PHYSICAL REVIEW D 100, 064061 (2019)
Ghostfree quadratic curvature theories with massive spin-2 and spin-0 particles
† Katsuki Aoki 1,* and Shinji Mukohyama1,2, 1Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502 Kyoto, Japan 2Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 277-8583 Chiba, Japan
(Received 31 July 2019; published 27 September 2019)
We consider generic derivative corrections to the Einstein gravity and find new classes of theories without ghost around the Minkowski background by means of an extension of the spacetime geometry. We assume the Riemann-Cartan geometry, i.e., a geometry with a nonvanishing torsion, and consider all possible terms in the Lagrangian up to scaling dimension four. We first clarify the number, spins, and parities of all particle species around the Minkowski background and find that some of those particle species are ghosts for generic choices of parameters. For special choices of the parameters, on the other hand, those would-be ghosts become infinitely heavy and thus can be removed from the physical content of particle species. Imposing the conditions on the coupling constants to eliminate the ghosts, we find new quadratic curvature theories which are ghostfree around the Minkowski background for a range of parameters. A key feature of these theories is that there exist a nonghost massive spin-2 particle and a nonghost massive spin-0 particle in the graviton propagator, as well as the massless spin-2 graviton. In the limit of the infinite mass of the torsion, the Riemann-Cartan geometry reduces to the Riemannian geometry and thus the physical content of particle species coincides with that of the well-known quadratic curvature theory in the metric formalism, i.e., a massive spin-2 ghost, a massive spin-0 particle and the massless spin-2 graviton. Ghostfreedom therefore sets, besides other constraints, an upper bound on the mass of the torsion. In addition to the above mentioned particle species (a massive spin-2 particle, a massive spin-0 particle and the massless spin-2 graviton), the ghostfree theory contains either the set of a massive spin-1 and a massive spin-0 (Class I) or a couple of spin-1 (Class II). These additional particle species mediate gravity sourced by the spin of matter fields.
DOI: 10.1103/PhysRevD.100.064061
I. INTRODUCTION a low energy EFT of gravity with quantum corrections to GR. One such example of this approach is the Starobinsky Although Einstein’s general relativity (GR) is one of the model [1], where the Ricci scalar squared term is added to most successful gravitational theories, it has been believed the Einstein-Hilbert action. This model yields a successful that GR is incomplete in the ultraviolet (UV) regime and is inflationary universe. Its predictions are in good agreement merely a low energy effective field theory (EFT). There are with the current observations [2] and future observations several attempts to unify quantum theory and gravity such as superstring theory, Horava-Lifshizˇ gravity, ghostfree such as LiteBIRD [3] can achieve the sensitivity sufficient nonlocal gravity and so on. While constructing a complete to detect its inflationary prediction, the primordial gravi- quantum theory of gravity is certainly important, as a tational waves. complementary attempt one should keep exploring phe- A common feature of quantum gravity in the low energy nomenological signatures of quantum gravity by means of regime is the appearance of derivative corrections to the Einstein-Hilbert action. The Ricci scalar squared men- tioned above is indeed one of them. However, not only *[email protected] is the Ricci scalar squared but also other generic higher † [email protected] curvature terms such as Ricci tensor squared are naturally expected to appear if there is no mechanism to prohibit Published by the American Physical Society under the terms of them. Typical reasons to ignore them in the literature are for the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to simplicity and due to the ghost problem. Except fðRÞ the author(s) and the published article’s title, journal citation, theories, higher curvature terms in the action generally lead and DOI. Funded by SCOAP3. to higher derivatives of the graviton and then yields a
2470-0010=2019=100(6)=064061(18) 064061-1 Published by the American Physical Society KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) massive spin-2 ghost as the Ostrogradsky ghost. If the ghost new signatures of higher curvature corrections to the gives a dominant contribution, the system is outside the Einstein gravity. regime of validity of EFT and the theoretical control is lost. The rest of the present paper is organized as follows. In Therefore, in order for a theory to be predictive, the mass of Sec. II, we first give a general discussion on a gravitational the ghost (if exists) should be sufficiently heavier than the Lagrangian receiving derivative corrections in the frame- energy scale of interest, e.g., the inflation scale in the work beyond the Riemannian geometry. We then assume case of the inflationary universe. In particular, in the limit the metric compatibility condition just for simplicity and where the ghosts due to generic higher curvature terms are provide the explicit Lagrangian relevant to the perturbation infinitely heavy, a higher curvature theory reduces to one of analysis around the Minkowski background. In Sec. III,we fðRÞ theories, which can be recast to the form of scalar- compute the quadratic Lagrangian around the Minkowski tensor theories [4]. For this reason, it appears that phe- background and show the kinetic structure of the theory. nomenologically interesting signatures of higher derivative The number, spins, and parities of particle species in the corrections to the Einstein gravity come only from the spin- theory are then identified. Introducing a matter field, we 0 particle, at least in the context of the standard Riemannian also compute the graviton propagator of the theory. The geometry. main result is shown in Sec. IV, where the stability (no In the present paper, we point out that derivative ghost, no tachyon) conditions around the Minkowski corrections to the Einstein gravity can yield not only the background are derived based on the 3 þ 1 decomposition. spin-0 particle but also a massive spin-2 particle, a massive The stability conditions and the particle contents of the spin-1 particle and so on without suffering from light ghostfree theories are summarized in Sec. IV D. Finally, we ghosts. The no-go argument in the previous paragraph is give concluding remarks in Sec. V. based on the Riemannian geometry, where the connection is computed by the metric and thus the metric is the only II. ACTION independent object that describes the spacetime geometry. A. General discussion From the first principle of geometry, on the other hand, the metric and the connection are in principle independent It is known that quantum corrections give rise to objects. We thus treat the metric and the connection as derivative corrections to the Einstein-Hilbert action, independent variables and study derivative corrections to 2 L L ∂ ∂ L L 2 the Einstein gravity. Imposing some conditions on the ¼ ðg; g; gÞ¼ R þ R þ ; ð2:1Þ coupling constants of the theory, we find a new classes of theories in which the massless spin-2 graviton, the mas- where sive spin-2 particle and the massive spin-0 particle, which M2 appear in the graviton propagator as in the usual quadratic L ¼ pl R; ð2:2Þ curvature theory, can coexist without any ghost instability R 2 at least around the Minkowski background. 