X ) E = ( − R )–( 2 8 X D 5.5 − . (

2

D ( O E ) 2 2 R D Θ 8

4 1 E component of − and then taking the lowest com- − 2 + 2

]. Recall that a chiral superfield Φ D F. D

( 40 O E O ) , )

2 R R continues to be chiral and has an exact by acting on it with the chiral projector Φ 8 + ΘΘ . As explained above, we can construct a 8 D to be a chiral superfield. Multiplication by 2 α 1 4 ψ − O D X − ] and [ X 2 − 1 4 E 2 39 2Θ D D − – = ( – 31 – ( √ = O 2 2 + 37 ) , D D F R A 35

8 E E − 1 4 1 4 then projects out the Θ Φ = 2 component of a chiral superfield can be obtained by first − − ”. For example, D 2 X

( = = = 1 supergravity, we are interested in constructing invariant E E Θ O Θ N 2 ) 2 d R d 8 R , requires the integrand Z − means that under local supersymmetry, the entire integral transforms out of any Lorentz scalar X 2 and then taking the lowest component. Choosing E E ) that the Θ D 2 X ( Θ D 2 E 1 4 A.2 d Θ − R 2 d component of Φ. . The integral 2 Z R 8 Within the context of − 2 written as Under the assumption that we ignore all fermions, including the gravitino, this can be chiral superfield have seen in ( acting with follows that superfield Lagrangians.superspace, This can bethe accomplished chiral density as follows.into a An total integral spaceexpansion derivative. over in the chiral The local product superspace coordinate Θ is the Θ has the following Θ expansion The components of Φponent, can which be we obtained denote by by acting “ with A constructing higher-derivative SUGRA Lagrangians We give a brief explanationwritten of how in the terms supergravity of Lagrangians superfields,here in can ( is be based expressed on in work component presented fields. in The [ formalism used like to thank hisderivative supersymmetry long and term supergravitation. collaborator Finally,and R. Jean-Luc Paul Deen Lehners Steinhardt is for for gratefulUniversity to his many of Anna joint discussions Pennsylvania Ijjas for work and their on the support. Center higher- for Particle Cosmology at the working group “Rethinking Cosmology” for many helpful conversations. Ovrut would also D JHEP08(2017)014  (A.9) (A.5) (A.6) (A.7) (A.8) (A.11) (A.10) ˙ δA CB conformal

R 3 ˙ † δ ¯ L , V Φ ) 2 b  . + D

A  c
( Φ a †

V ) b ˙ † δ D Φ µ − Φ 2 D Φ ˙ 2 D δ D µ D + ( A D a Φ CB . Φ V A α ie . D T c . D

1 6 D D Φ + operators appropriately, and

D Φ O CBδ ). To exhibit our methods, let + D
. D A + h ˙ D R 3 α R † µ 2 I 8 δ −

CB b V D Φ ) A.6
δ . µ D T − O V 2 † 3 b ) Lagrangian, we now apply the pre- ) , D

b − D

− 3 I δ R 1 ) , + 18 = 0 Φ † 8 − 3 c and O A CB ( ) ¯ 2 † are fermionic. In keeping with the thesis L ) D † T − − α Φ D Φ α − (Φ + Φ ∗ 2 ) in the curved superspace , + CBD D α ψ

D D α R D 2 A ) 3 ( + ( 5.9 (Φ + Φ D − D E V √ O ) MM A e 2 , † V R 1 9 – 32 – ) = ( D D b + 2 + (Φ + Φ Θ 2 2 e

2 + CBe c = d Φ D CB ( R 8

α d T R − e Φ = 0 )
Z  − ˙ D † α 1 V , − 12 ∗ 1 4 Φ 

− D = (Φ + Φ 2  − I O = ( = 2 D eM 2 O = 2 Φ A D , which is not hermitian, one writes I I Θ , 2 D V + Θ operators now give rise to terms which would not have been 3 2 1 O D ¯ Φ  L M C D D − , 3 1 6 e D − D 2 1 B − ) † D = = . bc

) E R O − 2 ( (Φ + Φ D − 2 B D D C D As an example, consider the first term (  example, expressions which contain of this paper, allthe such terms computation will is be thecommutators dropped. presence of However, the of the essential curvaturepresent difficulty in and involved the in torsion global in supersymmetric supergravity. case. Explicitly, Hence, we have anti- We calculate these termscommuting them by until distributing wechiral the are fields able to apply the defining expressions forMany chiral terms and that anti- arise in the intermediate stages of the calculation involve fermions. For us compute Having obtained the superfieldceding expression formalism for to the express itone in must terms evaluate of the component four fields. lowest It component follows terms from in the ( above that the higher-derivative superfield expression Then, the associated Lagrangianperspace is integration. obtained For by the appropriate chiral projection and su- by evaluating the following terms, Galileon. It was constructed using the formalism just described. That is, one begins with It follows that one can compute the component field expansion of a supergravity Lagrangian where JHEP08(2017)014 are = 1 D N (A.13) (A.12) CB

† T Φ ) 2 ∗  D and F

 4 † A

. ∗ Φ − .

