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PHYSICAL REVIEW D 98, 044043 (2018)

Ghost-free scalar- interactions

† ‡ Rampei Kimura,1,2,* Yuki Sakakihara,3, and Masahide Yamaguchi2, 1Waseda Institute for Advanced Study, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan 2Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan 3Department of Physics, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan

(Received 15 June 2018; published 28 August 2018)

We discuss a covariant extension of interactions between scalar fields and in a flat space-time. We show, in a covariant theory, how to evade fermionic ghosts appearing because of the extra degrees of freedom behind a fermionic nature even in the Lagrangian with first derivatives. We will give a concrete example of a quadratic theory with up to the first derivative of multiple scalar fields and a Weyl fermion. We examine not only the maximally degenerate condition, which makes the number of degrees of freedom correct, but also a supplementary condition guaranteeing that the time evolution takes place properly. We also show that proposed derivative interaction terms between scalar fields and a Weyl fermion cannot be removed by redefinitions.

DOI: 10.1103/PhysRevD.98.044043

I. INTRODUCTION scalar-tensor theories [4–6,13–20], vector-tensor theories [21–29], and form fields [30–32]. Construction of general theory without ghost degrees of On the other hand, generic theory with fermionic d.o.f. freedom (d.o.f.) has been discussed for a long time. has not yet been investigated well. If we could find such According to Ostrogradsky’s theorem, a ghost always appears in a higher (time) derivative theory as long as it fermionic theories which have not been explored, some is nondegenerate [1] (see also Ref. [2]). If a term with applications of them will be expected. One will be higher derivatives plays an important role in the dynamics, fermionic dark , the interaction of which could be the ghost d.o.f. associated with it must be removed because, constrained by the condition of the absence of ghost d.o.f. otherwise, the dynamics is unstable. This is a different Another application is inflation and its subsequent reheat- approach from , which allows a ghost ing, which needs interactions between an and as long as it appears above the scale we are interested in; , since their interactions are still that is, a term with higher derivatives can be treated as a unknown. The new interaction may affect the history perturbation. during the reheating era through production. The Such an Ostrogradsky ghost could be circumvented by effect of the interactions can also appear in observables introducing the degeneracy of the kinetic matrix, which such as non-Gaussianinty through the loop corrections of leads to the existence of a primary constraint and a series of standard model particles, as pointed out in Refs. [33,34]. subsequent constraints, which are responsible for elimi- The construction of a ghost-free (higher derivative) super- nating extra ghost d.o.f. associated with higher derivatives. symmetric theory is another direction, though some After the recent rediscovery [3–5] of Horndeski theory [6] attempts have already been made [35–38]. in the context of the Galileon [7], the pursuit of finding What is prominent in fermionic theory is that ghosts general theory without such ghost d.o.f. has been revived, appear even with no higher derivatives in the usual mean- especially for bosonic d.o.f., in the context of point ing, i.e., no second derivatives and higher. As mentioned in particles [8–11], their field theoretical application [12], Ref. [39] for a purely fermionic system, if N fermionic variables carry 2N d.o.f. in the phase space, then negative *[email protected] norm states inevitably appear. That is why we usually see † [email protected] that the canonical kinetic term of the fermionic field ‡ [email protected] linearly depends on the derivative of the field like the Dirac Lagrangian. Indeed, in the case of usual Weyl Published by the American Physical Society under the terms of fermions, such unwanted states are evaded by the existence the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of a sufficient number of primary constraints. In our the author(s) and the published article’s title, journal citation, previous paper [40], as a starting point, we studied point and DOI. Funded by SCOAP3. particle theories, the Lagrangian of which contains both

2470-0010=2018=98(4)=044043(21) 044043-1 Published by the American Physical Society KIMURA, SAKAKIHARA, and YAMAGUCHI PHYS. REV. D 98, 044043 (2018)

ψ α x bosonic and fermionic variables with their first derivatives, N Weyl fermionic fields I ðt; Þ, and their Hermitian ψ¯ α_ x α α_ 1 1 … and showed that the coexistence of fermionic d.o.f. and conjugates I ðt; Þ ( , ¼ or 2, I ¼ ; ;N). bosonic ones allows us to have some extension of their Following our previous work [40], we consider the kinetic terms. In the present paper, we apply the analysis Lagrangian of which the fields carry up to the first done in Ref. [40] to the field system with scalar fields and derivative. The most general action is symbolized by Weyl fermions, which can be even a preliminary step toward tensor-fermion or any other interesting theories Z 4 L ϕa ∂ ϕa ψ α ∂ ψ α ψ¯ α_ ∂ ψ¯ α_ including fermionic d.o.f. S ¼ d x ½ ; μ ; I ; μ I ; I ; μ I : ð1Þ This paper is organized as follows. In Sec. II, following the previous analysis for point particle theories, we give a general formulation of the construction of covariant theory Here, the Lorentzian indices are raised and lowered by the with up to first derivatives of n-scalar fields and N-Weyl Minkowski metric ημν, and the fermionic indices are raised fermions. We then derive 4N primary constraints by and lowered by the antisymmetric tensors εαβ and εα_ β_.In introducing maximally degenerate conditions. In Sec. III, addition, the capital latin indices ðI;J;K;…Þ are contracted we concretely write down the quadratic theory of n-scalar δ ψ αψ ≔ δ ψ αψ with the Kronecker delta IJ such as I I;α IJ I J;α. and one Weyl fermion fields and apply the set of the Throughout this paper, we use the maximally degenerate conditions, which we have proposed ðþ; −; −; −Þ. for removing the fermionic ghosts properly. In Sec. IV,we When the Lagrangian (1) consists only of scalar fields, perform Hamiltonian analysis of the quadratic theory and the absence of Ostrogradsky’s ghosts is automatically derive a supplementary condition such that all the primary ensured since the Euler-Lagrange equations contain no constraints are second class. In Sec. V, we obtain the more than second derivatives with respect to time. On the explicit counterparts in Lagrangian formulation to the other hand, second derivatives in fermionic equations of conditions in Hamiltonian formulation. In Sec. VI,we motion are generally dangerous because extra d.o.f. in the show that the obtained theories satisfying the maximally fermionic sector immediately lead to negative norm states degenerate conditions cannot be mapped into theories of (see Refs. [39,40] for the detail.). In order to avoid such which the Lagrangian linearly depends on the derivative of ghost d.o.f., the system must contain the appropriate the fermionic fields. In Sec. VII, we give a summary of our number of constraints, which originate from the degeneracy work. In Appendix A, we summarize definitions and of the kinetic matrix. In this section, we derive degeneracy identities of the Pauli matrices. In Appendix B, we show conditions and a supplementary condition for the the equivalence between the maximally degenerate con- Lagrangian (1). The basic treatment of Grassmann algebra ditions and the primary constraints obtained in Sec. II.In and Hamiltonian formulation, of which we are making use, Appendix C, we extend our analysis in Sec. III to multiple are summarized in, e.g., Refs. [39,40], and we use the left Weyl fermions and derive the primary constraints. derivative throughout this paper. We find degeneracy conditions of (1), which yield an II. GENERAL FORMALISM FOR CONSTRUCTING appropriate number of primary constraints, eliminating half DEGENERATE LAGRANGIAN of the d.o.f. of fermions in phase space. For deriving the In the present paper, we construct flat space-time conditions, we first take a look at the canonical momenta theories with n real scalar fields ϕaðt; xÞ (a ¼ 1; …;n), defined as

∂L ∂L ∂L πμ πμ πμ − πμ † ϕa ¼ ; ψα ¼ α ; ψ¯ α_ ¼ α_ ¼ ð ψ α Þ : ð2Þ ∂ ∂ ϕa I ∂ ∂ ψ ∂ ∂ ψ¯ I ð μ Þ ð μ I Þ I ð μ I Þ

Then, the variations of the 0th component of canonical momenta with respect to all canonical variables give the following set of equations, 0 1 0 1 0 1 _ b ðϕÞ δπϕa δϕ δz B C B C B a C β ðψÞ B δπψα C KB δψ_ C B δ C @ I A ¼ @ J A þ @ zα;I A; ð3Þ _ ψ¯ δπψ¯ α_ δψ¯_ β δ ð Þ I J zα_;I π ≡ π0 where we have omitted the superscript of the 0th component, i.e., A A, and defined the kinetic matrix,

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0 1 0 1 L _ a _ b −L _ a β −L β_ B B _ ϕ ϕ ϕ ψ_ ϕ_ aψ¯_ Aab aβ;J aβ;J J J B C B C B L L β L _ C K ¼ @ Cαb;I Dαβ;IJ Dαβ_ A ¼ ψ_ αϕ_ b ψ_ αψ_ ψ_ αψ¯_ β ; ð4Þ ;IJ @ I I J I J A

Cα_ Dαβ_ D _ L α_ L β L _ b;I ;IJ α_ β;IJ ψ¯_ ϕ_ b ψ¯_ α_ ψ_ ψ¯_ α_ ψ¯_ β I I J I J and 0 1 0 10 1 0 10 1 ϕ L −L β −L _ L −L β −L _ ð Þ ϕ_ a∂ ϕb ϕ_ a∂ ψ _ a β δ ∂ ϕb ϕ_ aϕb ϕ_ aψ _ a β b δz i i ϕ ∂ ψ¯ ð Þ ϕ ψ¯ δϕ a B J i J C i B J J C B ψ C B CB β C B CB β C B ð Þ C L α L β L _ B C L α L β L _ B C δ B ψ_ ∂ ϕb ψ_ α∂ ψ α β C δ ∂ ψ B ψ_ ϕb ψ_ αψ α β C δψ @ zα A ¼ I i i ψ_ ∂ ψ¯ @ ð i JÞ A þ I ψ_ ψ¯ @ J A: ð5Þ ;I @ I J I i J A @ I J I J A _ ðψ¯ Þ β_ β δ Lψ¯_ α_ ∂ ϕb Lψ¯_ α_ ∂ ψ β L α_ β_ δ ∂ ψ¯ Lψ¯_ α_ ϕb L _ α_ β L α_ β_ δψ¯ zα_ i i ψ¯_ ∂ ψ¯ ð i Þ ψ¯ ψ ψ¯_ ψ¯ J ;I I I I i J J I I J I J

ba β Here, we have introduced the shortcut notation, δπψ α ¼ðDαβ − Cα A B β Þδψ_ I ;IJ b;I a ;J J β_ 2 − C baB δψ¯_ ∂ L ∂ ∂L þðDαβ_;IJ αb;IA aβ_;JÞ J LXY ¼ ¼ : ð6Þ ∂Y∂X ∂Y ∂X ba δπ − δ ðϕÞ δ ðψÞ þ Cαb;IA ð ϕa za Þþ zα;I ; ð11Þ