2 μν μνρσ L 2 α α α The construction of the new classes of higher curvature R ¼ 1R þ 2RμνR þ 3RμνρσR ; ð2:3Þ theories that we shall explore in the present paper is inspired by the recent developments of modified gravity, and in (2.1) represent terms involving more than four which had revealed that some of no-go results of ghostfree derivatives acted on the metric. Here, we have set the theories can be overturned. One of such examples is cosmological constant to zero in order to admit the ghostfree massive gravity: although it was for a long time Minkowski background solution. In four dimensions, α believed that a massive spin-2 field cannot be ghostfree one of the parameters i can be set to zero by means of the Gauss-Bonnet theorem. From now on, we thus set beyond the linear order due to the Boulware-Deser ghost α 0 [5], the ghostfree nonlinear potential of the massive spin-2 3 ¼ so that the theory is parametrized by the two parameters ðα1; α2Þ in addition to the Planck mass. If terms field was found by de Rham et al. in 2010 [6,7].Another 2 example is ghostfree scalar-tensor theories with higher in in (2.1) are suppressed by Mpl then one may use (2.1) derivative terms: while it was for a long time supposed to discuss a classical dynamics of gravity as long as ∇ ∇ ≪ nþ2 ≥ 0 that equations of motion must be second order differential j α1 αn Rμνρσj Mpl for all n in tetrad basis. equations for a theory to avoid the Ostrogradsky ghost, the From naturalness it is then expected that jα1;2j¼Oð1Þ. discovery of the degenerate higher order scalar-tensor Nonetheless, in some cases jα1j or/and jα2j might be (DHOST) theories [8–15] opened up a new window on parametrically larger than Oð1Þ for some reasons. constructing ghostfree higher order theories via degen- Indeed, actual values of jα1j and jα2j depend on the UV eracy of the kinetic matrix. Therefore, we consider it completion of GR, i.e., quantum gravity. In the lack of worthwhile to revisit higher curvature theories in light of complete understanding of quantum gravity, we should not these recent developments, in particular by combining the a priori exclude the possibility of large jα1j or/and jα2j.If ideas behind these recent successes and seeking possible jα1j ≫ 1 or/and jα2j ≫ 1 and if terms in of (2.1) are still
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2 Γ suppressed by Mpl then the quadratic curvature corrections ν ν ν ρ ∇ μA ¼ ∂μA þ ΓρμA : ð2:9Þ can give relevant contributions even in the classical regime below Mpl and can give various interesting phenomenology We suppose that all gravitational degrees of freedom such as Starobinsky inflation [1]. However, it is known that (d.o.f.) except the standard massless spin-2 graviton are the theory (2.1) generically has a massive spin-2 ghost in massive so that the fifth force, existence of which is addition to a massive spin-0 degree and the massless spin-2 strongly constrained by the solar system experiments graviton. (see [16] for example), does not appear in the low energy A possible solution to the spin-2 ghost problem is to limit. In this case, it is useful to decompose the general assume the hierarchy jα1j ≫ jα2j (with α3 ¼ 0) as usually μ affine connection Γαβ into the Riemannian part and the postulated in the literature so that the mass of the spin-2 μ deviations from it, namely the Levi-Civita connection fαβg ghost is much heavier than that of the spin-0 degree. In this μ and the distortion tensor κ αβ: case, one can safely integrate out the massive spin-2 ghost in the regime of validity of the EFT describing the physics n μ o Γμ κμ : : of the spin-0 degree. In practice, if we assume α2 ¼ 0 then αβ ¼ αβ þ αβ ð2 10Þ the massive spin-2 ghost is infinitely heavy and the gravitational sector of the low energy EFT has the massless The torsion and the nonmetricity are then expressed by spin-2 graviton and the massive spin-0 degree only. In the present paper, on the other hand, we shall deform the theory μ 2κμ αβ 2κðαβÞ T αβ ¼ ½βα ;Qμ ¼ μ; ð2:11Þ so that the massive spin-2 mode becomes nonghost while keeping the stable massive spin-0 mode and the massless or, conversely, the distortion is given by spin-2 graviton. A new ingredient in the present paper that 1 makes such deformation possible is an extension of the κμ − μ − μ μ spacetime geometry. αβ ¼ 2 ðT αβ Tβ α þ Tαβ Þ We shall treat the metric and the connection as inde- 1 − μ − μ − μ pendent variables. The geometry with a metric and a 2 ðQ αβ Qβ α Qαβ Þ: ð2:12Þ generic affine connection is called the metric-affine geom- etry. In the low energy regime, it would be sufficient to Hence, the Lagrangian can be rewritten as include terms up to scaling dimension four. The Lagrangian is generally given by LG ¼ LGðg; R; T; Q; ∇T;∇QÞ; ð2:13Þ
2 μ ∇ LG ¼ LGðg; ∂g; ∂ g; Γ; ∂ΓÞ; ð2:4Þ where R αβγ and μ are the Riemann curvature and the covariant derivatives defined by the Levi-Civita connec- where the scaling dimensions of the metric and the tion. In the form (2.13), we can regard the torsion tensor α connection are supposed to be ½gαβ ¼0 and ½Γβγ ¼1, and the nonmetricity tensor as independent variables respectively. Since the partial derivative is not a covariant instead of parts of the connection. In energy scales below quantity, the diffeomorphism invariance then leads to the the masses of the torsion and the nonmetricity, all compo- requirement that the Lagrangian should be built out of nents of them may be integrated out; the gravitational geometrical quantities and their covariant derivatives as theory is then represented by the metric only. Therefore, in the context of particle physics, what we did here is Γ Γ Γ introducing new heavy particles in UV regime (see [17] L L ∇ ∇ G ¼ Gðg; R;T;Q; T; QÞ; ð2:5Þ for more details on this point). where B. Metric compatible theory Γ μ μ μ μ σ μ σ Hereafter, we impose the metric compatibility condition R ναβðΓÞ ≔ ∂αΓνβ − ∂βΓνα þ ΓσαΓνβ − ΓσβΓνα; ð2:6Þ αβ Qμ ¼ 0; ð2:14Þ μ μ μ T αβ ≔ Γβα − Γαβ; ð2:7Þ just for simplicity. The geometry with a metric and a metric Γ compatible connection is then called Riemann-Cartan αβ αβ Qμ ≔ ∇ μg ; ð2:8Þ geometry. This geometry particularly has gained attention a in the literature since one can choose the tetrad e μ and ab ½ab are the curvature, the torsion, and the nonmetricity tensors, the anti-symmetric spin connection ω μ ¼ ω μ, where respectively. The covariant derivative of a vector is a; b; are the Lorentz indices, as independent variables defined as and can regard them as gauge fields associated with the
064061-3 KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) local translation and the local rotation, respectively. In this ð2Þ 2 Γ T μνρ ¼ gμ½νTρ ; ð2:21Þ 1 ab μ ν 3 picture, the curvature two-form 2 R μνdx dx and the 1 a μ ν torsion two-form 2 T μνdx dx are interpreted as the field ð3Þ strengths of them (see [18,19] for reviews). Theories whose σ T μνρ ¼ ϵμνρσT ; ð2:22Þ Lagrangian is algebraically constructed by the curvature and the torsion are called the Poincar´e gauge theories (PGTs). On the other hand, in the present paper we shall include not only all the ingredients used in the Lagrangian ð1Þ ð2Þ ð3Þ of PGTs but also terms quadratic in derivatives of the T μνρ ¼ Tμνρ − T μνρ − T μνρ ð2:23Þ torsion in the Lagrangian because there would be no reason to exclude them since both the curvature squared and the with torsion derivative squared have scaling dimension four. We assume that the cosmological constant vanishes to ν Tμ ≔ T νμ; ð2:24Þ admit the Minkowski spacetime as a vacuum solution and study linear perturbations around the Minkowski back- 1 νρσ ground. The general parity preserving Lagrangian up to T μ ≔ ϵμνρσT : ð2:25Þ scaling dimension four is then given by the following 6 pieces ð1Þ The irreducible piece T μνρ satisfies L L L 2 L 2 L 2 L G ¼ R þ T þ R þ ð∇TÞ þ R∇T
