A Positive Energy Theorem for P(X,phi) Theories
Benjamin Elder, Austin Joyce, Justin Khoury, Andrew J. Tolley
aa r X i v : . [ h e p - t h ] J un A Positive Energy Theorem for P ( X, φ ) Theories
Benjamin Elder, Austin Joyce, Justin Khoury, and Andrew J. Tolley Center for Particle Cosmology, University of Pennsylvania, Philadelphia, PA 19104, USA Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637 CERCA/Department of Physics, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USA
We describe a positive energy theorem for Einstein gravity coupled to scalar fields with first-derivative interactions, so-called P ( X, φ ) theories. We offer two independent derivations of thisresult. The first method introduces an auxiliary field to map the theory to a lagrangian describingtwo canonical scalar fields, where one can apply a positive energy result of Boucher and Townsend.The second method works directly at the P ( X, φ ) level and uses spinorial arguments introduced byWitten. The latter approach follows that of arXiv:1310.1663 , but the end result is less restrictive.We point to the technical step where our derivation deviates from theirs. One of the more interestingimplications of our analysis is to show it is possible to have positive energy in cases where dispersionrelations following from locality and S-Matrix analyticity are violated.
In recent years there has been much interest in deriva-tively coupled scalar theories, particularly in cosmology,but also in other areas of high-energy physics and con-densed matter. The novelty of these theories is that, incertain cases, they can have large classical non-linearitieswhile remaining radiatively stable, allowing for a rangeof interesting phenomena. Ghost condensation [1] andgalileons [2] possess time-dependent solutions that canviolate the Null Energy Condition (NEC) [3–7] and yieldnovel cosmologies [8–13]. These examples are free ofghost or gradient instabilities, but have other unwelcomefeatures, such as superluminality and conflict with blackhole thermodynamics [14], casting doubt on whether theyadmit a local ultraviolet (UV) completion [15].It is natural to wonder if there are any statements onecan make about the viability of these theories in the pres-ence of gravity. One desirable property is that the vac-uum be classically stable. This will be the case if the thetheory admits a positive energy theorem for asymptot-ically flat solutions, i.e. , the ADM mass is always non-negative and is zero for Minkowski space only [16]. Itwas originally shown [17] that Einstein gravity plus mat-ter has positive energy if the matter obeys the domi-nant energy condition (DEC) [18]. This proof was latersimplified using a spinor technique due to Witten [19–21]. (Similar proofs exist for asymptotically anti-de Sit-ter [22–25] and de Sitter [26, 27] spacetimes.) The resultwas extended by Boucher and Townsend, who showedthat the DEC is not necessary to ensure positive en-ergy [28, 29]. See also [30]. For a nonlinear σ -modelwith N scalars, L = − f IJ ( φ ) ∂ µ φ I ∂ µ φ J − V ( φ I ) , (1)where f IJ is positive-definite, positivity is guaranteed solong as V ( φ I ) is derivable from a “superpotential” W ( φ I )obeying the equation [31]: V ( φ I ) = 8 f IJ W ,φ I W ,φ J − W , (2)assuming that V ( φ I ) admits a minimum with V ( ¯ φ I ) ≤
0. In this Letter, we further extend this result and derivea positive energy theorem for scalar theories of the form L = P ( X, φ ) , (3)by similarly constraining the functional form that P ( X, φ ) can take. Here X is the canonical kinetic term: X = − ( ∂φ ) . (We use the mostly-plus sign conven-tion.) This class of theories has a long history, especiallyin cosmology. They can be used for inflation [32–34],dark energy [35, 36], bouncing cosmologies [8, 9, 11], anddisplay screening around heavy sources [37–41].We establish the positive energy result in two differentways. First, at the classical level we map (3) to an equiv-alent two-derivative theory via an auxiliary field [42].Turning on a small kinetic term for this second field,the action takes the form (1). We can then apply the re-sult (2), which is translated to a statement about P ( X, φ )upon integrating out the auxiliary field.Second, we will reproduce this result directly at the P ( X, φ ) level using Witten’s spinor arguments. This ap-proach was taken in [43], although we will see that theirresult was slightly too restrictive. We will show that re-laxing a small technical assumption in their argumentallows for greater flexibility in choosing the functionalform of P ( X, φ ).This broader assortment of P ( X, φ ) theories consistentwith positive energy allows for interesting phenomena.In particular, consider P ( X ) = X + αX , arguably thesimplest P ( X, φ ) example. With α >
0, this theory obeysthe DEC and hence has positive energy. Even with α < P ,X >
0. Thisis remarkable since this theory with α <
Two-Field Description : A P ( X, φ ) theory can be mappedto a 2-derivative action by introducing an auxiliary field χ [42] so that the lagrangian takes the form L = − P ,χ ( ∂φ ) − χP ,χ + P , (4)where P = P ( χ, φ ). Indeed, the equation of motion for χ is P ,χχ ( X − χ ) = 0, which sets χ = X , as long as P ,χχ = 0. Substituting χ = X in (4) gives L = P ( X, φ ),establishing the classical equivalence of the two descrip-tions. To put it in the form (1), we simply turn on asmall kinetic term for χ : L = − P ,χ ( ∂φ ) − Z ( ∂χ ) − χP ,χ + P . (5)At this level, this is just a technical trick — at the endwe will take Z →
0. Upon making the identifications f χχ = Z ; f φφ = P ,χ ; V ( χ, φ ) = χP ,χ − P , (6)this is of form (1). Note f IJ must be positive-definite,imposing P ,χ >
0. After integrating out χ , this translatesto P ,X >
0, which is equivalent to the NEC [44]. Insome cases this will restrict the range of X , but this isacceptable because it is a Lorentz-invariant restriction onthe space of allowed solutions. The condition P ,X > χP ,χ − P = 8 W ,φ P ,χ + 8 W ,χ Z − W . (7)To have a smooth Z → W ( χ, φ ) = W ( φ ) + Z √ G ( χ, φ ) + O ( Z ) , where the factor of 2 √ χ → X , the positive energy condition becomes P − XP ,X + 8 W ,φ P ,X + G ,X − W = 0 . (8)This is our main result. It is the analogue of (2) fortheories of the P ( φ, X ) type. Positivity of the energy re-quires the existence of two functions, W ( φ ) and G ( φ, X ),related to P ( φ, X ) through (8). Asymptotically, we as-sume X → φ → φ such that P ,φ ( φ ) = 0.The proof generalizes to N scalar fields with P ( X IJ , φ K ), where following [43] we have defined thetensor X IJ = − ∂ µ φ I ∂ µ φ J . This generalization is par-ticularly interesting because the EFT of fluids [45] is atheory of this type. We introduce a matrix of scalar fields χ IJ , and the generalization of (5) becomes L = − P MN ∂ µ φ M ∂ ν φ N − Z P KM P LN ∂ µ χ KL ∂ ν χ MN + P − χ MN P MN , (9)where P IJ ≡ ∂P/∂χ IJ is positive definite and invertible.Again, integrating out χ and setting Z → X IJ = χ IJ . Following the same steps as before, we find thatthe superpotential must take the form W = W ( φ I ) + Z √ G ( φ I , χ MN ) + O ( Z ) . Writing the inverse of P IJ as P IJ , we arrive at the positivity condition P − X MN P MN + 8 P MN W ,φ M W ,φ N + P KM P LN G KL G MN − W = 0 . (10) Direct derivation : We now re-derive the positive en-ergy condition (8) directly at the level of P ( X, φ ). Thismethod generally follows the presentation of Witten’sproof of the positive energy theorem in [43], but witha crucial difference, which we will point out below.The starting point is the Nester 2-form [19, 21]: N µν = − i (cid:16) ¯ ǫγ µνρ ˆ ∇ ρ ǫ − ˆ ∇ ρ ǫγ µνρ ǫ (cid:17) . (11)where we have defined the super-covariant derivativeˆ ∇ µ ǫ = ( ∇ µ + A µ ) ǫ . (12)Some words on notation: ǫ is a commuting Diracspinor [29], with conjugate ¯ ǫ = iǫ † γ ; the Dirac matricesobey the Clifford algebra { γ µ , γ ν } = 2 g µν , and we havedefined the anti-symmetric product γ µνρ ≡ γ [ µ γ ν γ ρ ] .The virtue of N µν is that its integral is simply relatedto the energy of a gravitating system [19, 21, 29] E = Z ∂ Σ dΣ µν N µν = Z Σ dΣ ν ∇ µ N µν , (13)where Σ is an arbitrary space-like surface, with dΣ ν de-noting