β Here, A is a symmetric matrix, while Dαβ , Dαβ_ , ba ab ;IJ ;IJ δπψ¯ α_ ¼ðDαβ_ − Cα_ A B β Þδψ_ I ;IJ b;I a ;J J Dαβ_ ;IJ, and Dα_ β_ are antisymmetric matrices under the ;IJ _ − C baB δψ¯_ β exchange of the greek indices as þðDα_ β_;IJ α_b;IA aβ_;JÞ J ba δπ − δ ðϕÞ δ ðψ¯ Þ − − þ Cα_b;IA ð ϕa za Þþ zα_;I : ð12Þ Dαβ;IJ ¼ Dβα;JI;Dαβ_;IJ ¼ Dβα_ ;JI; − Dα_ β_;IJ ¼ Dβ_ α_;JI; ð7Þ In order to obtain the sufficient number of constraints, we _ need conditions that the velocity terms δψ_I and δψ¯ I cannot and B and C are Grassmann-odd matrices and related as be expressed in terms of the canonical variables. In the “ Cαb;I ¼ −Bbα;I and Cα_b;I ¼ −Bbα_;I. The Hermitian proper- present paper, we adopt the maximally degenerate μ μ ”1 ties of πϕa and the anti-Hermitian properties of πψα , i.e., conditions, i.e., I πμ † πμ πμ † −πμ ð ϕa Þ ¼ ϕa and ð ψα Þ ¼ ψ¯ α_ , lead to − C baB 0 − C baB 0 I I Dαβ;IJ αb;IA aβ;J ¼ ;Dαβ_;IJ αb;IA aβ_;J ¼ ;

† † † ð13Þ A ¼ A ; B β ¼ −B ; Cα ¼ −C ; ab ab a ;J aβ_;J b;I α_b;I − C baB 0 − C baB 0 † † Dαβ_ ;IJ α_b;IA aβ;J ¼ ;Dα_ β_;IJ α_b;IA aβ_;J ¼ : Dαβ_ ¼ −D ;Dαβ ¼ −D : ð8Þ ;IJ αβ_;IJ ;IJ α_ β_;IJ ð14Þ

In the present paper, we assume that the scalar submatrix Two of these four conditions are equivalent to the others of the kinetic matrix Aab is nondegenerate, i.e., invertible. since they are related through Hermitian conjugates, This assumption is equivalent to requiring ba † ba ðDαβ;IJ − Cαb;IA Baβ;JÞ ¼ −ðDα_ β_ − Cα_b;IA B β_ Þ; 0 ;IJ a ;J det Að Þ ≠ 0; 9 ab ð Þ ð15Þ

ð0Þ ba † ba D _ − Cα A B _ − Dαβ_ − Cα_ A B β : where det Aab is defined by setting all fermionic variables ð αβ;IJ b;I aβ;JÞ ¼ ð ;IJ b;I a ;JÞ and their derivatives to zero. Then, the first equation (3) can ð16Þ be solved for δϕ_ b, The maximally degenerate conditions lead to the following _ δϕ_ b ba δπ − δψ_ β − δψ¯_ β − δ ðϕÞ 4N primary constraints, ¼ A ð ϕa Baβ;J J Baβ_;J J za Þ; ð10Þ

1 where we have defined the inverse of the kinetic matrix in There should be other candidates for conditions eliminating ab −1 ab ghost d.o.f., but here we adopt the simplest case, in which all the the scalar sector as A ¼ðA Þ . Plugging this into the constraints eliminating extra d.o.f. are primary constraints. See infinitesimal momenta of ψ α and ψ¯ α_, we get Ref. [40] for the other possibilities.

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_ _ a a β β β β that we have the maximum number of physical d.o.f. in the Φψ α ≡ πψ α − Fα;Iðϕ ; πϕa ; ∂iϕ ; ψ ; ∂iψ ; ψ¯ ; ∂iψ¯ Þ¼0; I I J J J J maximally degenerate Lagrangian, as we have with usual ð17Þ Weyl fields. Then, we need to check that no secondary constraints appear by examining the consistency conditions † Φψ¯ α_ ¼ −ðΦψα Þ I I of the primary constraints. To this end, we first define the _ _ total Hamiltonian density as a a β β β β ¼ πψ¯ α_ − Gα_;Iðϕ ; πϕa ; ∂iϕ ; ψ ; ∂iψ ; ψ¯ ; ∂iψ¯ Þ¼0; I J J J J α ¯α_ HT ¼ H þ Φψα λ þ Φψ¯ α_ λ ; ð19Þ ð18Þ I I I I

† with Gα_;I ¼ −ðFα_;IÞ . The explicit proof of the equivalence where the Hamiltonian and the total Hamiltonian are between Eqs. (17) and (18) and Eqs. (13) and (14) is shown given by in Appendix B. As discussed in Ref. [40], the existence of Z Z these primary constraints (17) and (18) based on the 3 3 H ¼ d xH;H¼ d xH : ð20Þ maximally degenerate conditions is enough to remove T T the extra d.o.f. in the fermionic sector. However, if these primary constraints yield secondary constraints, we have an Now, we would like to calculate the Poisson bracket, even smaller number of physical d.o.f. Here, let us consider defined as

Z δF x δG y δF x δG y F x G y 3 ðt; Þ ðt; Þ − ðt; Þ ðt; Þ f ðt; Þ; ðt; Þg ¼ d z a a δϕ ðt; zÞ δπϕa ðt; zÞ δπϕa ðt; zÞ δϕ ðt; zÞ δFðt; xÞ δGðt; yÞ δFðt; xÞ δGðt; yÞ δFðt; xÞ δGðt; yÞ δFðt; xÞ δGðt; yÞ − εF þð Þ α þ α þ α_ þ α_ : δψ ðt; zÞ δπψ α ðt; zÞ δπψ α ðt; zÞ δψ ðzÞ δψ¯ ðt; zÞ δπψ¯ α_ ðt; zÞ δπψ¯ α_ ðt; zÞ δψ¯ ðt; zÞ I I I I I I I I ð21Þ

The Poisson brackets between the canonical variables are given by

a a 3 fϕ ðt; xÞ; πϕb ðt; yÞg ¼ δ bδ ðx − yÞ; ð22Þ

α α 3 ψ x π β y −δ δ δ x − y f I ðt; Þ; ψ ðt; Þg ¼ β IJ ð Þ; ð23Þ J

α_ α_ 3 ψ¯ x π _ y −δ δ δ x − y f I ðt; Þ; ψ¯ β ðt; Þg ¼ β_ IJ ð Þ; ð24Þ J while other Poisson brackets are zero. Then, the Poisson brackets between the primary constraints are Z δ y δ y δ y δFα;Iðt; xÞ Fβ;Jðt; Þ 3 δFα;Iðt; xÞ Fβ;Jðt; Þ δFα;Iðt; xÞ Fβ;Jðt; Þ Φ α x Φ β y − f ψ ðt; Þ; ψ ðt; Þg ¼ β þ α þ d z a a ; ð25Þ I J δψ y δψ t; x δϕ t; z δπϕa t; z δπϕa t; z δϕ t; z Jðt; Þ I ð Þ ð Þ ð Þ ð Þ ð Þ Z δFα;Iðt; xÞ δGα_;Jðt; yÞ 3 δFα;Iðt; xÞ δGα_;Jðt; yÞ δFα;Iðt; xÞ δGα_;Jðt; yÞ Φ α x Φ α_ y − f ψ ðt; Þ; ψ¯ ðt; Þg ¼ α_ þ α þ d z ; ð26Þ I J δψ¯ y δψ x δϕa z δπ z δπ z δϕa z Jðt; Þ I ðt; Þ ðt; Þ ϕa ðt; Þ ϕa ðt; Þ ðt; Þ Z δ _ y δ _ y δ _ y δGα_;Iðt; xÞ Gβ;Jðt; Þ 3 δGα_;Iðt; xÞ Gβ;Jðt; Þ δGα_;Iðt; xÞ Gβ;Jðt; Þ Φ α_ x Φ _ y − f ψ¯ ðt; Þ; ψ¯ β ðt; Þg ¼ β_ þ α_ þ d z a a : ð27Þ I J δψ¯ y δψ¯ ðt; xÞ δϕ ðt; zÞ δπϕa ðt; zÞ δπϕa ðt; zÞ δϕ ðt; zÞ Jðt; Þ I Then, the time evolution of the primary constraints are ! ! ! ! _ Z β Φψ α ðt; xÞ fΦψ α ðt; xÞ;HTg fΦψα ðt; xÞ;Hg λ ðt; yÞ I I I 3yC x y J ≈ 0 ¼ ¼ þ d IJðt; ; Þ _ ; ð28Þ _ Φ α_ x Φ α_ x β Φψ¯ α_ ðt; xÞ f ψ¯ ðt; Þ;HTg f ψ¯ ðt; Þ;Hg λ¯ y I I I Jðt; Þ where

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fΦψα ðt; xÞ; Φ β ðt; yÞg fΦψα ðt; xÞ; Φ β_ ðt; yÞg I ψ I ψ¯ B J J C CIJðt; x; yÞ¼@ A: ð29Þ Φ α_ x Φ β y Φ α_ x Φ _ y f ψ¯ ðt; Þ; ψ ðt; Þg f ψ¯ ðt; Þ; ψ¯ β ðt; Þg I J I J