þ L 2 þ L ∇ 2 þ L 4 : ð2:15Þ ð1Þ ð1Þ ð1Þ RT ð TÞT T 0 0 μ 0 T μðνρÞ ¼ ;T½μνρ ¼ ;Tμν ¼ ð2:26Þ The last three parts, which include terms schematically represented by the suffixes, are irrelevant to linear pertur- and thus has 16 independent components. As we have bation analysis around the Minkowski spacetime. By the already stated, in light of the Gauss-Bonnet theorem we set use of the equivalence upon integration by parts and α3 ¼ 0 without loss of generality. Then, the remaining 16 α β the Bianchi identity, the most general expressions of the dimensionless parameters ð i; i;ai;biÞ characterize linear relevant terms are perturbations in addition to the two mass parameters Mpl L 2 and MT. The terms T act as the mass terms of the torsion. 2 M Hence, we assume ai ≠ 0 throughout in order to recover L pl R ¼ 2 R; ð2:16Þ GR in the low energy limit. There are a few general remarks on the Lagrangian 2 μν μνρσ (2.15). (i) Solutions of (2.15) generally have a nonvanish- L 2 α α α R ¼ 1R þ 2RμνR þ 3RμνρσR ; ð2:17Þ ing torsion because LR∇T gives source terms of the torsion equation of motion. Since the source terms are ð1Þ μνρ μ proportional to LR∇T ¼ β1Rμν∇ρ T þ β2R∇μT ; ð2:18Þ ∇ ∇ 2 ð1Þ ð1Þ ρRμν; μR; ð2:27Þ MT μνρ μ μ L 2 T T T ¼ 2 ða1 T T μνρ þ a2T Tμ þ a3 νÞ; ð2:19Þ R0 an Einstein manifold Rμν ¼ 4 gμν with a constant R0 can be ð1Þ ð1Þ ð1Þ ð1Þ a torsionless solution. In particular, Ricci flat spacetimes μ νρσ μρσ ν L 2 ∇ ∇ ∇ ∇ ð∇TÞ ¼ b1 μ T νρσ T þ b2 μ T ν T ρσ with vanishing torsion are vacuum solutions of (2.15). ð1Þ ð1Þ (ii) In the case of β1 ¼ β2 ¼ 0, the metric perturbations and ρσμ ν μ ν þ b3∇μ T ∇ν T ρσ þ b4∇μTν∇ T the torsion perturbations are decoupled from each other at μ ν μ ν μ ν linear order around the Minkowski background. This case þ b5∇μT ∇νT þ b6∇μT ν∇ T þ b7∇μT ∇νT is less interesting because the massive spin-2 ghost exists as ð1Þ ð1Þ μνρ μνρσ α long as α2 ≠ 0. (iii) When we take the limit MT → ∞ with þ b8∇μ T ∇νTρ þ b9ϵ ∇α T μν∇ρT σ; finite βi and bi, all torsional d.o.f. become infinitely heavy ð2:20Þ and thus can be integrated out, and the theory is represented by the metric only. Then, the Lagrangian (2.1) is obtained where αi, βi, ai, and bi are dimensionless parameters. The as an effective theory. torsion tensor has been decomposed into three irreducible Before finishing this section, it would be worth mention- pieces, ing the relation to the parity preserving quadratic PGT
064061-4 GHOSTFREE QUADRATIC CURVATURE THEORIES WITH … PHYS. REV. D 100, 064061 (2019) Γ Γ Γ Γ Γ Γ Γ L λ μν − νμ r1 r2 μνρσ PGTþ ¼ R þðr4 þ r5ÞR μνR þðr4 r5ÞR μνR þ 3 þ 6 R R μνρσ 2 2 Γ Γ Γ Γ r1 − r2 μνρσ r1 r2 − μνρσ þ 3 3 R R μρνσ þ 3 þ 6 r3 R R ρσμν λ λ 2 t1 t2 μνρ − − t1 t2 μνρ −λ − t1 t3 μ ν þ 4 þ 3 þ 12 T Tμνρ þ 2 3 þ 6 T Tνρμ þ 3 þ 3 T μρT νρ; ð2:28Þ Γ where the term R2 is omitted by the use of the Gauss-Bonnet theorem. We follow the parametrization used in [20,21]. The Lagrangian has 8 parameters in addition to the gravitational constant λ. This Lagrangian contains up to the first derivative of the tetrad and the antisymmetric spin connection, while the previous Lagrangian (2.15) contains the second derivative of the tetrad due to the presence of the derivatives of the torsion. Therefore, PGTs are clearly a subset of the general Lagrangian (2.15). Indeed, (2.28) is obtained from (2.15) by setting
M2 4 λ pl α − α 2 2 − 2 β −4 2 − 2 β − − 2 ¼ 2 ; 1 ¼ r1 þ r3; 2 ¼ ð r1 r3 þ r4Þ; 1 ¼ ð r1 r3 þ r4Þ; 2 ¼ 3 ðr1 r3 þ r4Þ; 1 4 3 2λ − 2λ − − 2λ a1 ¼ 2 ðt1 þ Þ;a2 ¼ 3 2 ðt3 Þ;a3 ¼ 2 ðt2 Þ; MT MT MT 1 −4 4 − 2 − 2 b1 ¼ r1;b2 ¼ 2 ð r1 þ r3 r4 þ r5Þ;b3 ¼ ðr1 r3 þ r4Þ; 4 4 3 2 − 3 5 − −2 − − 2 b4 ¼ 9 ðr1 þ r4 þ r5Þ;b5 ¼ 9 ð r1 r3 þ r4 r5Þ;b6 ¼ r3 r5;b7 ¼ 2 r2 þ r3 þ r5; 4 − 2 b8 ¼ 3 ðr1 þ r4 þ r5Þ;b9 ¼ r3 þ r5; ð2:29Þ
μ TT TTμ μ T and appropriately choosing the remaining Lagrangian ∂ hμν ¼ 0;hμ ¼ 0; ∂ hμ ¼ 0: ð3:3Þ L 2 L 2 L 4 RT , ð∇TÞT , and T , where we have used the Gauss- Bonnet theorem to eliminate the Riemann squared term. The torsion can be also decomposed into
T III. GRAVITATIONAL DEGREES OF FREEDOM Tμ ¼ Aμ þ ∂μϕ; ð3:4Þ AND CRITICAL CONDITIONS T A. Quadratic Lagrangian T μ ¼ Aμ þ ∂μφ; ð3:5Þ We study perturbations around the Minkowski back- 1 ð Þ ∂μ∂ ν 1 ground TT ½ T T μνρ 2∂ νt 2 − ημν B ¼ ½ ρ μ þ □ 3 ρ gμν ¼ ημν þ δgμν; ð3:1Þ ∂ ∂ 1 αβ TT β μ T þ ϵνρ ∂ατ þ − ηβμ Bα : ð3:6Þ where we consider the torsion itself as a perturbation βμ □ 3 quantity since the background is torsionless. The metric perturbation can be decomposed into These decompositions classify the metric perturbations and 1 1 the torsion with respect to their spins and parities. TT T δgμν hμν 2∂ μh ∂μ∂ν − ημν□ σ ημνh; The quadratic action around the Minkowski background ¼ þ ð νÞ þ 4 þ 4 is then given by ð3:2Þ ð2Þ L L L − L L − L L − where the suffix TT of a tensor stands for the transverse- G ¼ 2þ þ 2 þ 1þ þ 1 þ 0þ þ 0 ; ð3:7Þ traceless and the suffix T of a vector stands for the transverse, that is, where
1 2 1 1 αβ 1 M þ α2□ β1□ TT TT TT 4 pl 2 2 h L2þ ¼ ðh ;t Þ □ ; ð3:8Þ 2 αβ αβ −2 2 2 2 □ TTαβ a1MT þ ð b1 þ b3Þ t
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1 2 αβ L − τTT 2 − 4 □ □τTT 2 ¼ 2 αβ ð a1MT b1 Þ ; ð3:9Þ
2 4 α 1 a3M − 2b6□ − b9□ AT T T T 3 L1þ ¼ ðAα ; Bα Þ ; ð3:10Þ 2 − 4 2 4 6 4 □ BTα 3 a1MT þ 9 ð b1 þ b2 þ b3Þ
2 2 α 1 a2M − 2b4□ − b8□ T T T T 3 A L1− ¼ ðAα ;Bα Þ ; ð3:11Þ 2 4 2 − 8 3 2 □ Tα 3 a1MT 9 ð b1 þ b2 þ b3Þ B pffiffiffi − 1 2 2 3α α □ − 3β □ Φ 1 2 Mpl þ ð 1 þ 2Þ 2 L0þ ¼ ðΦ; ϕÞ □ ; ð3:12Þ 2 − 2 2 □ ϕ a2MT þ ðb4 þ b5Þ
1 2 L − φ − 2 □ □φ 0 ¼ 2 ð a3MT þ ðb6 þ b7Þ Þ ; ð3:13Þ with
4 1 μν Φ ¼ pffiffiffi ðh − □σÞ¼pffiffiffi θμνh ð3:14Þ 3 3
∂μ∂ν and θμν ≔ ημν − □ . The symbol * is attached to omit the symmetric parts of the matrices. In the momentum space, we define the following quantities 1 2 − 1 α 2 − 1 β 2 2 4 Mpl 2 2q 2 1q K2þ ≔−q ; ð3:15Þ −2 2 − 2 2 2 a1MT ð b1 þ b3Þq
2 2 2 − ≔− 2 4 K2 q ð a1MT þ b1q Þ; ð3:16Þ 2 2 2 4 2 a3MT þ b6q 3 b9q K1þ ≔ ; ð3:17Þ − 4 2 − 4 6 4 2 3 a1MT 9 ð b1 þ b2 þ b3Þq 2 2 2 2 2 a2MT þ b4q 3 b8q K1− ≔ ; ð3:18Þ 4 2 8 3 2 2 3 a1MT þ 9 ð b1 þ b2 þ b3Þq pffiffiffi − 1 2 − 2 3α α 2 3β 2 2 2 Mpl ð 1 þ 2Þq 2q K0þ ≔−q ; ð3:19Þ − 2 − 2 2 a2MT ðb4 þ b5Þq
2 2 2 − ≔− − − 2 K0 q ð a3MT ðb6 þ b7Þq Þ: ð3:20Þ
−1 þ Then, the number of poles of Ka ða ¼ 0 ; 1 ; 2 Þ 2 ∶ 2 þ 1 particle species ð2 massive þ 1 masslessÞ; basically correspond to the number of particle species in 2−∶ 1 particle species ðmassiveÞ; each sector although there are unphysical poles at q ¼ 0 in −1 −1 −1 −1 −1 1þ∶ 2 K2− ;K0− ;K0þ . The unphysical poles of K2− ;K0− are due particle species ðmassiveÞ; to the presence of the derivative in the expressions (3.5) and 1−∶2 −1 particle species ðmassiveÞ; (3.6) while that of K þ is due to not only the derivatives but 0 0þ∶ 2 also the remaining gauge mode. A more rigorous counting particle species ðmassiveÞ; of the number of particle species will be performed based 0−∶ 1 particle species ðmassiveÞ: on the 3 þ 1 decomposition in the next section. For generic choices of parameters, the number of particle species in Here, s on the left denotes the spin s and the parity .In each sector is as follows: any Lorentz-invariant theories the number of local physical
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TABLE I. Critical conditions in general theory. TABLE II. Critical conditions in Poincar´e gauge theory.