( 2 auxiliary fields of Φ ∗
β  D † CBD take the values 0 or ) are fermionic and, ) F AF D

∗ Φ ν M R 2 A ˙ 2 α ) Φ ν ) A.9 D D ∂ D A ∂ A AF ν ˙ ∂A

Φ α ν µ b b, c, d ν and ( e ∂ ∂ Φ D ∗ D ˙ 3 ν ν α α µ ) b b

Φ b A ∂ β b ∗ e e D µ Φ ˙ D ˙ β µ AF β iσ A ∂ a b β ν D 1 2 ν 2 e D ˙ b β ) yields + − iσ e D  − D 2 ˙ A ∂ µ α A )( 3 a µ ( b β A.9 −

e − A − ∂ σ ) µ )( ν ˙ Φ

 64 β ). ∗

b ) ˙ ∂ β α a † α A e † − µ A b β µ σ D µ a Φ Φ ˙ a σ ∂ e β 2 2 = 5.13 + e ˙ − µ β ˙ a

D D αβ D a α ˙  A β e
a

, and the exponents b ˙ † β Φ α ab σ iσ ˙ – 33 – Φ α Φ σ a α Φ η 2 D = ( ˙ α α 2 2 + D ∗ − Φ iσ

a α ˙ ˙ D β ( D β
2 F σ − ˙ D α † b β a, α, Φ D ( 2  − − σ ˙ ) Φ D ( ab α

D 2 ) can be evaluated using similar methods. Putting these η  ˙  βα ββ D Φ Φ 2 D  ˙ a ∂A β ˙ D α βα βα D − σ Φ ˙ ˙ A.6   D β α 3   ˙ ˙ ˙  contribution to ( D α α α − 64 ( are bosonic or fermionic respectively. ˙ ˙ ˙ D ) and then inserting in ( β β β ) I Φ , †     − 3 D ¯ L = 16 = = = = = 16 = 16 = 2 A.12

D indices can be †

B,C,D (Φ + Φ Φ 3 2 2 − ) D D † Φ and used throughout the paper. D Φ A A, B, C, D D (Φ + Φ 2 D ], chapter 15. 40 Using these results, we determine that the first two terms in ( B Useful supergravity identities Here we present a non-exhaustive listin of appendix identities necessary for the computations described The remaining three terms infour ( component fieldsupergravity terms , together, yields the and eliminating the Putting this back into ( hence, are taken to vanish. The third term is given by We compute the lowest component term on the right-hand-side as follows. 1 when the indices superfields which respectively contain componentssupergravity, of the these curvature and superfields torsion. andin For [ their component expansions are given, for example, where the JHEP08(2017)014  ˙  ”-we (B.2) (B.4) (B.5) (B.7) (B.8) (B.9) ˙ βe

(B.16) (B.10) (B.12) (B.13) (B.14) (B.15) T † ∗ Φ ˙  F F D 4 4 . − − + † † = = Φ Φ 

} ˙ † † δ D Φ  Φ Φ 2

D

2 2 † ˙ , † † βe † † D α Φ D D T Φ Φ Φ Φ ˙ ˙ β α 2 2 2 2 + {D D D † D D D ˙ D γ α β 2 2

Φ D a D D D Φ Φ D ˙ ∗ α D

D D

F a D ˙ ˙ Φ Φ Φ α α ∗ Φ ˙ Φ βe + D D D D D β D † T ˙ M ˙ 2 α α β β

D  Φ ˙ 3 D D D D δ ˙ α D Φ 32 γ e

α M

D D D ∗ + a ˙ } Φ Φ Φ Φ α Φ iσ A ˙ γ F ∗ D 2 µ D D D D )Φ (B.1) D D ˙ ˙ D ˙ ∂ A 2 α β β β β 3 β − , 16 + a µ † a a D D D D D α D