In order not to have secondary constraints, all the Lagrange multipliers should be fixed by the equation (28). This can be realized if the coefficient matrix of the Lagrange multipliers (29) has the inverse after integrating over y, and then the primary constraints (17) and (18) are second class. In this case, the number of d.o.f. is 2 2 4 − 4 × nðbosonÞþ × NðfermionsÞ NðconstraintsÞ 2 d.o.f. ¼ 2 ¼ nðbosonÞþ NðfermionsÞ; ð30Þ as desired. To find explicit expressions of these obtained nonzero. This implies that an arbitrary function ¯ ¯ conditions, one needs a concrete Lagrangian form. Here- Aðϕa; Ψ; ΨÞ can be expanded in terms of Ψ and Ψ as after, we will consider the most simplest case N ¼ 1. The a ¯ a a a ¯ extension to multiple Weyl fields can be done in the same Aðϕ ; Ψ; ΨÞ¼a0ðϕ Þþa1ðϕ ÞΨ þ a2ðϕ ÞΨ way, although the analysis will be tedious. A brief ϕa ΨΨ¯ introduction for constructing Lagrangians for arbitrary N þ a3ð Þ : ð33Þ is given in Appendix C. † When the arbitrary function A is real, i.e., A ¼ A , then a0 III. DEGENERATE SCALAR-FERMION and a3 are also real, and a1 ¼ a2. The inverse of ¯ THEORIES Aðϕa; Ψ; ΨÞ is given by Since the fermionic fields obey the Grassmann algebra, a a a1 ϕ a2 ϕ A−1 ϕa Ψ Ψ¯ −1 ϕa 1 − ð Þ Ψ − ð Þ Ψ¯ one can dramatically simplify the analysis by restricting the ð ; ; Þ¼a0 ð Þ a a a0ðϕ Þ a0ðϕ Þ number of fermionic fields. Hereafter, we confine the 2 ϕa ϕa − ϕa ϕa theory to n scalar fields and one Weyl field to simplify a1ð Þa2ð Þ a0ð Þa3ð Þ ΨΨ¯ þ a 2 ; the analysis. In this section, we construct the most general a0ðϕ Þ scalar-fermion theory of which the Lagrangian contains up 34 to quadratic in first derivatives of scalar and fermionic ð Þ fields. To this end, we first construct Lorentz scalars and A ϕa Ψ Ψ¯ vectors without any derivatives, which consist of the scalar and the condition that ð ; ; Þ has an inverse is ϕa ≠ 0 fields ϕa and the fermionic fields ψ α, ψ¯ α_. The fermionic simply a0ð Þ . indices can be contracted with the building block matrices, As we will show in the following subsection, the most μ μν μν 2 general action up to quadratic in first derivatives of the εαβ, ε _, σ , ðσ εÞαβ, and ðεσ¯ Þ _, and we have three α_ β αα_ α_ β scalar fields and the Weyl fermion can be written as possibilities, Z α ¯ α_ μ α_ μ α 4 Ψ ¼ ψ ψ α; Ψ ¼ ψ¯ α_ψ¯ ;J¼ ψ¯ σαα_ψ ; ð31Þ S ¼ d xðL0 þ L1 þ L2Þ; ð35Þ

† ¯ μν where Ψ ¼ Ψ. Note that any contractions of ðσ εÞαβ and μν where ðεσ¯ Þα_ β_ with the fermionic fields always vanish due to the 3 Grassmann properties. One can also construct a scalar L0 ¼ P0; ð36Þ quantity from the square of Jμ contracted with the Minkowski metric. However, it reduces to ð1Þ a μ ð2Þ α_ μ α α_ μ α L1 ¼ Pa ∂μϕ J þ P ðψ¯ σαα_∂μψ − ∂μψ¯ σαα_ψ Þ μ ν η −2ΨΨ¯ ð3Þ μ α ð3Þ† μ α_ μνJ J ¼ ; ð32Þ þ P J ψ α∂μψ þ P J ∂μψ¯ ψ¯ α_; ð37Þ where we have used (A4). Furthermore, the square of Ψ 1 2 1 2 1 2 μν a b μν a α a α_ μν † Ψ ∝ ψ ψ ψ ψ L2 V ∂μϕ ∂νϕ S α∂μϕ ∂νψ ∂μϕ ∂νψ¯ S α vanishes because of the fact that and the ¼ 2 ab þ a þ ð a Þ 2 ¯ 2 ¯ Grassmann property, i.e., Ψ ¼ Ψ ¼ 0, while ΨΨ is 1 1 μν ∂ ψ α∂ ψ β ∂ ψ¯ β_∂ ψ¯ α_ μν † þ 2 Wαβ μ ν þ 2 ν μ ðWαβÞ 2 See Appendix A for the detailed definition of each matrix. α_ μν α 3 μν μν þ ∂μψ¯ Qαα_∂νψ ; ð38Þ For N ≠ 1, ðσ εÞαβ and ðεσ¯ Þα_ β_ can be contracted with different Weyl fields, and therefore rank-2 tensors exist. See Appendix C for the detail. and

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μν ημν 1 Vab ¼ Vab ; − ð2Þ ð2Þ† ∂ ϕa μ ð2Þ ð2Þ† μψ ∂ ψ α 2 ðP þ P Þ;ϕa μ J þðP þ P Þ;ΨJ α μ μν ð1Þ μν ð2Þ μν β α η ψ σ ε ψ Sa ¼ Sa α þ Sa ð Þαβ ; ð2Þ ð2Þ† μ α_ þðP þ P Þ Ψ¯ J ∂μψ¯ ψ¯ α_; ð44Þ μν μν μν μν ; Wαβ ¼ W1η εαβ þ W2η ψ αψ β þ W3ðσ εÞαβ † † μν μν γ where P ≡∂P=∂z and P;z ≡∂P =∂z. Thus, the real part þ W4½ψ αðσ εÞβγ þ ψ βðσ εÞαγψ ; ;z ð2Þ L μν † _ of P can be absorbed into the other terms in 1, and thus ημνψ ψ¯ σμνε ψ βψ¯ − ψ ψ¯ β εσ¯ μν 2 2 † Qαα_ ¼ Q1 α α_ þ Q2ð Þαβ α_ Q2 α ð Þβ_ α_ we can impose Pð Þ ¼ −Pð Þ without loss of generality. ρμ β β_ ρν þ Q3ðσ εÞαβψ ψ¯ ðεσ¯ Þβ_ α_: ð39Þ 3. Quadratic terms in the derivatives ðiÞ ðiÞ Here, P0, P , V, S , Wi, and Qi are arbitrary functions of Next, the Lagrangian containing two derivatives is ϕa; Ψ, and Ψ¯ . These arbitrary functions satisfy the follow- ing properties: 1 μν a b μν a α a α_ μν † L2 V ∂μϕ ∂νϕ S α∂μϕ ∂νψ ∂μϕ ∂νψ¯ S α ¼ 2 ab þ a þ ð a Þ V ¼ V ;V¼ V †; ab ba ab ab 1 μν α β 1 β_ α_ μν † † þ Wαβ∂μψ ∂νψ þ ∂νψ¯ ∂μψ¯ ðWαβÞ † ð1Þ ð1Þ ð2Þ ð2Þ† 2 2 P0 ¼ P0;Pa ¼ Pa ;P¼ −P ; α_ μν α † † þ ∂μψ¯ Qαα_∂νψ ; ð45Þ Q1 ¼ Q1;Q3 ¼ Q3: ð40Þ μν μν μν μν a α α_ where V , Saα, Wαβ, and Qαα_ consist of ϕ , ψ , ψ¯ , and ab μν constant matrices. The coefficient V should take the form A. Construction of Lagrangian up to quadratic ab in the derivatives μν 1 2 V ¼ Vð Þημν þ Vð ÞJμJν; ð46Þ In this subsection, we derive the most general ab ab ab Lagrangian (35)–(40) one by one. ðiÞ ϕa Ψ Ψ¯ where Vab are arbitrary functions of ; , and . 1. No derivative term However, the identity (A6) leads to The Lagrangian without any derivatives should be the 1 μ ν − ημνΨΨ¯ arbitrary function of the form in (33); thus, J J ¼ 2 : ð47Þ

a ¯ L0 ¼ P0ðϕ ; Ψ; ΨÞ; ð41Þ ð2Þ ð1Þ Therefore, the Vab term can be absorbed into the Vab term, a ¯ ð2Þ where P0 is an arbitrary function of ϕ , Ψ, and Ψ, and the and we can generally set V ¼ 0. As a result, we have † ab Hermitian property of the Lagrangian requires P0 ¼ P0. μν ημν Vab ¼ Vab ; ð48Þ 2. Linear terms in the derivatives ϕa Ψ Ψ¯ The candidate for the Lagrangian including terms where Vab is an arbitrary real function of ; , and , which is symmetric under the exchange of a and b. proportional to the first derivatives is μν Second, the general form of Saα is L ð1Þ∂ ϕa μ ð2Þψ¯ α_σμ ∂ ψ α ð2Þ†∂ ψ¯ α_σμ ψ α 1 ¼ Pa μ J þ P αα_ μ þ P μ αα_ μν ð1Þ μν ð2Þ μ α_ ν ð3Þ ν α_ μ Saα ¼ Sa η ψ α þ Sa J ψ¯ σαα_ þ Sa J ψ¯ σαα_ ð3Þ μ α ð3Þ† μ α_ þ P J ψ α∂μψ þ P J ∂μψ¯ ψ¯ α_; ð42Þ ð4Þ μ ν þ Sa J J ψ α; ð49Þ where PðiÞ are arbitrary functions of ϕa, Ψ, and Ψ¯ and ð1Þ ðiÞ ϕa Ψ Ψ¯ the Hermitian property of the Lagrangian requires Pa ¼ where Sa are arbitrary functions of ; , and .By ð1Þ† using (A5) and (A6), the second and third terms can be Pa . The second and third terms can be decomposed as rewritten as ð2Þψ¯ α_σμ ∂ ψ α ð2Þ†∂ ψ¯ α_σμ ψ α P αα_ μ þ P μ αα_ 1 μ α_ ν ¯ μν ¯ μν β 1 J ψ¯ σαα_ ¼ Ψη ψ α þ Ψðσ εÞαβψ ; ð50Þ ð2Þ ð2Þ† ∂ μ 2 ¼ 2 ðP þ P Þ μJ μ 1 1 † μ μ ν α_ ¯ μν ¯ μν β ð2Þ − ð2Þ ψ¯ α_σ ∂ ψ α − ∂ ψ¯ α_σ ψ α J ψ¯ σαα_ ¼ Ψη ψ α − Ψðσ εÞαβψ : ð51Þ þ 2 ðP P Þð αα_ μ μ αα_ Þ: ð43Þ 2

If we integrate the first term in the right-hand side by parts, Therefore, the second and third terms can be compactly ð2Þ μν β it becomes expressed as Sa ðσ εÞαβψ , which is antisymmetric under