Critical conditions Critical conditions þ 2 þ 2 4α2ð2b1 þ b3Þ − β1 ¼ 0 (primary) 2 ð2r1 − 2r3 þ r4Þðt1 þ λÞ¼0 2 2 − 2 − 2 α 0 2 r1 0 ð b1 þ b3ÞMpl a1 2MT ¼ (secondary) ¼ þ 2− 0 1 ð2r3 þ r5Þðt1 þ t2Þ¼0 b1 ¼ − þ 2 1 r1 r4 r5 t1 t3 0 1 b6 6b1 4b2 b3 2b 0 (primary) ð þ þ Þð þ Þ¼ ð þ þ Þþ 9 ¼ þ 6 4 6 0 0 ðr1 − r3 þ 2r4Þðt3 − λÞ¼0 a3ð b1 þ b2 þ b3Þþ a1b6 ¼ (secondary) − − 2 0 r2 ¼ 0 1 4b4ð3b1 þ 2b2 þ b3Þ − b8 ¼ 0 (primary) a2ð3b1 þ 2b2 þ b3Þþ3a1b4 ¼ 0 (secondary) þ 2 0 4ð3α1 þ α2Þðb4 þ b5Þ − 3β2 ¼ 0 (primary) 2 2 3α α 2 0 2þ∶ 1 1 1 1 ðb4 þ b5ÞMpl þ a2ð 1 þ 2ÞMT ¼ (secondary) þ particle species ð massive þ masslessÞ; 0− 0 b6 þ b7 ¼ 0þ∶1 particle species ðmassiveÞ;
and there is no particle species in the 2−, 1 and 0− sectors. m≠0 d.o.f. for each particle species for s ¼ð2; 1; 0Þ is Ns ¼ This is, needless to say, consistent with the well-known m¼0 ð5; 3; 1Þ for massive cases, and Ns ¼ð2; 2; 1Þ for mass- content of particle species for the Riemannian quadratic less cases. Therefore the total number of local physical curvature theory (2.1). d.o.f. is ð2 × 5 þ 1 × 2Þþð1 × 5Þþð2 × 3Þþð2 × 3Þþ 2 1 1 1 32 ð × Þþð × Þ¼ . For the massive cases the masses B. Matter scattering via gravitational interaction of these particles can be evaluated by the locations of a δa δ a the poles. By the use of the tetrad eμ ¼ μ þ eμ and the spin ωab When the coupling constants satisfy a critical condition, connection μ, the matter coupling to gravity is given by a pole vanishes correspondingly to a reduction of the 1 1 number of d.o.f. In the 2þ; 1 ; 0þ sectors, after assuming L δ a Θμ − ωab μ int ¼ e μ a μS ab ð3:21Þ the critical condition, the d.o.f. can be further reduced by 2 imposing an additional critical condition. We shall call μ μ where Θ and S are the canonical energy-momentum these critical conditions the primary critical condition and a ab tensor and the canonical spin tensor defined by the secondary critical condition in order. All critical conditions are summarized in Table I. 1 δ 2 δ Θμ ≔ Sm μ ≔− Sm In the case of PGT, the coupling constants satisfy all a a ;Sab ab : ð3:22Þ det e δe μ det e δω primary critical conditions and then the number of particle μ species in each sector is The canonical spin tensor has the antisymmetric indices Sμ ¼ −Sμ while the canonical energy-momentum ten- 2þ∶ 1 1 1 1 ab ba þ particle species ð massive þ masslessÞ; sor is not symmetric, in general. Around the Minkowski 2−∶ 1 particle species ðmassiveÞ; background, one does not need to distinguish the spacetime α β μ ν þ indices, ; ; ; ; , and the Lorentz indices, a; b; , 1 ∶ 1 particle species ðmassiveÞ; a since they are just transformed by δα. The energy, momen- 1−∶ 1 particle species ðmassiveÞ; tum and angular momentum of the matter are supposed to be conserved on shell: 0þ∶ 1 particle species ðmassiveÞ; − μ μ 0 ∶ 1 particle species ðmassiveÞ; ∂μΘ ν ¼ 0; ∂μS αβ þ Θαβ − Θβα ¼ 0: ð3:23Þ when no additional assumptions are imposed. Other critical On the other hand, after changing the variables a ab conditions in PGT are summarized in Table II. In the limit ðδeμ; ω μÞ → ðδgμν;TμνρÞ, we find M → ∞, the locations of some poles go to infinity and T 1 1 thus the dynamical ones in this limit are L δ μν − μνρ − νρμ ρνμ int ¼ 2 gμνT 4 TμνρðS S þ S Þð3:24Þ
1 The limit to satisfy a critical condition, say b1 → 0, leads to where Tμν is the Belinfante tensor, the infinite mass of the corresponding particle specie. Therefore, in the sense of EFT, the critical conditions are not necessary to μν μν 1 ρμν μρν νμρ hold exactly for the reduction of the number of particle species. It T ¼ Θ þ ∂ρðS − S þ S Þ; ð3:25Þ suffices to satisfy the critical condition approximately, as far as 2 the mass of the corresponding particle species is sufficiently heavier than the energy scale of interest. which is symmetric and conserved
064061-7 KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) μν νμ ∂ μν 0 β2 −1 1 T ¼ T ; μT ¼ : ð3:26Þ −1 11 → 8 4α − 1 −6 ðK2þ Þ 2 4 þ Oðq Þ; ð3:35Þ 2b1 þ b3 q 2þ 0þ Due to the conservation law, only the sectors and μν 2 are mediators of gravity via the coupling δgμνT at tree 3β −1 1 −1 11 → 2 4 3α α − 2 −6 ðK0þ Þ ð 1 þ 2Þ 4 þ Oðq Þ; level. A matter particle with a nonvanishing canonical spin b4 þ b5 q tensor is scattered by exchanging not only the graviton but also the torsion, called tordion, via the coupling to the spin ð3:36Þ tensor. As an example, let us consider the Dirac field which suggests there would be Ostrogradsky ghosts. On the Γ Γ other hand, when the primary critical conditions are L i ψγ¯ μ∇ ψ − ∇ ψ¯ γμψ − ψψ¯ D ¼ 2 ð μ ð μ Þ Þ m ; ð3:27Þ imposed 2 4α2 2b1 b3 − β 0; where γa is the gamma matrix with fγa; γbg¼−2ηab. The ð þ Þ 1 ¼ covariant derivative of a Dirac spinor is 2 4ð3α1 þ α2Þðb4 þ b5Þ − 3β2 ¼ 0; ð3:37Þ Γ 1 ab the Ostrogradsky ghost modes may be eliminated and then ∇ μψ ¼ ∂μ þ ω μ½γa; γb ψ: ð3:28Þ 8 the propagator recovers a q2 behavior.2 We recall that these conditions are automatically satisfied in PGTs. This could The canonical spin tensor of a Dirac field is given by be understood by the fact that PGTs do not have the second derivatives of the tetrad in the Lagrangian (although this μνρ 1 μνρσ 5 S ¼ ϵ jσ; ð3:29Þ property is certainly spoiled by radiative corrections), and 2 thus they are not higher derivative theories. The graviton 5 5 propagator is then where jμ ¼ ψγ¯ μγ ψ is the axial current which is not con- served in general. As a result, the gravitational coupling is 2 4 8m þ α2 −i 2 − Δhh D 2 ð Þ i μν;ρσ ¼ 2 μν;ρσ þ 4 2 2 Pμν;ρσ 1 3 M M q þ m þ L δ μν − T μ 5 pl pl 2 int ¼ 2 gμνT 4 jμ; ð3:30Þ 2 8m0− ð3α1 þ α2Þ −i ð0Þ þ 4 2 2 Pμν;ρσ; ð3:38Þ and then the particle species with 1þ and 0− in the gravity Mpl q þ m0þ sector are mediators of gravity as well. D For simplicity, we consider a matter field whose canoni- where μν;ρσ is the graviton propagator of the Einstein cal spin tensor vanishes, e.g., a minimal scalar field. The gravity, tree level scattering amplitude is then − 1 D i θ θ − θ θ μν;ρσ ¼ 2 μðρ σÞν μν ρσ : ð3:39Þ ˜ μν hh ˜ ρσ q 2 M ¼ T ðp1;p2ÞΔμν;ρσðqÞT ðp3;p4Þ; ð3:31Þ where T˜ μν is the Fourier transform of the source with the The masses of the massive spin-2 mode and of the massive spin-0 mode are external momenta pi (i ¼ 1,2,3,4)andq is the internal momentum. The gauge independent part of the graviton 2 2 4a1α2M M propagator is m2 pl T ; 2þ ¼ 2 β2 − 8 α2 2 Mpl 1 a1 2MT 2 0 − Δhh −1 11 ð Þ −1 11 ð Þ 2 2 i μν;ρσ ¼ iðK2þ Þ Pμν;ρσ þ iðK0þ Þ Pμν;ρσ; ð3:32Þ 2a2ð3α1 þ α2ÞM M m2 pl T : : 0þ ¼ 8 3α α 2 2 3β2 2 ð3 40Þ −1 11 −1 11 a2ð 1 þ 2Þ MT þ 2Mpl where ðK2þ Þ and ðK0þ Þ are the upper left component −1 −1 ð2Þ ð0Þ 2 μν ρσ μν ρσ ≫ → of K2þ ;K0þ . The projection operators P ; ;P ; are In the limit MT Mpl, the masses become m2þ − 2 2α 2 → 2 4 3α α defined by Mpl= 2 and m0þ Mpl= ð 1 þ 2Þ, and then the 1 propagator becomes ð2Þ ≔ θ θ − θ θ Pμν;ρσ μðρ σÞν 3 μν ρσ; ð3:33Þ 2Precisely, the second condition of (3.37) is not required to be 1 free from the Ostrogradsky ghost. There is a massless spin-0 ð0Þ ≔ θ θ Pμν;ρσ 3 μν ρσ: ð3:34Þ ghost but this massless ghost is harmless since this is just related to the remaining gauge mode as mentioned before (see [22] for −4 example). It will, however, turn out in the next section that the In general, the propagator has a q behavior in the high second condition is indeed required in order that all dynamical energy limit, modes are ghostfree. We therefore assume the second one as well.