the normal-pointing volume form. The divergenceof N µν is given by [43] ∇ ν N µν = 2 i ˆ ∇ ν ǫγ µνρ ˆ ∇ ρ ǫ − T µν M i ¯ ǫγ µ ǫ − i ¯ ǫγ µνρ F νρ ǫ , (14)where F νρ = ∇ ν A ρ − ∇ ρ A ν + [ A ν , A ρ ] is the curvatureof the connection A µ . The stress tensor for (3) is T µν = P ,X ∂ µ φ∂ ν φ + P g µν . (15)The term 2 i ˆ ∇ ν ǫγ µνρ ˆ ∇ ρ ǫ , gives a positive contribu-tion to the energy, after imposing the Witten condition γ i ˆ ∇ i ǫ = 0 [19]. The other two terms are not manifestlypositive. To proceed, we follow [43] and make the ansatz A µ = W ( φ ) γ µ , (16)for some W ( φ ). The last term in (14) becomes − i ¯ ǫγ µνρ F νρ ǫ = − i ¯ ǫγ µν ǫ W ,φ ∂ ν φ + 12 i ¯ ǫγ µ ǫ W . (17)Our goal is to write this as a sum of squares of spinors,plus a remainder piece. To do this, we define δλ = 1 √ (cid:18)p P ,X γ µ ∂ µ φ − W ,φ p P ,X (cid:19) ǫ ; δλ = G ,X ǫ , (18)so that − i ¯ ǫγ µνρ F νρ ǫ = i X i =1 δλ i γ µ δλ i + i ¯ ǫγ ν ǫP ,X ∂ µ φ∂ ν φ + i ¯ ǫγ µ ǫ (cid:18) XP ,X − W ,φ P ,X − G ,X + 12 W (cid:19) . (19)This is the key difference from the derivation in [43]. Inthat calculation, the authors only used one δλ spinorfield, which led to a restricted class of solutions. Insteadwe expressed − i ¯ ǫγ µνρ F νρ ǫ as the sum of two squares ofspinors. The second spinor introduces a new function G = G ( X, φ ), which allows us to derive a more generalpositivity constraint than [43].Combining (14), (15) and (19), we obtain ∇ ν N µν = 2 i ˆ ∇ ν ǫγ µνρ ˆ ∇ ρ ǫ + i X i =1 δλ i γ µ δλ i + i ¯ ǫγ µ ǫ (cid:18) XP ,X − P − W ,φ P ,X − G , X +12 W (cid:19) . (20)The first line is positive-definite, whereas the second lineis not. To ensure positivity of E , it is sufficient to set thesecond line to zero. This yields (8), which is preciselythe energy condition obtained from the 2-field approach.The mass vanishes for ˆ ∇ µ ǫ = δλ a = 0, which impliesMinkowski or AdS space-time [29]. Having derived thisconstraint on the functional form of P , we now turn tosolving this equation in a few situations of interest [46]. Pure P ( X ): One simple but nontrivial case to consider is P = P ( X ), i.e. , a field with purely derivative couplingsand no potential. We simply assume that W ≡ W isconstant, and take G = G ( X ). In this case, the positiveenergy condition (8) reduces to an ordinary differentialequation for G , which can be integrated: G ( X ) = Z d X (cid:0) XP ,X − P + 12 W (cid:1) / . (21)In order for this integral to be real-valued, we must have XP ,X − P ≥ − W . Note that this condition is weakerthan the dominant energy condition: XP ,X − P ≥ P ( X ) = X − βX ; β ≥ . (22)This theory violates the DEC for all X : XP ,X − P = − βX <
0. Recall that our derivation requires P ,X ≥ | X | ≤ / √ β .In this case, (21) can be integrated, ensuring the exis-tence of a suitable superpotential, and guaranteeing thatthe theory has positive energy in the allowed X range.This theory with “wrong-sign” X term is well-knownto violate the standard dispersion relations followingfrom local S-matrix theory [15], at least at tree level.Nevertheless, we have shown that the theory does allowpositive energy, at least over the range of X where theNEC is satisfied. This may seem paradoxical from theperspective of the 2-field action discussed earlier; afterall, (5) describes two healthy scalars with some potential,and therefore should have an analytic S-matrix. The res-olution is that the vacuum state X = 0 or, equivalently, χ = 0, is tachyonic in the two-field language, hence itsS-matrix is ill-defined. Separable P ( X, φ ): A slightly more complicated case iswhere P is a separable function: P ( X, φ ) = K ( φ ) ˜ P ( X ) − V ( φ ) , (23)with K ( φ ) ≥ G ( X, φ ) via G ,X = H ( X, φ ) + 8 W ,φ K ( φ ) (cid:18) − P ,X (cid:19) . (24)Inserting this into (8), we find that P must satisfy˜ P − X ˜ P ,X + H ( X, φ ) K ( φ ) = 1 K ( φ ) (cid:18) W − W , φ K + V ( φ ) (cid:19) . For this to be separable, H must factorize as H ( X, φ ) = K ( φ ) H ( X ). The above then implies two equations H ( X ) = X ˜ P ,X − ˜ P ( X ) − E ; V ( φ ) = 8 W , φ K ( φ ) − W + EK ( φ ) . (25)We must ensure that through all these redefinitions wemaintain G , X ≥