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μ ν and , by absorbing the symmetric remaining parts into 1 ð1Þ ð4Þ 1 1 2 2 μ β_ ν ¯ μν ¯ μν γ ψ ψ ψ ψ J ψ αψ¯ σ Ψη ψ αψ β Ψψ α σ ε ψ ; 56 Sa . The last term with Sa is proportional to or ββ_ ¼ 2 þ ð Þβγ ð Þ in a component expression; thus, this automatically van- ishes. As a result, we can assign 1 ν β_ μ ¯ μν ¯ μν γ J ψ αψ¯ σ Ψη ψ αψ β − Ψψ α σ ε ψ : 57 μν ð1Þ μν ð2Þ μν β ββ_ ¼ 2 ð Þβγ ð Þ Saα ¼ Sa η ψ α þ Sa ðσ εÞαβψ : ð52Þ μν e e Third, the general form of Wαβ is Thus, the symmetric parts of W3, W4 and W5, W6 can be μν μν expressed in terms of η εαβ and η ψ αψ β. On the other μν ημνε ημνψ ψ e ψ¯ α_σμ ψ¯ β_σν Wαβ ¼ W1 αβ þ W2 α β þ W3ð αα_Þð ββ_Þ hand, antisymmetric parts of them are taken into account by μν μν γ σ ε and ψ α σ ε ψ . As before, the last term with e α_ ν β_ μ μ β_ ν ð Þαβ ð Þβγ þ W4ðψ¯ σαα_Þðψ¯ σ _ÞþW5J ψ αψ¯ σ _ 1 1 ββ ββ W7 is proportional to ψ ψ in a component expression; νψ ψ¯ β_σμ μ νψ ψ thus, this automatically vanishes. As a result, we can þ W6J α ββ_ þ W7J J α β: ð53Þ μν express Wαβ as μψ ψ¯ β_σν ≃− νψ ψ¯ β_σμ νψ ψ¯ β_σμ ≃ We note that J β αβ_ J α ββ_ and J β αβ_ μν μν μν μν W ¼ W1η εαβ þ W2η ψ αψ β þ W3ðσ εÞ − μψ ψ¯ β_σν αβ αβ J α ββ_ hold as long as they are contracted with μν μν γ W4 ψ α σ ε ψ β σ ε ψ : α β ð2Þ ð3Þ þ ½ ð Þβγ þ ð Þαγ ð58Þ ∂μψ ∂νψ . Similarly to the case of S and S , we can e e rewrite W3, W4, W5, and W6 as The W4 term is manifestly symmetrized by making μν γ μν γ 1 use of the fact that ψ αðσ εÞβγψ ≃ ψ βðσ εÞαγψ in the α_ μ β_ ν ¯ μν ¯ μν ψ¯ σ ψ¯ σ Ψη εαβ − Ψ σ ε ; 54 ð αα_Þð ββ_Þ¼2 ð Þαβ ð Þ Lagrangian, which holds again since they are contracted α β with ∂μψ ∂νψ . μν _ 1 α_ ν β μ ¯ μν ¯ μν Finally, let us take a look at Qαα_. The general form is ðψ¯ σ Þðψ¯ σ Þ¼ Ψη εαβ þ Ψðσ εÞαβ: ð55Þ αα_ ββ_ 2 given by

μν ημνψ ψ¯ ˜ ψ¯ β_σμ σν ψ β ˜ ψ¯ β_σν σμ ψ β μσν νσμ Qαα_ ¼ Q1 α α_ þ Q2ð αβ_Þð βα_ ÞþQ3ð αβ_Þð βα_ ÞþQ4J αα_ þ Q5J αα_ μ νψ ψ¯ μψ ψ βσν νψ ψ βσμ μψ¯ ψ¯ β_σν νψ¯ ψ¯ β_σμ þ Q6J J α α_ þ Q7J α βα_ þ Q8J α βα_ þ Q9J α_ αβ_ þ Q10J α_ αβ_: ð59Þ

1 2 ¯ 1˙ ¯ 2˙ The last five terms, Q6;7;8;9;10, contain more than one ψ , ψ , ψ ,orψ , and therefore they automatically vanish. We can ˜ ˜ rewrite Q2, Q3, Q4, and Q5 as

β_ μ ν β 1 μν μν β μν β_ ρμ β ρν β_ ψ¯ σ σ ψ − η ψ αψ¯ α_ σ ε ψ ψ¯ α_ − ψ α εσ¯ _ψ¯ − 2 σ ε ψ εσ¯ _ψ¯ ; 60 ð αβ_Þð βα_ Þ¼ 2 þð Þαβ ð Þα_ β ð Þαβ ð Þα_ β ð Þ

β_ ν μ β 1 μν μν β μν β_ ρμ β ρν β_ ψ¯ σ σ ψ − η ψ αψ¯ α_ − σ ε ψ ψ¯ α_ ψ α εσ¯ _ψ¯ − 2 σ ε ψ εσ¯ _ψ¯ : 61 ð αβ_Þð βα_ Þ¼ 2 ð Þαβ þ ð Þα_ β ð Þαβ ð Þα_ β ð Þ 1 μσν ημνψ ψ¯ σμνε ψ βψ¯ ψ εσ¯ μν ψ¯ β_ − 2 σρμε ψ β εσ¯ ρν ψ¯ β_ J αα_ ¼ 2 α α_ þð Þαβ α_ þ αð Þα_ β_ ð Þαβ ð Þα_ β_ ; ð62Þ 1 νσμ ημνψ ψ¯ − σμνε ψ βψ¯ − ψ εσ¯ μν ψ¯ β_ − 2 σρμε ψ β εσ¯ ρν ψ¯ β_ J αα_ ¼ 2 α α_ ð Þαβ α_ αð Þα_ β_ ð Þαβ ð Þα_ β_ : ð63Þ

Note that the last terms are symmetric under the replacement of μ and ν, as shown in (A8). Thus, four functions from Q1;4;5 ˜ μν μν β μν β_ ρμ β ρν β_ and Q2;3 are independent. We adopt η ψ αψ¯ α_, ðσ εÞαβψ ψ¯ α_, ψ αðεσ¯ Þα_ β_ψ¯ , and ðσ εÞαβψ ðεσ¯ Þα_ β_ψ¯ as independent functions, and then we have

μν μν μν β † β_ μν ρμ β β_ ρν Qαα_ ¼ Q1η ψ αψ¯ α_ þ Q2ðσ εÞαβψ ψ¯ α_ − Q2ψ αψ¯ ðεσ¯ Þβ_ α_ þ Q3ðσ εÞαβψ ψ¯ ðεσ¯ Þβ_ α_: ð64Þ

Since the first and the fourth terms are Hermite, the coefficients are required to be real.

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Collecting all the results, we obtain the most general second derivatives of the fermionic fields. Thus, one should Lagrangian (35)–(40). choose the arbitrary functions appearing in the Lagrangian with care in order to have the correct number of d.o.f. In B. Degeneracy conditions this subsection, we derive the degeneracy conditions for the In the previous subsection, we have obtained the most Lagrangian (38) with (39). (Note that the linear Lagrangian general Lagrangian n-scalar and one Weyl fields which in derivatives (37) is irrelevant to the degeneracy condi- contains up to quadratic in first derivatives. As explained tions.) The maximally degenerate conditions (13) and (14) earlier, the Euler-Lagrange equations, in general, contain lead to

− C baB ε ð1Þ ba ð1Þ ψ ψ 0 Dαβ αbA aβ ¼ W1 αβ þðW2 þ Sb V Sa Þ α β ¼ ; ð65Þ

− C baB − ð1Þ ba ð1Þ † ψ ψ¯ − σi0ε ψ γψ¯ γ_ εσ¯ i0 0 Dαβ_ αbA aβ_ ¼ ðQ1 þ Sb V ðSa Þ Þ α β_ Q3ð Þαγ ð Þγ_ β_ ¼ ; ð66Þ where Vab ¼ðV−1Þab. In general, they give us four conditions,4

0 ð1Þ ba ð1Þ ψ ψ 0 ð1Þ ba ð1Þ † ψ ψ¯ 0 ψ γψ¯ γ_ 0 W1 ¼ ; ðW2 þ Sb V Sa Þ α β ¼ ; ðQ1 þ Sb V ðSa Þ Þ α β_ ¼ ;Q3 ¼ : ð67Þ

The functions included in L2 are simplified after inserting these four conditions:

μν ημν Vab ¼ Vab ; μν ð1Þ μν ð2Þ μν β Saα ¼ Sa η ψ α þ Sa ðσ εÞαβψ ; μν − ð1Þ ba ð1Þημνψ ψ σμνε ψ σμνε ψ σμνε ψ γ Wαβ ¼ Sb V Sa α β þ W3ð Þαβ þ W4½ αð Þβγ þ βð Þαγ ; μν − ð1Þ ba ð1Þ †ημνψ ψ¯ σμνε ψ βψ¯ − †ψ ψ¯ β_ εσ¯ μν Qαα_ ¼ Sb V ðSa Þ α α_ þ Q2ð Þαβ α_ Q2 α ð Þβ_ α_: ð68Þ

† IV. HAMILTONIAN FORMULATION πψ¯ α_ ¼ −ðπψ α Þ : ð71Þ In the previous section, we derived the Lagrangian _ a satisfying the degeneracy conditions which lead to four Solving (69) for ϕ and plugging it into (70) and (71),we primary constraints, as explicitly shown in the following. indeed obtain primary constraints (17) and (18), where Fα They suggest that the extra d.o.f. associated with fermionic and Gα_ are given by Ostrogradsky’s ghosts are removed; however, one needs to ð2Þ α_ 0 ð3Þ 0 make sure of the condition that no more (secondary) Fα ¼ −P ψ¯ σαα_ − P J ψ α constraints arise in order to have four physical d.o.f. in 00 ab ð1Þ 0 0i β β_ 0i † − S αV π b − P J − S ∂ ψ − ∂ ψ¯ S the phase space. Following Sec. II, we derive such a a ½ ϕ b bβ i i ð bβÞ condition. i0 a 0i β β_ i0 − Saα∂iϕ þ Wαβ∂iψ − ∂iψ¯ Q _; ð72Þ The momenta can be directly calculated from their αβ definition, † Gα_ ¼ −ðFαÞ : ð73Þ ð1Þ 0 _ b 0ν α α_ 0ν † πϕa ¼ Pa J þ Vabϕ þ Saα∂νψ þ ∂νψ¯ ðSaαÞ ; ð69Þ Then, plugging these expression into (25) and (26) and ð2Þ α_ 0 ð3Þ 0 ν0 a picking up the bosonic parts, we obtain πψα ¼ −P ψ¯ σαα_ − P J ψ α − Saα∂νϕ 0ν β β_ ν0 ∂ ψ − ∂ ψ¯ ð0Þ þ Wαβ ν ν Qαβ_; ð70Þ fΦψα ðt;xÞ;Φψβ ðt;yÞg

ð2Þ ð0Þ a 0i ¼ 2ðSa Þ ∂iϕ ðσ εÞαβδðx− yÞ 4 As far as we seek a healthy theory where we have no ghost not ∂ ∂ 0i ð0Þ ð0Þ only in the background but also in the perturbations, these four þðσ εÞαβ W3 ðxÞ δðx− yÞþW3 ðyÞ δðx − yÞ ; conditions are reasonable; e.g., even if the background evolution ∂xi ∂yi is like ψ αψ β ∝ εαβ as a result of the equations of motion, the perturbations on it will face serious instabilities. ð74Þ

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0 2 0 0 Φ α x Φ _ y ð Þ −2 ð Þ ð Þσ δ x − y f ψ ðt; Þ; ψ¯ β ðt; Þg ¼ ðP Þ αβ_ ð Þ; ð75Þ where (0) represents their bosonic parts. When we calculate the time derivative of the primary constraints, we integrate the product of the constraints matrix and the Lagrange multipliers over y. Focusing on the second term in (74), Z ∂ ∂ 3 0i ð0Þ ð0Þ β d yðσ εÞαβ W ðxÞ δðx − yÞþW ðyÞ δðx − yÞ λ ðt; yÞ 3 ∂xi 3 ∂yi Z ∂ 3 0i ð0Þ β ¼ − d yðσ εÞαβ ðW ðyÞÞδðx − yÞλ ðt; yÞð76Þ ∂yi 3 holds as far as we drop total derivative terms, where we have used the fact that ð∂=∂xiÞδðx − yÞ¼−ð∂=∂yiÞδðx − yÞ. Using this result and the following identities,