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4 4 − 2 hh i ð Þ ð1Þ ð1Þ −iΔμν;ρσj →∞ ¼ Dμν;ρσ − Pμν;ρσ MT 2 2 2 2 ϵ Bk Mpl Mpl q þ m2þ T 00i ¼ Bi;T0ij ¼ ijk ; ð1Þ ð1Þ 2 −i ð0Þ 1 Pμν ρσ: 3:41 − ϵ Bk δ ϵ lτ þ 2 2 2 ; ð Þ T ij0 ¼ tij 2 ijk ;Tijk ¼ i½jBk þ jk il; ð4:4Þ Mpl q þ m0þ τ This is indeed the well-known graviton propagator of the where tij and ij are symmetric and traceless. The metric δ − η quadratic curvature theory (2.1) with a massive spin-2 perturbations gμν ¼ gμν μν are ghost. We note that the spin-2 part of (3.41) has a q−4 δ −2α δ β δ 2 ζδ behavior in the high energy limit which makes the theory g00 ¼ ; g0i ¼ i; gij ¼ ðhij þ ijÞ; renormalizable although the corresponding energy scale ð4:5Þ goes beyond the mass of the ghost mode [23]. −4 One may recover the q behavior if one removes i ij where h i ¼ δ hij ¼ 0. the critical conditions (3.37). However, in this case the ð2Þ When α2 ≠ 0, the quadratic order Lagrangian L Ostrogradsky ghosts appear below the scale at which the −4 contains the second time derivative of hij. It is then useful q behavior is realized. Therefore, in the present paper, we to eliminate second time derivatives and to reduce the shall not discuss the renormalizability of graviton loops any action to the form that depends on perturbation variables more. Instead, we simply interpret the scale at which a 3 and their first derivatives only by introducing auxiliary ghost appears as the cutoff scale of the theory. fields and by writing the action as
2 ̈ IV. STABILITY CONDITIONS L0 Lð Þ ̈ H Φij − hij ; 4:6 ¼ jhij¼Φij þ ijð Þ ð Þ The propagator (3.38) implies that the theory can be 2 0 2 0 α 0 Φ ghostfree if the inequalities m2þ > , m0þ > and 2 > , where Hij; ij are symmetric and traceless. The variation 3α α 0 ̈ 1 þ 2 > are satisfied. However, gravity is also medi- with respect to Hij yields the constraint Φij ¼ hij and thus ated by particles of other sectors when the canonical spin the new Lagrangian is equivalent to the original one. tensor does not vanish. We thus investigate ghostfree Instead, one can take the integration by part so that conditions for all sectors of the Lagrangian (2.15) under 2 _ _ij the assumptions L0 ¼ Lð Þj ̈ þ H Φij þ H h þðtotal derivativeÞ; hij¼Φij ij ij
α1 ≠ 0; α2 ≠ 0: ð4:1Þ ð4:7Þ
We have already set α3 ¼ 0 by using the Gauss-Bonnet and can take the variation with respect to Φij. One then theorem. obtains the constraint which determines Φij in terms of As discussed in the previous section, there still exists a other variables. After substituting the constraint into the remaining gauge mode when the metric perturbations are Lagrangian, one finds the Lagrangian with up to the first decomposed as (3.2). We thus give up the explicit Lorentz order time derivative of h . 3 1 ij invariance here; instead, we adopt the þ decomposi- In the SVT decompositions, a scalar, a vector and a tions and then decompose perturbations into scalar-vector- traceless tensor are decomposed into scalar type (S), vector tensor (SVT) components commonly used in the context of type (V), and tensor type (T) perturbations in terms of the the cosmological perturbation theory. Although we discuss spatial rotation: the perturbations around the Minkowski background, the 3 þ 1 decomposition can be straightforwardly extended to ðVÞ ϕ ¼ ϕðSÞ;A¼ ∂ AðSÞ þ A ; an arbitrary spacetime (see Appendix A). i i i We denote the 3 þ 1 components of the torsion as 1 2 ðVÞ ðTÞ t ¼ ∂ ∂ − δ ∂ tðSÞ þ ∂ t þ t ; ð4:8Þ follows: ij i j 3 ij ði jÞ ij ϕ T0 ¼ ;Ti ¼ Ai; ð4:2Þ where
T 0 ¼ φ; T ¼ A ; ð4:3Þ i i ∂i ðVÞ ∂i ðVÞ 0 ∂i ðTÞ 0 ðTÞi 0 Ai ¼ ti ¼ ; tij ¼ ;ti ¼ ; ð4:9Þ
3Alternatively, there are attempts to obtain the Lorentzian and SVT perturbations are decoupled from each other at the metric signature from a Euclidean metric as a macroscopic linear order in perturbations. The spatial dependence of the effective description [24,25]. In this case, higher derivatives do variables can be Fourier transformed and then each Fourier not necessarily lead to the ghost problem. Another attempt is a i modification of a quantization prescription to turn a ghost particle sector is characterized by the spatial momentum k .For into a fake particle [26]. each ki, the vector type and the tensor type perturbations
064061-9 KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) have two different modes which can be chosen as the the (true) scalar perturbations, and the pseudoscalar pertur- helicity 1 modes and the helicity 2 modes, respectively. bations. In Appendix B, we summarize how to confirm i i Let us denote two helicity basis YRðkÞ;YLðkÞ for vectors absence or existence of a ghost instability and how to ij ij evaluate masses of the particles from the quadratic and YR ðkÞ;YL ðkÞ for tensors, respectively. The vector ðVÞ ðVÞ ðVÞ Lagrangian of each sector. perturbations (Ai , Bi , ti ) and the pseudovector per- turbations (AðVÞ, BðVÞ, τðVÞ) can be decomposed as i i i A. Tensor perturbations ðVÞi i i A ¼ ALYL þ ARYR; As discussed in Sec. III, there generally exists the Ostrogradsky ghost due to the presence of higher time BðVÞi ¼ B Yi þ B Yi ; L L R R derivatives. However, this can be removed by imposing the ðVÞi ðVÞ i ðVÞ i critical condition t ¼ tL YL þ tR YR; ð4:10Þ
2 AðVÞi A i − A i 4α2 2b1 b3 − β 0: 4:17 ¼ LYL RYR; ð þ Þ 1 ¼ ð Þ BðVÞi B i − B i ¼ LYL RYR; This kind of conditions is called degeneracy conditions in τðVÞi τðVÞ i − τðVÞ i the context of higher order scalar-tensor theories [11] since ¼ L YL R YR: ð4:11Þ these are conditions under which the kinetic matrix is ðTÞ degenerate. In Appendix C, we study a toy model that Similarly, tensor perturbations t and pseudotensor per- ij explains the reason why higher time derivatives lead to the τðTÞ turbations ij are decomposed as Ostrogradsky ghost and the reason why the ghost can be removed by the degeneracy condition. In this sense, what ðTÞij ij ij t ¼ tLYL þ tRYR ; ð4:12Þ we seek here is a degenerate higher order spin-2 theory. After integrating out Φij from (4.7), we confirm that τðTÞij τ ij − τ ij ¼ LYL RYR : ð4:13Þ there exists the Ostrogradsky ghost when (4.17) is not imposed. We then impose (4.17) under which the tensor Note that there are couplings such as perturbations of tij are nondynamical. After integrated out tij, the quadratic Lagrangian of the tensor perturbations ijk ijk l ϵ Ai∂jAk; ϵ til∂jτk ; ð4:14Þ becomes
2 due to the presence of the pseudovectors and the pseudo- M 2 L pl ˜_ − 2 ˜ 2 tensors which give mixings between vectors and pseudo- T ¼ 2 ½hij k hij vectors and those between tensors and pseudotensors, e.g., 2 2 1 A ð b1 þ b3Þ _ 2 2 2 2 couplings between AL and L. Needless to say, the L sector − H − k m H þ β2 2 2 2 ½ ij ð þ 2þ Þ ij and the R sector are still decoupled from each other. Also, a1 1MT Mpl the L sector and the R sector obey the same equations of 2 2 − a1b1MT τ˜_2 − 2 a1 2 τ˜2 motion because of the absence of parity violating operators. 2 2 ij k þ MT ij ; ð4:18Þ 2b1k þ a1M 2b1 In order to make this fact manifest, the minus sign has been T inserted in front of the R modes in the above decomposition with new variables of the pseudovector and pseudotensor perturbations so that 1 1 ˜ − ϵijk∂ AðVÞ A i A i hij ¼ hij þ 2 Hij; ð4:19Þ j ¼ LY þ RY ; ð4:15Þ M k k L R pl