ˆ b e }D D 2 2 4 2 Φ 2 = ˙ ˙ Φ α a ˙ γ α – 34 – γ 

{D a − φ a α − − − + 4 − ˙ † ib D α D D , e iσ Φ = α 3 D α ˙ F σ δ 32 2 D

D 2 3 ˙
−
β D

− † ∗ † +  ˙ e α γ Φ − ˙  Φ (B.3) ∗ Φ α Φ A = −{D βa D ˙ iσ 2 δ } µ ˙ D γ

A T β F 2 aα φ  ∂ a D D µ Φ = 0 (B.6) σ µ D ˙ ˙ − γ D − ˆ αa ˙ ∂ a a Φ }D ˙ β D

˙ , α γ e T µ ˙ a † φ β , ˙ D  µ α D a D ˙ ˙ ˙ ∗ F, α D α } β a Φ e ˙ D D δ a a α ˙ φ α D Φ α ˙ α ˙ ˙ , ˙ F α α α γ µ αβ { ˙ a D a α  φ β α D R iσ  iσ D β ˙ D ˙ ˙ e † β α β , 2 2 T 2 MF σ , ˙ a iσ  φ 2 α D α Φ {D 2 3  − − − − ˙  α ( D iσ 2 1 3 D D {D = = = 8 = = 0 = = 2 = = = 0= 16 (B.11) D

= = † † † † † Φ Φ Φ Φ Φ † † Φ = Φ = 2 Φ = Φ Φ Φ Φ Φ φ φ β α α ˙ ˙ ˙ Φ Φ 2 2 γ φ φ β β α ˙ ˙ δ γ D D D D D ˙ D D D ˙ D D ˙ D D D β α α α ˙ D ˙ 2 2 D ˙ ˙ β β α α α α ˙ ˙ γ β D D D D D D D ˙ D D D D D β α D ˙ D α β α α ˙ α D α D D D D D α α D D α α α D D Additionally, we find that D D D drop all fermions and presentfor the purely the bosonic relevant superfields result. are The given lowest component by expressions When calculating the lowest component of a superfield expression-indicated by “ The purely superfield results of interest are JHEP08(2017)014 , , , , , (B.17) ]. ]. D 90 Found. , , . SPIRE

SPIRE † † † † IN IN

Φ Φ Φ Φ † ][ ][ D D D D Φ (2013) 241303 ˙ ˙ (2010) 015 β β α α Phys. Rev. D D D D D α , 05

110 D † † † †

Φ Φ Φ Φ ]. † D D D D Φ ˙ ˙ ˙ β β β α ]. JCAP D , D D D D α ]. SPIRE

Dirac- Born-Infeld Genesis: An D IN Φ Φ Φ Φ arXiv:1103.6029

SPIRE [ arXiv:1512.03807 ][ D D D D Φ ˙ ˙ IN [ β β α α D SPIRE ][ ]. ˙ D D D D ]. ]. Phys. Rev. Lett. β IN ].

,

D ][ Towards a Nonsingular Bouncing Cosmology Φ Φ Φ Φ

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][ ) arXiv:0811.2197 D 93 Bouncing Cosmologies: Progress and Problems † [ A smooth bouncing cosmology with scale invariant spectrum hep-th/0005016 Φ ), which permits any use, distribution and reproduction in [ DBI and the Galileon reunited ]. ]. D ]. † Φ (2011) 017 D SPIRE SPIRE arXiv:1007.0027 hep-th/0702165 arXiv:1206.2382 Φ arXiv:1603.05834 The Matter Bounce Alternative to Inflationary Cosmology SPIRE Phys. Rev. [ [ [ 07 IN IN [ D , IN (2000) 208 Phys. Rev. Lett. [ ][ ][ Φ CC-BY 4.0 (2009) 064036 , D arXiv:1310.7577 This article is distributed under the terms of the Creative Commons [ 2 JCAP , D B 485 D 79 2 (2010) 021 (2007) 010 (2012) 020 (2017) 797 D 11 11 08 47 arXiv:1212.3607 arXiv:1003.5917 Phys. (2014) 025005 effective description JCAP arXiv:1206.4196 Improved Violation of the[ Null Energy Condition JCAP Embedded Branes JCAP curved space [ Phys. Rev. Phys. Lett. M. Koehn, J.-L. Lehners and B.A. Ovrut, M. Koehn, J.-L. Lehners and B. Ovrut, R.H. Brandenberger, R. 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