† † Φ x Φ _ y − Φ α x Φ β y Φ _ x Φ β y − Φ α x Φ _ y f ψ¯ α_ ðt; Þ; ψ¯ β ðt; Þg ¼ f ψ ðt; Þ; ψ ðt; Þg ; f ψ¯ α ðt; Þ; ψ ðt; Þg ¼ f ψ ðt; Þ; ψ¯ β ðt; Þg ; ð77Þ we can write the bosonic part of the constraint matrix (29) as 0 1 ð0Þ 2 ð2Þ∂ ϕa − ∂ σ0iε −2 ð2Þσ0 B ð Sa i iW3Þð Þαβ P αβ_ C Cð0Þðt; x; yÞ¼@ A δðx − yÞ: ð78Þ ð2Þ 0 ð2Þ† a † 0i −2P σβα_ −ð2Sa ∂iϕ − ∂iW3Þðεσ¯ Þα_ β_

After performing the integration in (28), we can define the A. Equations of motion for fermions with Jð0Þ x the maximally degenerate conditions matrix H ðt; Þ, thanks to the delta function, as Z We derive the equations of motion for the fields and 0 Jð Þðt; xÞ¼ d3yCð0Þðt; x; yÞ: ð79Þ see the EOMs for fermions become the first-order (non- H linear) differential equations after applying the maximally degenerate conditions. Jð0Þ x If this matrix H ðt; Þ is invertible, i.e., the determinant is The variations with respect to the scalar and Weyl fields nonzero,5 yield a set of Euler-Lagrange equations, 0 10 1 0 00 00 00 † 0 1 Jð Þ x ≠ 0 V S β −ðS βÞ ϕ̈b det ðt; Þ ; ð80Þ ab a a Ea H B CB C B C B − 00 00 − 00 CB ψ̈β C @ Sbα Wαβ Qαβ_ A@ A ¼ @ Eα A; ð81Þ all the Lagrange multipliers introduced in the total β_ 00 † 00 − 00 † ψ̈¯ Eα_ Hamiltonian (19) are determined by solving the simulta- ðSbαÞ Qβα_ ðWαβÞ neous equation (28). In this case, the theory does not have secondary constraints, and all the (primary) constraints are where the coefficient matrix in the left-hand side corre- second class. Therefore, the number of d.o.f. is n þ 2 as sponds the kinetic matrix in (4) and we have defined counted in (30), and the extra d.o.f. are properly removed. ∂L − ∂ ð1Þ μ − ∂ μν ∂ ϕb Ea ¼ a μðPa J Þ μVab ν V. EULER-LAGRANGE EQUATIONS ∂ϕ − ∂ μν ∂ ψ α − ∂ ψ¯ α_∂ μν † In this section, we show that the degeneracy conditions μSaα ν ν μðSaαÞ and the supplementary condition obtained in the − ij ∂ ∂ ϕb − ij ∂ ∂ ψ α − ∂ ∂ ψ¯ α_ ij † Vab i j Saα i j i j ðSaαÞ ; ð82Þ Hamiltonian formulation can be also derived in the Lagrangian formulation. In addition, we show that if ∂L E ∂ ð2Þψ¯ α_ σμ ∂ ð3Þ μψ the supplementary condition is satisfied, the equations of α ¼ ∂ψ α þ μðP Þ αα_ þ μðP J αÞ motion for fermions can always be solved in terms of the ∂ νμ ∂ ϕa − ∂ μν ∂ ψ β ∂ ψ¯ α_∂ νμ first derivative of fermionic fields. þ μSaα ν μWαβ ν þ ν μQαα_ ij a ij β ¯_ α_ ð0iÞ þ Saα∂i∂jϕ − Wαβ∂i∂jψ þ 2∂iψ Qαα_ 5 ϕa ϕa Allowing the theory to have solutions with ¼ ðtÞ ∂ ∂ ψ¯ α_ ij requires nonvanishing Pð2Þ. þ i j Qαα_; ð83Þ

044043-9 KIMURA, SAKAKIHARA, and YAMAGUCHI PHYS. REV. D 98, 044043 (2018) where ð0iÞ means symmetrized with respect to 0 and i, and The condition that the dependence of the second derivative † ̈ Eα_ ¼ −ðEαÞ . Solving (82) for ϕ, we obtain of fermions vanishes is exactly the same as the degeneracy conditions (65), (66), and the third line of (81) is also ̈a ab 00 α ̈α_ 00 † reduced to the Hermitian conjugate. Imposing the max- ϕ ¼ V ½Eb − S αψ̈ − ψ¯ ðS αÞ : ð84Þ b b imally degenerate conditions, we obtain the first-order differential equations for ψ α and ψ¯ α_, Eliminating ϕ̈from the second lines of (81), we find the equations of motion for fermions, 00 ab Yα ≡ Eα þ SaαV Eb ¼ 0; ð86Þ

00 00 ba 00 ψ̈β − 00 00 ba 00 † ψ̈¯ β_ Y ≡ − Y † E − 00 † ab 0 ½Wαβ þ SbαV Saβ ½Qαβ_ þ SbαV ðSaβÞ α_ ð αÞ ¼ α_ ðSaαÞ V Eb ¼ ; ð87Þ E 00 ba ¼ α þ SbαV Ea: ð85Þ and by substituting (82) and (83), Yα is written as

∂L Y ∂ ð2Þψ¯ α_ σμ ∂ ð3Þ μψ ∂ νμ ∂ ϕa − ∂ μν ∂ ψ β ∂ ψ¯ α_∂ νμ α ¼ ∂ψ α þ μðP Þ αα_ þ μðP J αÞþ μSaα ν μWαβ ν þ ν μQαα_ ∂L ð1Þ ab ð1Þ μ μ c μν β β_ μν † þ S V ψ α − ∂μðP J Þ − ∂μV ∂ ϕ − ∂μS ∂νψ − ∂νψ¯ ∂μðS Þ : ð88Þ a ∂ϕb b bc bβ bβ

_ We note that the dependence on ∂iψ¯ α_ and the second spatial maximally degenerate conditions, the equations of motion derivatives of the fermions vanish after applying (68), and for the fermionic variables should be the first-order differ- they have become the first-order differential equations for ential equations after appropriately combining the bosonic both time and spatial derivatives. equations of motion. Let us assume that we have already As we saw, thanks to the maximally degenerate con- done this process, and we can generally write down the ditions, one can remove the second derivative terms of the reduced equations of motion for fermionic variables as Weyl field in the Euler-Lagrange equations by combining _ _ _ _ the equations of motion for the scalar fields, suggesting a1θ1 þ a2θ2 þ a3θ1θ2 ¼ a4; that the number of initial conditions to solve the EOMs _ _ _ _ is appropriate. Thus, we have confirmed even in the b1θ1 þ b2θ2 þ b3θ1θ2 ¼ b4; ð89Þ Lagrangian formulation that the extra d.o.f. do not appear in the theory with the maximally degenerate conditions. where a1, a2, b1, and b2 (a3, a4, b3, and b4) are Grassmann- i Note that the second derivative terms of the Weyl field in even (Grassmann-odd) functions depending on θα, q , and i ð0Þ ð0Þ ð0Þ ð0Þ the equations of motion for the scalar fields (84) can be q_ . We first assume at least one of a1 , a2 , b1 , and b2 is removed by using the time derivative of (86) and (87),as ð0Þ nonzero. For instance, if a1 ≠ 0, we can solve the first discussed in Ref. [40]. _ equation of (89) for θ1: B. Solvable condition for nonlinear equations θ_ −1 −1 − −1 θ_ ≕ θ_ including first derivatives 1 ¼ a1 a4 þ a1 ð a2 þ a3a1 a4Þ 2 A1 þ A2 2: ð90Þ

So far, we have seen that the maximally degenerate Then, plugging this into the second equation of (89),we conditions lead to the first- (second-)order differential have equations of motion for the Weyl (scalar) fields. As discussed in Ref. [40], it is not clear whether the equations _ b1A2 b2 b3A1 θ2 b4 − b1A1: 91 of motion for fermions can be solved or not due to the ð þ þ Þ ¼ ð Þ Grassmann properties. Here, we derive a condition, under which the equations of motion even nonlinear in the time Thus, if the uniqueness condition, derivatives can be explicitly solved. We start with the ð0Þ −1 − ð0Þ ≠ 0 simplest example in a point particle system, and then we ðb1A2 þ b2 þ b3A1Þ ¼½a1 ð b1a2 þ a1b2Þ ; extend this analysis to the theory we focus on in the ð92Þ present paper. is satisfied, one can solve the set of nonlinear equation (89). 1. Example: Two Grassmann-odd variables In the other case, where the equations of motion have no i _ Suppose that we have a system composed of q linear terms in θα, there is no way to express each time and two fermionic variables θ1 and θ2. Due to the derivative in terms of nonderivative variables, and the

044043-10 GHOST-FREE SCALAR-FERMION INTERACTIONS PHYS. REV. D 98, 044043 (2018) equations of motion are unsolvable. Therefore, the nonzero X2 X2 β β γ_ γ_ Y X ψ_ 1 ψ_ n ψ¯_ 1 ψ¯_ m 0 determinant of the coefficient matrix of the linear terms in α ¼ αβ1βnγ_1γ_m ¼ ; _ n m θα is the necessary and sufficient condition for the equations to be uniquely solved for the time derivatives ð93Þ of the fermionic variables. † Yα_ ¼ −ðYαÞ 2. More general argument X2 X2 γ_ γ_ β β † In this subsection, we will extend the previous analysis to − ψ¯_ m ψ¯_ 1 ψ_ n ψ_ 1 X 0 ¼ ð αβ1βnγ_1γ_m Þ ¼ ; our degenerate theory and show the equivalence between n m the condition (80) and the condition that the fermionic 94 equations (86) and (87) are uniquely solved. ð Þ The equations of motion for fermions (86) and (87) X ψ α ψ¯ α_ ϕa ϕ_ a include up to first time derivatives, and due to the where αβ1β γ_1γ_ consists of , , , , and their n m α Grassmann properties, they can be therefore generally spatial derivatives. If these equations can be solved for ψ_ written as [see also (88)] and ψ¯_ α_, they should have the forms