1 2b1 þ b3 − ϵðijjk∂ τðTÞ lÞ ¼ τ Yil þ τ Yil; ð4:16Þ τ˜ τ − ϵ ∂kH l; : j k L L R R ij ¼ ij β 2 ðijkl jÞ ð4 20Þ k a1 1MT and so on. Contrary to the vector and tensor perturbations, where the suffix (T) has been dropped for simplicity of the scalar perturbations have only one helicity mode. The 2 2 notation. In the limit k ≫ a1M =b1, H or τ˜ is a ghost parity preservation then guarantees that the (true) scalar T ij ij depending on whether a1 < 0 or a1 > 0. In the case a1 < 0, perturbations and the pseudoscalar perturbations are one cannot assume the critical condition of 2þ to remove decoupled. As a result, the general perturbations can be the ghost H . Therefore, we have to assume a1 > 0 and the classified into four decoupled sectors, namely the tensor ij 2− perturbations (which include both the truetensor and pseu- critical condition of the sector, dotensor perturbations), the vector perturbations (which 0 include both truevector and pseudovector perturbations), b1 ¼ : ð4:21Þ
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Regardless of whether the massive 2þ mode is dynamical primary critical conditions and thus they are automatically or not, the 2− sector must be nondynamical in order to be classified into Class I. − þ ghostfree if α2 ≠ 0. In Class I theories, the critical conditions of 2 and 1 As a result, the stability conditions of the tensor have been assumed and thus the pseudoscalar perturbations perturbations require the conditions (4.17) and (4.21), must have the particles with 1þ and 0− only. Indeed, we namely the primary critical condition of 2þ and the critical confirm that there are two dynamical variables with masses condition of 2−. The ghostfree and tachyonfree conditions 3 are then 2 a1a3 2 m1þ ¼ MT; a3ð4b2 þ b3Þþ6a1b6 2 2 β1 2 2 a3 2 α 0 0 m − ¼ M ; ð4:25Þ 2 > ;
2 1. Pseudoscalar perturbations a3 < 0; ð4b2 þ b3Þb6 þ 2b9 < 0; 4b2 þ b3 > 0:
There are four variables from Ai; Bi; φ and τij in the ð4:28Þ pseudoscalar perturbations. We have seen that the ghostfree condition of the tensor perturbations requires b1 ¼ 0 so that The masses of these particles are given by the roots of τ ij is nondynamical. After integrating out pseudoscalar 2 4 f4b9 þ 2b6ð4b2 þ b3Þgm perturbations of τij, we still find a ghost instability in the − 4 6 2 2 3 4 0 limit k → ∞. We thus assume one of the conditions fa3ð b2 þ b3Þþ a1b6gm MT þ a1a3MT ¼ :
2 ð4:29Þ b6ð4b2 þ b3Þþ2b9 ¼ 0; ð4:23Þ In particular, when b9 ¼ 0, T μ is decoupled and the masses or are 3 b6 þ b7 ¼ 0; ð4:24Þ 2 a3 2 2 a1 2 mT ¼ MT;m1− ¼ MT: ð4:30Þ 2b6 4b2 þ b3 namely the primary critical condition of 1þ or the critical condition of 0−. In either case, the pseudoscalar perturba- It is worth mentioning that both Class I and Class II tions can be stable in an appropriate parameter space. Let us theories have to satisfy the inequality call theories with (4.23) Class I and theories with (4.24) 4 0 Class II, respectively. We recall that PGTs satisfy all b2 þ b3 > ð4:31Þ
064061-11 KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) to be free from the ghost. Combining the ghostfree ghost d.o.f. On the other hand, when the condition (4.33) is condition of the tensor mode b3 > 0, one obtains an firstly assumed without imposing (4.34), there is a non- inequality dynamical variable which is a linear combination of the variables ðHij;hij;Ai;BiÞ. The variation with respect to it 2b2 þ b3 > 0 ð4:32Þ yields a constraint equation which generally determines the nondynamical variables in terms of other variables. which will be imposed hereafter, especially for the analysis However, after the nondynamical variable is eliminated by of stability conditions of the scalar perturbations. using the constraint equation, the ghost cannot be removed. The only way to remove the ghost is to impose the 2. Scalar perturbations conditions (4.34) and (4.35) simultaneously so that the 5 We then analyze (true) scalar perturbations which gen- nondynamical variable becomes a Lagrangian multiplier. erally contain d.o.f. from the 2þ; 1−; 0þ sectors. Note that Imposing three critical conditions (4.33), (4.34), and there are gauge d.o.f. in scalar perturbations. We choose the (4.35), we find that there are two nondynamical variables, gauge βi ¼ 0, ζ ¼ 0 so that the quadratic order Lagrangian one of which is a Lagrangian multiplier. Therefore, three of 4 ϕ (4.7) contains time derivatives up to first order. After five variables ðhij;Hij;Ai;Bi; Þ can be eliminated by eliminating nondynamical variables ðα;tij; ΦijÞ under the solving the constraint equations. We finally obtain the ghostfree conditions obtained from tensor perturbations, quadratic order Lagrangian which contains two dynamical we find a quadratic Lagrangian in terms of the variables d.o.f. from the 2þ; 0þ sectors. The stability conditions are, 2 0 2 0 α 0 ðhij;Hij;Ai;Bi; ϕÞ and confirm that there is still a ghost, in as expected, the inequalities m2þ > , m0þ > , 2 > , general. 3α1 þ α2 > 0. Note that a1 > 0, 2b2 þ b3 > 0 have to be We do not assume the secondary critical condition of 2þ satisfied from the stability conditions of the tensor and so that the massive spin-2 particle is dynamical. There are pseudoscalar perturbations which then lead to b4 > 0, then two possibilities to get a degenerate kinetic matrix: a2 < 0 via (4.33) and (4.35). As a result, the inequality 1− 2 0 imposing the critical condition of m0þ > is guaranteed by
2 2 4b4 2b2 b3 − b 0; 4:33 3 β ð þ Þ 8 ¼ ð Þ 2 − 2 2 a2MT < 2 Mpl: ð4:37Þ 8 ð3α1 þ α2Þ or imposing the critical condition of 0þ,
2 4ð3α1 þ α2Þðb4 þ b5Þ − 3β2 ¼ 0; ð4:34Þ C. Vector perturbations We assume the ghost and tachyonfree conditions both of which lead to a degenerate kinetic matrix. Contrary obtained from the tensor, scalar, and pseudoscalar pertur- to the pseudoscalar perturbations, it will turn out that both bations. It is then relatively straightforward to confirm that conditions (4.33) and (4.34) have to be simultaneously there is no unstable modes in the vector perturbations in this imposed to obtain a ghostfree theory. Furthermore, we have case. We also confirm that Class I theory has two dynamical 1− to also impose the secondary critical condition of the d.o.f. in the vector perturbations while Class II theory has sector three dynamical d.o.f. This is consistent with the discussion of Sec. III: vector perturbations of Class I theory are 3 2 0 a1b4 þ a2ð b2 þ b3Þ¼ : ð4:35Þ originated from 2þ and 1þ while those of Class II are from 2þ and two of 1þ. For instance, let us assume (4.33) is satisfied so that ϕ becomes nondynamical. After integrating out ϕ, we find that there still exists a ghost mode in the limit k → ∞. D. Summary of the stability conditions Therefore, we further require the degeneracy of the kinetic Let us summarize the conditions for Class I and Class II matrix. There are again two possibilities to degenerate the theories and their particle contents. The common critical kinetic matrix; one is, of course, the condition (4.23) and conditions are the other is the secondary critical condition of 0þ, b1 ¼ 0; ð4:38Þ 2 2 3α α 2 0 ðb4 þ b5ÞMpl þ a2ð 1 þ 2ÞMT ¼ : ð4:36Þ 2 4α2b3 − β1 ¼ 0; ð4:39Þ Even if (4.36) is assumed, the ghost cannot be removed. Therefore, we have to assume (4.33) in order to remove the 5The linearity of the action with respect to a nondynamical variable is indeed crucial for the removal of the Boulware-Deser 4 Even if choosing another gauge, say βi ¼ 0, hij ¼ 0, the ghost in massive spin-2 theories (see [27,28] for reviews and second time derivatives of ζ disappear after integrating out α. references therein).