0. Combining (24)–(25), we find X ˜ P ,X − ˜ P ( X ) ≥ E − W ,φ K ( φ ) (cid:18) − P ,X (cid:19) . (26)This allows for DEC-violation through the kinetic partof the action whenever the right-hand side is negative.A few limiting cases of these results: • If ˜ P = X , corresponding to the two-derivative la-grangian L = K ( φ ) X − V ( φ ), we can set E = 0 and G = 0. The second of (25) reduces to the standardresult (2) for a single scalar field V ( φ ) = 8 W , φ K ( φ ) − W . (27) • For the pure P ( X ) case, corresponding to K ( φ ) =1 and V ( φ ) = 0, the second of (25) allows us tochoose W = W = constant, with E = 12 W . Thefirst of (25), combined with (24), then implies G ,X = H ( X ) = XP ,X − P + 12 W , (28)whose integral reproduces (21). Conclusions : We derived, following two different meth-ods, an extension of the positive energy theorem of Gen-eral Relativity to the class of P ( X, φ ) scalar field theo-ries. We found that as long as it is possible to write P interms of two arbitrary superpotential-like functions, posi-tive energy is guaranteed. This derivation generalizes theresult of [28, 29] for two-derivative scalar theories witharbitrary potential, and reduces to the known conditionas a particular case. This result allows for more general P than the recent result of [43], and we highlighted thetechnical step where our derivation deviates from theirs.By examining a few special classes of P we showed thatin the P ( X ) context it is possible to have positive energywhile violating the DEC. The derivation does howeverrequire that the NEC to be satisfied. More interestingly,it is possible to have positive energy in cases where the S-matrix fails to satisfy the usual analyticity requirementsfor a local theory. It will be interesting to extend our results to more general derivative interactions, such asgalileons or massive gravity [47]. Acknowledgments : We thank L. Berezhiani, R. Deen,G. Goon, M. Nozawa,T. Shiromizu and R. Wald for help-ful discussions and comments. This work is supportedin part by NASA ATP grant NNX11AI95G (B.E. andJ.K.) and NSF PHY-1145525 (J.K.); the Kavli Institutefor Cosmological Physics at the University of Chicagothrough grant NSF PHY-1125897 and by the Robert R.McCormick Postdoctoral Fellowship (A.J.). A.J.T. issupported by DOE Early Career Award DE-SC0010600. [1] N. Arkani-Hamed, H. -C. Cheng, M. A. Luty andS. Mukohyama, JHEP , 074 (2004).[2] A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D , 064036 (2009).[3] P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore,JHEP , 080 (2006).[4] A. Nicolis, R. Rattazzi and E. Trincherini, JHEP ,095 (2010) [Erratum-ibid. , 128 (2011)].[5] K. Hinterbichler, A. Joyce, J. Khoury and G. E. J. Miller,Phys. Rev. Lett. , no. 24, 241303 (2013).[6] V. A. Rubakov, Phys. Rev. D , 044015 (2013).[7] B. Elder, A. Joyce and J. Khoury, Phys. Rev. D ,044027 (2014).[8] E. I. Buchbinder, J. Khoury and B. A. Ovrut, Phys. Rev.D , 123503 (2007).[9] P. Creminelli and L. Senatore, JCAP , 010 (2007).[10] P. Creminelli, A. Nicolis and E. Trincherini, JCAP ,021 (2010).[11] C. Lin, R. H. Brandenberger and L. Perreault Levasseur,JCAP , 019 (2011).[12] P. Creminelli, K. Hinterbichler, J. Khoury, A. Nicolis andE. Trincherini, JHEP , 006 (2013).[13] K. Hinterbichler, A. Joyce, J. Khoury and G. E. J. Miller,JCAP , 030 (2012).[14] S. L. Dubovsky and S. M. Sibiryakov, Phys. Lett. B ,509 (2006).[15] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolisand R. Rattazzi, JHEP , 014 (2006).[16] For the ADM mass to be well-defined, we focus on flat (orAdS) asymptotics, where φ → const . at spatial infinity.This immediately rules out time-dependent asymptotics, φ → φ ( t ), which may be more realistic for cosmology.[17] R. Schon and S. -T. Yau, Commun. Math. Phys. , 45(1979).[18] The DEC states that: i ) for any time-like u , T µν u µ u ν ≥
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0. Since thisis positive-definite, we can write is as the square of somefunction. Calling this function G ,X yields (8).yields (8).