ψ_ α ¼ðψ¯_ α_Þ† X αi1i j1j β β β_ β_ δ δ δ_ δ_ m n ψ 1 ψ k ψ¯ 1 ψ¯ l ∂ ψ 1 ∂ ψ m ∂ ψ¯ 1 ∂ ψ¯ n ¼ C _ _ _ _ i1 im j1 jn ; ð95Þ β1βkβ1βlδ1δmδ1δn k;l;m;n

αi1 ϕa ϕ_ a ∂ ϕa ψ α ψ¯ α_ 6 where the coefficients Cβ1 consist of , , and i and has no dependence on and . The nonlinear equations (93) and (94) can be solved for the first time derivatives of the fermion if we could uniquely determine the αi1 coefficients Cβ1 by substituting (95) into the them. After the substitution, the EOMs should become trivial at each order of γ γ_ γ γ_ γ δ γ_ γ γ_ δ_ γ γ_ (ψ , ψ¯ ), (∂iψ , ∂iψ¯ ), (ψ ψ ψ¯ , ψ ψ¯ ψ¯ ), and so on. At the lowest order, where the terms are linear in ψ and ψ¯ ,wehave 0 1 0 1 0 0 ! ∂X ð0Þ ∂X ð0Þ X ð Þ X ð Þ β β γ α α γ B αβ αβ_ C C γ C γ_ ψ B ∂ψ γ ∂ψ¯ γ_ C ψ @ A ¼ −@ A : ð96Þ ð0Þ ð0Þ β β γ_ ð0Þ ð0Þ γ_ − X − X ðC γ_Þ ðC γÞ ψ¯ − ∂X α − ∂X α ψ¯ ð αβ_ Þ ð αβ Þ ∂ψ¯ γ_ ∂ψ γ

Therefore, if the matrix 0 1 X ð0Þ X ð0Þ B αβ αβ_ C Jð0Þ x − L ðt; Þ¼ @ A ð97Þ − X ð0Þ − X ð0Þ αβ_ αβ

β β has a nonzero determinant, we can uniquely determine C γ and C γ_ from (96). Similarly, the coefficients associated with the γ ¯ γ_ spatial derivative terms (∂iψ , ∂iψ ) are determined by 0 1 0 1 ð0Þ ð0Þ ! ∂X ð0Þ ∂X ð0Þ X X β β γ α α γ αβ _ i i ∂ ψ ∂ ∂ ψγ ∂ ∂ ψ¯ γ_ ∂ ψ B αβ C C γ C γ_ i B ð i Þ ð i Þ C i @ A ¼ −@ A : ð98Þ ð0Þ ð0Þ βi βi ∂ ψ¯ γ_ ∂X ð0Þ ∂X ð0Þ ∂ ψ¯ γ_ − X − X ðC γ_Þ ðC γÞ i − α − α i αβ_ αβ γ_ ∂ ∂ ψγ ∂ð∂iψ¯ Þ ð i Þ

Once the coefficients at the linear order in the fermionic fields are determined, the coefficients at the cubic order can be determined by the equation 0 1 ð0Þ ð0Þ ! ! X X β β γ δ η_ γ δ η_ B αβ αβ_ C C γδη_ C γδ_ η_ ψ ψ ψ¯ Gαγδη_ Gαγδ_ η_ ψ ψ ψ¯ − @ A _ ¼ _ : ð99Þ ð0Þ ð0Þ β β γ δ η_ − G − G γ δ η_ − X − X ðC ηδ_ γ_Þ ðC ηδγ_Þ ψ ψ¯ ψ¯ ð αηδ_ γ_Þ ð αηδγ_Þ ψ ψ¯ ψ¯ αβ_ αβ

6(k þ l þ m þ n) is a finite odd number.

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1 Here, the coefficient matrices Gαγδη_ and Gαγδ_ η_ are expressed 2 i α_ μ α α_ μ α L ¼ ð∂μϕÞ þ ðψ¯ σ ∂μψ − ∂μψ¯ σ ψ Þ: ð103Þ X 2 2 αα_ αα_ in terms of αβ1βnγ_1γ_m and their derivatives with respect to ψ α, ψ¯ α_, and their spatial derivatives, as well as the linear β β βi βi Note that the kinetic matrix of (103) is trivially degene- coefficients C γ, C γ_, C γ, and C γ_. Again, we can solve B C 0 β β rate since ¼ ¼ D ¼ . Under the following field the above equations for C γδη_ and C γδ_ η_ and successively redefinition, determine all the coefficients appearing in (95) with the Jð0Þ i i ¯ same procedure as long as the matrix L has an inverse. ϕ ¼ φ − Ψ þ Ψ; ð104Þ Jð0Þ 2 2 Now, it is straightforward to calculate the matrix L from the concrete expression of the equations of motion where φ is a new , while keeping the Weyl field Jð0Þ Y (86) and (87). The definition of L is rewritten with α the same, the Lagrangian is transformed as and Yα_ as 0 1 0 0 1 ∂Yα ð Þ ∂Yα ð Þ 2 μ α α_ β _ L ¼ ð∂μφÞ − i∂ φðψ ∂μψ α − ψ¯ α_∂μψ¯ Þ 0 B ∂ψ_ ∂ψ¯_ β C 2 Jð Þ x − L ðt; Þ¼ @ A; ð100Þ ∂Y ð0Þ ∂Y ð0Þ i μ μ α_ α_ ψ¯ α_σ ∂ ψ α − ∂ ψ¯ α_σ ψ α ∂ψ_ β ∂ψ¯_ β_ þ 2 ð αα_ μ μ αα_ Þ

1 α 2 α μ α_ 1 α_ 2 and the components are explicitly given by − ðψ ∂μψ αÞ þðψ ∂μψ αÞðψ¯ α_∂ ψ¯ Þ − ðψ¯ α_∂μψ¯ Þ : 2 2 0 † 0 ∂Yα ð Þ ∂Yα_ ð Þ ¼ − ð105Þ ∂ψ_ β ∂ψ¯_ β_ As one can see, the submatrices B, C,andD in the kinetic − 2 ð2Þ ð0Þ∂ ϕa − ∂ ð0Þ σ0iε ¼ ðSa Þ i iW3 ð Þαβ; ð101Þ matrix become nonzero due to the transformation (104),and 0 † 0 the mapped Lagrangian apparently yields the second-order ∂Yα ð Þ ∂Yα_ ð Þ ¼ − ¼ 2ðPð2ÞÞð0Þσ0 : ð102Þ differential equations for the fermionic fields. One can, β_ ∂ψ_ β αβ_ ∂ψ¯_ however, easily check that this mapped Lagrangian satisfies the maximally degenerate conditions (65) and (66) as well as Jð0Þ It is manifest that L agrees with the matrix (79) with (78), the supplementary condition (80); therefore, the number of obtained in Hamiltonian formulation. Therefore, the con- d.o.f. is the same as the original Lagrangian (103).8 This is dition that the fermionic equations can be uniquely solved because the field redefinition (104) is an invertible trans- in the Lagrange formulation is equivalent to the condition formation, keeping the number of d.o.f. under the trans- that all the primary constraints are second class. formation [41]. Now, we would like to know whether our degenerate VI. FIELD REDEFINITION Lagrangian can be mapped into theories of which the So far, we have successfully obtained the most general kinetic matrix is trivially degenerate by the covariant field ϕa ψ α ψ¯ α_ → A ϕ ψ ψ¯ ηΛ ϕ ψ ψ¯ quadratic Lagrangian for n-scalar and one Weyl fields redefinition ð ; ; Þ ðf ð ; ; Þ; ð ; ; Þ; Λ_ 9 satisfying the maximally degenerate conditions including η¯ ðϕ; ψ; ψ¯ ÞÞ, i.e., if it is possible that up to first derivatives. However, one needs to carefully check 0 1 0 1 A B β B _ ˜ whether the obtained theories can be mapped into a known ab a ;J aβ;J Aab 00 7 B C B C theory, which trivially satisfies the degeneracy conditions. K B C C⟶field redefinition ¼ @ αb;I Dαβ;IJ Dαβ_;IJ A @ 000A; For an example, let us consider the Lagrangian containing a C 000 canonical scalar field and a Weyl field, α_b;I Dαβ_ ;IJ Dα_ β_;IJ ð106Þ 7Degeneracy of fermionic fields is always necessary to avoid negative norm states even if it includes only first derivatives. This fact is in sharp contrast with the cases of bosonic fields, which 8The Hamiltonian analysis of this Lagrangian has already been can be healthy as far as the Lagrangian does not include second or examined in Ref. [40]. higher derivatives even if it is nondegenerate. When the La- 9One can consider field redefinitions including derivatives of grangian does not have mixing between bosonic and fermionic the scalar and fermionic fields, but, for instance, redefinitions like fields, the maximally degenerate conditions (13) and (14) are ðϕ;ψ;ψ¯ Þ → ðfðϕ;∂ϕ;ψ;ψ¯ Þ;ηðϕ;ψ;ψ¯ Þ;η¯ðϕ;ψ;ψ¯ ÞÞ, ðϕ; ψ; ψ¯ Þ → equivalent to the degeneracy of the kinetic matrix of the fermionic ðfðϕ; ψ; ψ¯ Þ; ηðϕ; ψ; ψ¯ ; ∂ψÞ; η¯ðϕ; ψ; ψ¯ ; ∂ψ¯ ÞÞ, ðϕ; ψ; ψ¯ Þ → fields, det D ¼ 0. In a special case where each field has no mixing ðfðϕ; ψ; ψ¯ Þ, ηðϕ; ψ; ψ¯ ; ∂ψ¯ Þ, η¯ðϕ; ψ; ψ¯ ; ∂ψÞÞ,orðϕ; ψ; ψ¯ Þ → with the others, all the components of the submatrix D vanish; ðfðϕ; ψ; ψ¯ ; ∂ψ; ∂ψ¯ Þ; ηðϕ; ψ; ψ¯ Þ; η¯ðϕ; ψ; ψ¯ ÞÞ result in the appear- i.e., the kinetic matrix is trivially degenerate. One of the simplest ance of the higher derivatives, implying that they might not keep examples is a no-derivative interacting system of a canonical the Lagrangian including just up to the first derivatives. To scalar field and a Weyl field like (103), which is linear in the first introduce such transformations, we also need to check the derivative of fermions. invertibility of them.