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2 same order of the magnitude, the modulus of ai can be 4ð3α1 þ α2Þðb4 þ b5Þ − 3β2 ¼ 0; ð4:40Þ Oð1Þ by normalizing MT. The ghostfree conditions then α ∼ β2 ∼ α−1β 2 constrain b and MT Mpl. As a result, all 4b4ð2b2 þ b3Þ − b ¼ 0; ð4:41Þ 8 α−1=2 massive particles have masses of the order of Mpl. Most of the critical conditions are the primary ones. In 3a1b4 þ a2ð2b2 þ b3Þ¼0; ð4:42Þ particular, all primary critical conditions have to be and both theories must satisfy the inequalities imposed in Class I theory. Recall that the primary critical conditions are satisfied in PGTs and there are following relations 3α1 þ α2 > 0; ð4:43Þ
b1 ¼ r1; ð4:51Þ α2 > 0; ð4:44Þ
3b1 þ 2b2 þ b3 ¼ r1 þ r4 þ r5; ð4:52Þ 4b2 þ b3 > 0; ð4:45Þ 3 β2 2 and 2 MT 2 ¼ ; ð4:53Þ −8a2 ð3α1 þ α2Þ 4λ − 2t3 2 2 3 β2 1 β1 0 < M2
064061-13 KATSUKI AOKI and SHINJI MUKOHYAMA PHYS. REV. D 100, 064061 (2019) particles species that generically appear in the propagator without the spin tensor feel different gravitational forces (3.38) cannot be simultaneously nonghost. However, in the due to the 1þ and 0− particle species that mediate gra- extended theories introduced in the present paper, thanks to vitational forces via the spin tensor. This aspect of the no- the new terms in the Lagrangian that depend on derivatives ghost torsionful theory is completely different from the of the torsion, i.e., second derivatives of the tetrad, both torsionless theory. Therefore new types of experimental massive spin-2 and massive spin-0 modes in (3.38) can constraints may be obtained via the tests of the equivalence coexist. principle if the extra particle species in the gravitational sector are light enough. V. CONCLUDING REMARKS Even if the extra particle species in the gravitational sector are relatively heavy, signatures of derivative correc- The present paper has discussed derivative corrections tions to the Einstein gravity in this framework may be to the Einstein gravity in the framework beyond the found in the early universe. For example, it would be Riemannian geometry and found new classes of higher interesting to investigate implications of the extra massive curvature theories with nonghost massive spin-2 and spin-0 particle species to the cosmological collider physics [34] particle species around the Minkowski background. We in the context of the Starobinsky inflationary model. firstly assume that the metric and the connection (or, the However, for quantitative analysis, some extensions of tetrad and the spin connection) are independent. In the the present result are required in order to apply it to cos- particle physics sense, this extension of the spacetime mological scenarios. This is because we have only con- geometry corresponds to the introduction of massive par- sidered the linear perturbations around the Minkowski ticles encoded in the torsion and the nonmetricity tensors. If background and thus our results cannot be applied directly these massive particles are heavy enough to be integrated to other backgrounds, around which terms such as L 2 ; out then the Riemannian description of the spacetime is RT L 2 and L 4 may contribute to the quadratic action for effectively recovered. Just for simplicity, we have consid- ð∇TÞT T ered theories with a metric compatible connection. We then perturbations. For example, to consider the inflationary L 2 L 4 take into account all possible terms in the Lagrangian up to universe, although ð∇TÞT and T could be negligible as scaling dimension four and identify the conditions under long as a torsionless de Sitter background is considered, L 2 which ghosts and tachyons are avoided around the RT has to be taken into account to study quantum Minkowski background. The stability conditions and the fluctuations. The quantitative analysis on the cosmological particle contents of the ghostfree theories are summarized perturbations with such new contributions is certainly in Sec. IV D. A key feature of the new classes of theories is important to seek observational signatures of the theory. that both massive spin-2 and spin-0 particles can coexist However, this is beyond the scope of the present paper and with the massless spin-2 graviton without any instability. we hope to study it in future publications. This cannot be achieved by either the metric theories or the Under the critical conditions that we have found, ghosts PGTs. As a result, the matter is gravitationally scattered by are absent around the flat background in the sense that their exchanging not only the massless spin-2 graviton but also masses are infinitely heavy. In a generic curved back- the additional massive spin-2 and spin-0 particles sourced ground, the ghosts may reappear but should remain heavy by the energy and momentum of the matter. The ghostfree as far as the deviation of the background from the flat one is theories are classified into two classes where the difference not too large. In particular, it is expected that the masses of appears when a matter has a nonvanishing spin tensor like a the ghosts should be roughly proportional to the inverse of Dirac field. Class I theory has additional massive spin-1 the background curvature scale and that the ghosts are and spin-0 particle species which mediate the force sourced therefore harmless in the context of EFT as long as the by the spin of matter fields. On the other hand, Class II curvature is small enough so that the masses of the ghosts theory has two spin-1 particle species which are also are sufficiently heavier than the energy scale of physical the mediators via the coupling to the spin. All massive processes of interest. There is also a possibility that all particles would have the masses of the same order of the ghost modes can be removed even around a curved back- L 2 L 2 magnitude when no additional hierarchy of the para- ground by tuning remaining terms such as RT , ð∇TÞT L 4 meters is assumed. For instance, the masses are of order and T . In particular, it is quite interesting if the torsionful of 1013 GeV if the coefficients of the quadratic curvature quadratic curvature theory can be made ghostfree at fully terms are of the same order of the magnitude as the nonlinear order by tuning a series of higher curvature/ Starobinsky inflationary model. derivative terms and resumming them, similarly to the case Therefore, the result of the present paper potentially of massive gravity where the de Rham-Gabadadze-Tolley opens up a new window to phenomenological signatures of (dRGT) theory provides a nonlinear completion of the derivative corrections to the Einstein gravity within the Fierz-Pauli linear theory. regime of validity of an effective field theory EFT. In Theoretically, it is important to investigate the robustness particular, the torsionful quadratic curvature theory predicts of the ghostfreeness against radiative corrections. In the that a matter field with a nonvanishing spin tensor and one context of the scalar-tensor theories, it was recently found
064061-14 GHOSTFREE QUADRATIC CURVATURE THEORIES WITH … PHYS. REV. D 100, 064061 (2019) that an additional local symmetry of the connection is Japan Society for the Promotion of Science Grants-in-Aid associated with the ghostfree structure when the spacetime for Scientific Research No. 17H02890, No. 17H06359, and geometry is extended from Riemannian one [35,36].Similar by World Premier International Research Center Initiative, considerations are certainly worthwhile in the context of MEXT, Japan. higher curvature theories. Another possibility to provide the robustness is a consequence of unitarity, Lorentz invariance, APPENDIX A: 3 + 1 DECOMPOSITION causality, and locality, called the positivity bounds [37]. Introducing a unit timelike vector nμ, one can define the For example, although the dRGT potential is originally spatial metric γμν via discovered by just assuming the ghostfreeness up to the scale Λ3, the positivity bounds do not accept the presence of the γμν ≔ gμν þ nμnν: ðA1Þ ghost at the scale Λ5ð< Λ3Þ and then enforce the structure of the dRGT potential at least up to quartic order [38]. The three dimensional Levi-Civita tensor ϵμνρ is defined by Furthermore, the recent paper [39] argues that the same assumptions predict a leading order deviation to the coupling ϵ ≔ ϵ σ of gravity to fermions that can be explained by the existence μνρ μνρσn ; ðA2Þ of the torsion. Although the present paper has used a bottom- ϵ up approach and thus has not discussed any fundamental where μνρσ is the four dimensional Levi-Civita tensor. 3 1 μ origins of the theory, it would be interesting to investigate Then, the þ components of the vector T are the connections between candidate fundamental theories of ϕ ≔ μ ≔ γμ gravity and their low energy predictions about derivative Tμn ;Aα Tμ α; ðA3Þ corrections to the Einstein gravity in the framework beyond α where Aα is a spatial vector, Aαn ¼ 0. The pseudovector the Riemannian geometry. In particular, it is worthwhile to T 3 1 see if there are any relations between the requirements from μ is similarly decomposed while the þ components of 1 UV theories (such as the positivity bounds) and the critical ð Þ T μνρ are defined by conditions summarized in Table I. Yet another approach to the robustness of the ghostfreeness is to keep the masses of ð1Þ ð1Þ α β ρ μ ν ρ possible Ostrogradsky ghosts to be heavy enough. Namely, ≔ γα ≔ γ γ Bα T μνρn n ;tαβ T μνρ ðα βÞn ; even if radiative corrections introduce failures of the critical 1 ð1Þ 1 ð1Þ conditions and thus reintroduce Ostrogradsky ghosts, one μ νρ μ Bα ≔ T μνρn ϵ α; ταβ ≔ T μνργ αϵβ νρ; ðA4Þ can still make robust predictions as far as the deviations are 2 2 ð Þ small enough to ensure that the masses of the Ostrogradsky where tαβ and ταβ are symmetric and traceless by defini- ghosts are heavier than the energy scale of phenomenologi- cal interest. tion. Conversely, the four dimensional quantities are In summary, we provide two new phenomenologically expressed by viable parameter spaces of the quadratic curvature theory, − ϕ Class I and Class II, by means of the extension of the Tμ ¼ nμ þ Aμ; ðA5Þ spacetime geometry in which not only the massive spin-0 ð1Þ particle and the massless graviton but also other massive 2 − ϵ Bσ − 2 T μνρ ¼ nμn½νBρ nμ νρσ tμ½νnρ spin-0,1,2 particles exist without any instability at least B ϵ σ γ ϵ στ around the Minkowski background. It would be worth þ σ μ½ν nρ þ μ½νBρ þ νρ μσ: ðA6Þ mentioning again that the present theory is not a class of PGTs since the derivatives of the torsion are explicitly included in the Lagrangian which are naturally expected APPENDIX B: GHOSTFREE CONDITIONS from the dimensional analysis. It is thus interesting to AND MASSES explore phenomenological and theoretical aspects of the After expanding the spatial dependence in terms of new type of derivative corrections to the Einstein gravity harmonics and eliminating nondynamical variables, the beyond the Riemannian geometry, which are left for quadratic action of each SVT components is generally future works. given by the form, Z ACKNOWLEDGMENTS X 1 q_TKq_ 2q_TMq − qTVq S2 ¼ dt L; L ¼ 2 ½ þ ; We would like to thank Hideo Kodama for useful comments. The work of K. A. was supported in part ðB1Þ by Grants-in-Aid from the Scientific Research Fund q 1 2 … of the Japan Society for the Promotion of Science where ¼fqigði ¼ ; ; ;NÞ is the set of dynamical (No. 19J00895). The work of S. M. was supported by variables and K, M, V are N × N constant matrices with
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K ≠ 0 V ≠ 0 q 1 2 … q 1 det , det . Here, the suffix T stands for trans- where 1 ¼fqigði ¼ ; ; ;nÞ and 2 ¼fqigði ¼ n þ ; pose as usual. The sum runs over different helicity modes as n þ 2; …;NÞ. We introduce new variables Q1 and write well as spatial momenta. The corresponding Hamiltonian is 1 2QTK q_ K M q − QTK Q 1 L ¼ 2 ½ 1 11ð q þ 11 12 2Þ 1 11 1 H ¼ ½ðp − MqÞTK−1ðp − MqÞþqTVq ; ðB2Þ 2 1 1 q_TK q_ − qTV q þ 2 2 22 2 2 1 11 1 where 1 T T −1 − q ðV22 þ M K M12Þq2; ðB8Þ ∂L 2 2 12 11 p : B3 ¼ ∂q_T ð Þ which is equivalent to the original Lagrangian when Q1 are Therefore, the stability conditions are that K and V are integrated out [40]. We instead take integration by part to _ positive definite in high k limit. The equation of motion is eliminate q1 from the Lagrangian and then integrate q1 out. The Lagrangian is then Kq̈þðM − MTÞq_ þ Vq ¼ 0; ðB4Þ 1 1 _ T −1 _ L ¼ Q1 K11V K11Q1 þ q_2K22q_2 from which we can obtain the dispersion relation via 2 11 2 1 − Q − K−1M q TK Q − K−1M q YN 2 ð 1 11 12 2Þ 11ð 1 11 12 2Þ ω2K ω M − MT − V ∝ ω2 − 2 2 0 det½ þ i ð Þ ½ ðk þ mi Þ ¼ ; 1 i − qTV q 2 2 22 2; ðB9Þ ðB5Þ
2 2 2 which does not yield the friction term. The variables can be ω − 0 where the proportionality to the product of ðk þ mi Þ further changed Q ≔ fQ1; q2g → P Q so that both the is a consequence of the Lorentz invariance. The tachyon kinetic matrix and the mass matrix are diagonalized. 2 ≥ 0 free conditions are obtained from mi . Since we are studying a Lorentz invariant theory, the final One may worry about the existence of the friction terms Lagrangian must be in the equations of motion, that breaks the Lorentz invariance, since the equations of motion must be given 1 1 1 Q_ TIQ_ − 2QTIQ − QT TI Q by forms of the Klein-Gordon equation. However, this is L ¼ 2 2 k 2 M M ðB10Þ just due to the choice of the variable. By means of a change I λ λ … λ λ 1 of the variables q → Pq where P is a nondegenerate where ¼ diagð 1; 2; ; NÞ with i ¼ and M ¼ … matrix, the matrices can be given by following element diagðm1;m2; ;mNÞ. The ghostfreeness of the theory is 6 λ block matrices the positiveness of i and the masses of the particles are mi. However, this requires cumbersome calculations. K11 0 0 M12 The easiest way to evaluate the ghostfreeness and the K M ¼ ; ¼ ; masses is, as explained before, just to check the positive- 0 K22 00 ness of K and V in high k limit and to compute the nodes of V11 0 V ¼ : ðB6Þ the determinant (B5). 0 V22 Note that only the positiveness of K does not guarantee the ghostfreeness. This is because one needs to change the The Lagrangian is thus variables to obtain the Klein-Gordon form (B10) after λ V 1 which the sign of i depends on as well. The positiveness T −1 L ¼ ðq_1 þ K11M12q2Þ K11ðq_1 þ K11 M12q2Þ of the kinetic matrix of (B10) is guaranteed by the 2 positiveness of not only K but also V. 1 1 q_TK q_ − qTV q þ 2 2 22 2 2 1 11 1 1 APPENDIX C: A TOY MODEL OF − qT V MT K−1M q DEGENERATE THEORY 2 2 ð 22 þ 12 11 12Þ 2 ðB7Þ We study a toy model 6A systematic way is that one can first take an orthogonal 1 1 _2 c1 ̈2 ̈ c3 2 2 2 transformation to diagonalize K and then normalize the variables L ¼ ϕ þ ϕ þ c2ϕ q_ þ q_ − m q ; ðC1Þ so that the elements of K become 1. Then, V can be 2 2 2 2 diagonalized via an indefinite orthogonal transformation. The matrix M can be an upper triangular matrix via integration in classical mechanics. We introduce a new variable Φ and by part. reduce the Lagrangian into the first order system
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1 1 _2 c1 2 c3 2 ̈ 2 2 This is the idea of the degenerate higher order theories [11]. L ¼ ϕ þ Φ þ c2Φq_ þ q_ þ λðϕ − ΦÞ − m q : 2 2 2 2 Note that the kinetic matrix of (C3) has at least one negative ðC2Þ eigenvalue for any choice of parameters ci. Nonetheless, the integrating-out of q changes the kinetic matrix of the Since Φ is nondynamical, it can be eliminated by the use of Lagrangian due to the coupling λq_ and then all eigenvalues its equation of motion. We then obtain of the kinetic matrix of (C4) can be positive. The degeneracy condition indeed corresponds to the 1 1 _2 _ _ 2 2 condition that the mass of ghost becomes infinity. When the L ¼ ϕ − ϕ λ þ ðc1c3 − c2Þq_ 2 2c1 parameter is assumed to be 1 1 c2 λ _ − λ2 − 2 2 pffiffiffiffiffiffiffiffiffi þ q m q : ðC3Þ c2 ¼ c1c3ð1 þ ϵÞ; ðC6Þ c1 2c1 2
The kinetic matrix always has a negative eigenvalue and with ϵ ≪ 1, the masses of the normal mode and the ghost thus there exists a ghost. However, if the Lagrangian is mode are pffiffiffiffiffiffiffiffiffi degenerate, that is if c2 ¼ c1c3, then the variable q becomes nondynamical and then solving it gives m2 m2 O ϵ1 ; normal ¼ 2 − 2 þ ð Þ 2 ðc3 c1m Þ 1 2 c3 2 λ L ϕ_ − ϕ_ λ_ λ_ − ; C4 − 2 1 ¼ 2 þ 2 2 2 ð Þ 2 c3 c1m ϵ0 c1m c1 mghost ¼ þ Oð Þ; ðC7Þ 4c1c3 ϵ which is free from ghost and tachyon when respectively. Therefore, even if the degeneracy condition c3 holds only approximately but not exactly, one can obtain a c3 > 0; >c1 > 0: ðC5Þ m2 stable theory below a cutoff.
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