044043-12 GHOST-FREE SCALAR-FERMION INTERACTIONS PHYS. REV. D 98, 044043 (2018) ˜ 1 1 where A is the kinetic matrix of the redefined scalar c ð1Þ ð1Þ † ¯ ab L ¼ V ∂μϕ − ðS ∂μΨ þðS Þ ∂μΨÞ A rel 2 ac 2 a a fields f . We note that, under the field redefinition, the Lagrangians, (36), (37), and (38), are not mixed since the 1 1 1 Vab V ∂μϕd − Sð Þ∂μΨ Sð Þ †∂μΨ¯ : transformation does not include the derivatives of the fields. × bd 2 ð b þð b Þ Þ From the definition of the kinetic matrix, the quadratic terms in derivatives in the Lagrangian (38) can be responsible for ð107Þ the kinetic matrix, while linear or lower ones (37), (36) are obviously not. Furthermore, among the quadratic terms, antisymmetrically contracted (with respect to Lorentzian ð1Þ indices of the derivatives) ones are irrelevant. The only If the coefficients Vab and Sa are constants, the square contribution to the kinetic matrix is coming from the brackets ½ can be redefined as the derivatives of new symmetrically contracted ones in L2. When all the subma- scalar fields. Then, the nontrivial terms in (107), which yield ˜ second time derivatives of the fermionic fields in Euler- trices except Aab in the kinetic matrix vanish, the corre- sponding symmetrically contracted covariant terms in the Lagrange equations, can be removed just as a nontrivial Lagrangian also should vanish simultaneously because of Lagrangian (105) is reduced to a trivial Lagrangian (103) by the covariance of the theory. As a result, we can just focus on the transformation (104). Under the field redefinition, a α α_ A Λ Λ_ the symmetrically contracted terms in L2 in this section. ðϕ ; ψ ; ψ¯ Þ → ðf ðϕ; ψ; ψ¯ Þ; η ðϕ; ψ; ψ¯ Þ; η¯ ðϕ; ψ; ψ¯ ÞÞ, Picking up these, we rewrite the relevant Lagrangian with the above Lagrangian is rewritten in terms of the new the degeneracy conditions as variables as

1 a b A μ B a b A μ Λ a ¯ b A μ Λ_ L G V G ∂μf ∂ f − G V G ∂μf ∂ η − G V G ∂μf ∂ η¯ rel ¼ 2 A ab B A ab Λ A ab Λ_ 1 1 a b Λ μ Σ a ¯ b Λ μ Σ_ ¯ a ¯ b Λ_ μ Σ_ − G V G ∂μη ∂ η − G V G ∂μη ∂ η¯ − G V G ∂μη¯ ∂ η¯ ; 2 Λ ab Σ Λ ab Σ_ 2 Λ_ ab Σ_ ð108Þ where we have defined 1 a a ab ð1Þ ð1Þ † ¯ G ¼ ϕ A − V ½S Ψ A þðS Þ Ψ A ; ð109Þ A ;f 2 b ;f b ;f 1 a a ab ð1Þ ð1Þ † ¯ G ¼ ϕ Λ − V ½S Ψ ηΛ þðS Þ Ψ ηΛ ; ð110Þ Λ ;η 2 b ; b ;

G¯ a − Ga † Ga Ga † Λ_ ¼ ð ΛÞ , and they satisfy A ¼ð AÞ . Now, let us seek the field redefinition such that the quadratic terms in the first _ derivative of ηΛ and η¯Λ, the second line of (108), are removed. This becomes possible if we find a transformation realizing

a GΛ ¼ 0: ð111Þ

ab ð1Þ To find such a transformation, we first expand V and Sa as

ab abð0Þ abð1Þ abð1Þ ¯ abð2Þ ¯ ð1Þ ð0Þ ð1Þ ¯ V ¼ v þ v Ψ þðv Þ Ψ þ v ΨΨ;Sa ¼ sa þ sa Ψ; ð112Þ

abð0Þ ðiÞ a where v and sa depend only on ϕ . Then, Eq. (111) can be rewritten as 1 h i a abð0Þ ð0Þ abð0Þ ð0Þ ¯ abð0Þ ð1Þ abð1Þ ð0Þ ¯ abð0Þ ð1Þ abð1Þ ð0Þ ¯ ϕ Λ ¼ v s Ψ ηΛ þ v ðs Þ Ψ ηΛ þðv s þðv Þ s ÞΨ ηΛ Ψ þðv ðs Þ þ v ðs Þ ÞΨΨ ηΛ ;η 2 b ; b ; b b ; b b ; ð113Þ

1 h abð0Þ ð0Þ abð0Þ ð0Þ ¯ abð0Þ ð1Þ abð1Þ ð0Þ ¯ v s Ψ Λ v s Ψ Λ ℜ v s v s ΨΨ Λ ¼ 2 b ;η þ ð b Þ ;η þ ð b þð Þ b Þð Þ;η i 0 ð1Þ 1 ð0Þ ℑ abð Þ abð Þ Ψ Λ Ψ¯ − ΨΨ¯ Λ þ i ðv sb þðv Þ sb Þð ;η ;η Þ : ð114Þ

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Ψ Λ Ψ¯ − ΨΨ¯ Λ It is obvious that the first three terms can have their integral forms, but the last term of (114), proportional to ;η ;η , cannot be integrated as we will see below. Let us explicitly see the transformation where ϕa coincides with the integral of the right-hand side of (114) except the last term. Such a transformation is realized through 1 A A ϕb − bΨ b Ψ¯ bΨΨ¯ ηΛ ηΛ ϕb ψ β ψ¯ β_ η¯Λ_ η¯Λ_ ϕb ψ β ψ¯ β_ f ¼ f 2 ½A þðA Þ þ B ; ¼ ð ; ; Þ; ¼ ð ; ; Þ; ð115Þ where Ab and Bb are functions of ϕa, determined properly in the following. For keeping the equivalence between the former and the latter Lagrangians, we assume the field redefinition invertible. The assumption of the invertible transformation requires the bosonic part of the whole Jacobian matrix, 0 1 ∂fA ∂fA ∂fA B ∂ϕb ∂ψ β ∂ψ¯ β_ C B C B ∂ηΛ ∂ηΛ ∂ηΛ C B β _ C Jacobian matrix ¼ B ∂ϕb ∂ψ ∂ψ¯ β C; ð116Þ @ A ∂η¯Λ_ ∂η¯Λ_ ∂η¯Λ_ ∂ϕb ∂ψ β ∂ψ¯ β_ to have a nonzero determinant. The off-diagonal bosonic parts of the Jacobian matrix are inevitably zero because of the Grassmann-odd property. Therefore, the above requirement is equivalent to 0 1 ∂ηΛ ∂ηΛ ð0Þ ∂fA ð0Þ B ∂ψ β ∂ψ¯ β_ C det ≠ 0; and det @ A ≠ 0: ð117Þ ∂ϕb ∂η¯Λ_ ∂η¯Λ_ ∂ψ β ∂ψ¯ β_

Since we have assumed the transformation is invertible, we can invert the definition of fA in (115) for ϕa,

1 ϕa aΨ a Ψ¯ aΨΨ¯ ϕað0Þ ¼ 2 ½A þðA Þ þ B þ ðfÞ; ð118Þ where ϕað0ÞðfÞ is the inverse function of fA in (115). The derivative of (118) with respect to ηΛ gives

1 h i 1 h i a a Λ a ¯ Λ a Λ ¯ ¯ Λ b a a ¯ a ¯ ϕ Λ ¼ A Ψ η þðA Þ Ψ η þ B ðΨ η Ψ þ ΨΨ η Þ þ ϕ Λ A Ψ þðA Þ b Ψ þ B ΨΨ ;η 2 ; ; ; ; 2 ;η ;ϕb ;ϕ ;ϕb 1 1 1 a a ¯ a b a ¯ a b a ¯ ¼ A Ψ ηΛ þðA Þ Ψ ηΛ þ B þ A ðA Þ Ψ ηΛ Ψ þ B þ ðA Þ A ΨΨ ηΛ ; ð119Þ 2 ; ; 2 ;ϕb ; 2 ;ϕb ;

Ψ Λ Ψ 1 2 ΨΨ 0 Ψ¯ Λ Ψ¯ 1 2 Ψ¯ Ψ¯ 0 where we have recursively solved the first line and have used ;η ¼ð = Þð ÞηΛ ¼ and ;η ¼ð = Þð ÞηΛ ¼ ϕb to find the second line without ;ηΛ . Now, comparing (113) and (119), we can easily determine the coefficients Aa and Ba as

a abð0Þ ð0Þ A ¼ v sb ; ð120Þ 1 a a a abð0Þ ð1Þ abð1Þ ð0Þ abð0Þ ð0Þ cdð0Þ ð0Þ B ℜ C ; C v s v s − v s c v s : ¼ ½ where ¼ b þð Þ b 2 ð ð b Þ Þ;ϕ d ð121Þ

Introducing the transformation to the definitions (109) and (110), we then have

i Ga − ℑ a Ψ Λ Ψ¯ − ΨΨ¯ Λ Λ ¼ 2 ½C ð ;η ;η Þ; ð122Þ

044043-14 GHOST-FREE SCALAR-FERMION INTERACTIONS PHYS. REV. D 98, 044043 (2018) 1 1 a a a a ¯ a a c ¯ bð0Þ i a ¯ ¯ G ¼ δ þ ðA Ψ þðA Þ ΨÞþ ðB þ ℜ½A c ðA Þ ÞΨΨ ϕ A − ℑ½C ðΨ A Ψ − ΨΨ A Þ: ð123Þ A b 2 ;ϕb ;ϕb 2 ;ϕb ;ϕ ;ϕb ;f 2 ;f ;f

a a Since we cannot pick up the imaginary part of C as one can see from (121), we still have nonvanishing GΛ. We cannot Ψ Λ Ψ¯ − ΨΨ¯ Λ remove the dependence on ;η ;η in any way as we expected from the expression of (114). Nevertheless, the quadratic derivative interactions of fermions in (108) is the square of (122) (and its Hermitian), and thus they accidentally vanish due to the Grassmann properties,

Ga Gb ∝ Ψ Λ Ψ¯ − ΨΨ¯ Λ Ψ Σ Ψ¯ − ΨΨ¯ Σ 0 Λ Σ ð ;η ;η Þð ;η ;η Þ¼ ; ð124Þ ¯ ¯ ¯ ¯ ¯ Ga Gb ∝ Ψ Λ Ψ − ΨΨ Λ Ψ _ Ψ − ΨΨ _ 0; Λ Σ_ ð ;η ;η Þð ;η¯Σ ;η¯Σ Þ¼ ð125Þ whereas, the cross term of the redefined scalar field and fermionic field cannot be removed because

a b A μ Λ i ð0Þ að0Þ b ¯ ¯ A μ Λ G V G ∂μf ∂ η ¼ − v ϕ ℑ½C ðΨ ηΛ Ψ − ΨΨ ηΛ Þ∂μf ∂ η ≠ 0: ð126Þ A ab Λ 2 ab ;fA ; ;

As a result, the final Lagrangian with the field redefinition (115) with (120) and (121) is given by 1 a b A μ B a b A μ Λ a ¯ b A μ Λ_ L G V G ∂μf ∂ f − G V G ∂μf ∂ η − G V G ∂μf ∂ η¯ : rel ¼ 2 A ab B A ab Λ A ab Λ_ ð127Þ

Λ μ Σ Therefore, we conclude that the fermionic derivative interactions like ∂μη ∂ η can be eliminated by the field redefinition, A μ Λ but the cross terms between scalar fields and fermions like ∂μf ∂ η cannot be removed. Such cross terms indicate that the Euler-Lagrange equations apparently contain the second derivatives of the fermionic field, while they are reduced to first- order equations because the maximally degenerate conditions are satisfied,

− C BAB Ga Gb BAGc Gd ∝ Gb Gd 0 DΛΣ ΛBA AΣ ¼ BVab ΛA AVcd Σ Λ Σ ¼ ; ð128Þ

− C BAB Ga Gb BAGc G¯ d ∝ Gb G¯ d 0 DΛΣ_ ΛBA AΣ_ ¼ BVab ΛA AVcd Σ_ Λ Σ_ ¼ : ð129Þ

To summarize, the kinetic matrix cannot be transformed to the form in (106) with any field redefinition, but it can be reduced to 0 1 0 1 B B ˜ B˜ B˜ Aab aβ;J aβ_;J Aab aβ;J aβ_;J B Cfield redefinitionB C K B Cα Dαβ D _ C⟶B ˜ C ¼ @ b;I ;IJ αβ;IJ A @ Cαb;I 00A: ð130Þ Cα_ Dαβ_ D _ ˜ b;I ;IJ α_ β;IJ Cα_b;I 00

This fact indicates that the cross terms are really newly involve any derivatives of the fermionic field. Thus, the found derivative interactions. conformal transformation, which mixes the metric and the The crucial difference between the purely bosonic Weyl field, takes the following form10: degenerate system (such as Horndeski theory [4–6] and Degenerate Higher-Order Scalar-Tensor (DHOST) theories ¯ Ψ Ψ¯ [17,20]) and our scalar-fermion degenerate system is the gμν ¼ Að ; Þgμν: ð131Þ highest derivatives appearing in the Lagrangian. This is because the Lagrangian containing first derivatives of the 10A candidate for the disformal transformation including the ¯ ¯ fermionic fields in general yields second-order differential Weyl field is g¯μν ¼ AðΨ; ΨÞgμν þ BðΨ; ΨÞJμJν; where Jμ is now a equations of motion, which is a signal of fermionic appropriately mapped with tetrad fields eμ , defined by a bη (Ostrogradsky’s) ghosts. Therefore, this situation corre- gμν ¼ eμ eν ab, from a flat tangent space. However, one can sponds to the bosonic Lagrangian containing up to second easily show that the arbitrary function B can be absorbed into the conformal factor A by using (47). On the other hand, derivatives. On the analogy of the conformal and disformal the identity (C3) tells us that the disformal transformation transformation in the scalar-tensor theory, the fermionic with multiple Weyl fields is independent from the conformal version of conformal (disformal) transformations does not transformation.

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Such a transformation will be significantly useful in finding scalar fields and the fermions remain, which suggests that new theories of “tensor-fermion theories,” and we will they are strictly new terms. As shown in Ref. [10],ina leave this interesting issue to future work. bosonic case, any healthy theory even with arbitrary higher derivative terms can be reduced to a nondegenerate VII. SUMMARY AND DISCUSSION Lagrangian written in terms of (unconstrained) variables with at most the first-order time derivatives through canoni- As usually discussed in the Lagrangian composed cal transformation. On the other hand, in a fermionic case, of bosonic d.o.f., we have to avoid the appearance of any healthy theories require constraints to reduce extra d.o.f. ghosts even in -fermion coexisting Lagrangians. of fermionic variables. Then, it would be interesting to Fermions easily suffer from the fermionic ghost as pointed investigate what kind of a simpler Lagrangian can be out in Ref. [39]; i.e., fermionic d.o.f. should be constrained by the same number of constraints as the (physical) generally found through canonical transformation from d.o.f. That is true even when we additionally have bosonic healthy fermionic theories even with arbitrary higher deriva- d.o.f. tive terms. Our purpose is the construction of covariant derivative There are several directions for applying the method we interactions between fermions or between scalar fields and have developed: inclusion of nonlinear derivative inter- fermions, free from fermionic ghosts. We have given the actions, second or higher derivatives, gauge invariance and/ prescription to find new interactions in the Lagrangian with or gravity, and so on. Some of these issues will be discussed up to the first derivative of fields, made up of a set of in future publications. conditions imposed on the Lagrangian. One of the con- ditions is the maximally degenerate condition which ACKNOWLEDGMENTS produces the sufficient number of primary constraints. In We would like to thank Teruaki Suyama for fruitful this paper, all of the constraints are simply assumed to be discussions. This work was supported in part by second class. If we have gauge invariance, that is, first class JSPS KAKENHI Grants No. JP17K14276 (R. K.), constraints, the number of primary constraints can be No. JP15H05888 (M. Y.), No. JP25287054 (M. Y.), and smaller. However, it should be notice that, after the gauge No. JP18H04579 (M. Y.). fixing, all of the primary constraints can be turned into second class. In the Lagrangian formulation, the equations of motion for fermions can be reduced to the first-order APPENDIX A: IDENTITIES OF differential equations by imposing the maximally degen- PAULI MATRICES erate condition, which should be solved for the first derivative of each fermionic variable. Whether they are In this Appendix, we summarize notations and identities solvable is not a priori because of the Grassmann-odd of Pauli matrices [42]. We have used the following matrices nature, and the requirement is exactly the same with the defined as invertibility of the constraint matrix. σ¯ μαα_ εα_ β_εαβσμ The coefficients of the derivative of fields in the ¼ ββ_; ðA1Þ Lagrangian are complicated but classified thanks to the 1 covariance, and they become much simpler in case of one σμν β σμ σ¯ ναβ_ − σν σ¯ μαβ_ ð Þα ¼ 4 ð αα_ αα_ Þ; Weyl field with multiple scalar fields. One of the prominent 1 features which we can learn from the concrete analysis is μν α_ μαα_ ν ναα_ μ ðσ¯ Þ β_ ¼ ðσ¯ σ _ − σ¯ σ _Þ; ðA2Þ that the first condition, that is, the maximally degenerate 4 αβ αβ condition, affects the symmetric part of the Lagrangian μν μν γ μν μν γ_ ðσ εÞαβ ¼ðσ Þα εγβ; ðεσ¯ Þα_ β_ ¼ εα_ γ_ðσ¯ Þ β_: ðA3Þ with respect to the space-time indices of the derivatives, but the supplementary second condition affects the anti- Note that the last two matrices are symmetric under the μν μν symmetric part. exchange of the fermionic indices, i.e., ðσ εÞαβ ¼ðσ εÞβα The proposed Lagrangian is possible to be used as μν μν and ðεσ¯ Þα_ β_ ¼ðεσ¯ Þβ_ α_. They satisfy the following useful a model for the interaction between scalar fields and a properties: Weyl field, but some may ask what happens when we perform field redefinitions. Transformation including μ ββ_ β β_ μ σ σ¯ μ ¼ −2δα δα_ ; σ σ _ ¼ −2ε _εαβ; derivatives seems to introduce cubic or higher nonlinear αα_ αα_ μββ α_ β μαα_ ββ_ α_ β_ αβ derivative terms as well as second or higher derivatives σ¯ σ¯ μ ¼ −2ε ε ; ðA4Þ in the Lagrangian. In a simple setup where the whole Lagrangian is expressed by the proposed quadratic σμ σν − σν σμ 2 σμνε ε εσ¯ μν ε αα_ ββ_ αα_ ββ_ ¼ ½ð Þαβ α_ β_ þð Þα_ β_ αβ; ðA5Þ Lagrangian, the quadratic terms in the derivative of fermions are absorbed by certain invertible field redefinitions, but the μ ν ν μ μν μρ ν σ σ þσ σ ¼ −η εαβε _ −4ðσ εÞαβðεσ¯ ρ Þ _ : ðA6Þ derivative interaction terms between the derivative of the αα_ ββ_ αα_ ββ_ α_ β α_ β

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In addition, one can easily derive the following useful after we locally solve the canonical momenta of the scalar identities: fields for ϕ_ a, which can be justified by our assumption that b ∂πa=∂ϕ ¼ Aab has an inverse. Taking the derivative of 1 _ σμνε σ ε σμ ε σμ ψ_ β ψ¯_ β ð Þαβ νγγ_ ¼ 2 ð αγ βγ_ þ βγ αγ_Þ; ðA7Þ (B1) with respect to J and J,wehave ∂ ∂ ∂ 1 μ μ μ μ Fα;I Fα;I Fα;I μρ ν ν ν ν ν L β L β ⇔ ðσ εÞαβðεσ¯ ρ Þγ_ δ_ ¼ − ðσαγ_σ _ þ σ _σβγ_ þ σαγ_σ _ þ σ _σβγ_Þ; ψ_ αψ_ ¼ ϕ_ aψ_ þ β β 8 βδ αδ βδ αδ I J J ∂πϕa ∂ψ_ ∂ψ_ J J ∂ ðA8Þ Fα;I ba ¼ Dαβ;IJ þ Baβ;J ¼ Dαβ;IJ − Cαb;IA Baβ;J; ∂π a μρ ν ϕ ðσ εÞαβðσρ εÞγδ ðB2Þ 1 −ημν ε ε ε ε σμνε ε ¼ 4 ½ ð αγ βδ þ βγ αδÞþð Þαγ βδ ∂Fα;I ∂Fα;I ∂Fα;I μν μν μν L _ L _ ⇔ þðσ εÞβγεαδ þðσ εÞαδεβγ þðσ εÞβδεαγ: ðA9Þ ψ_ αψ¯_ β ¼ ϕ_ aψ¯_ β þ β_ β_ I J J ∂πϕa ∂ψ¯_ ∂ψ¯_ J J ∂Fα The contraction of the Weyl indices is given as B ;I − C baB ¼ Dαβ_;IJ þ aβ_;J ¼ Dαβ_;IJ αb;IA aβ_;J; ∂πϕa αβ μ ν μν μν ε σ σ ¼ −2ðεσ¯ Þ _ þ η ε _; ðA10Þ αα_ ββ_ α_ β α_ β ðB3Þ εα_ β_σμ σν −2 σμνε ημνε αα_ ββ_ ¼ ð Þαβ þ αβ: ðA11Þ where we have used the relations derived from the derivative of (B1) with respect to ϕ_ a, Therefore, any contraction of the building block matrices, ε ε σμ σμνε εσ¯ μν αβ, α_ β_, αα_, ð Þαβ, and ð Þα_ β_, with respect to the ∂Fα;I ∂Fα;I Lψ_ αϕ_ a ¼ Lϕ_ bϕ_ a ⇔ Cαa;I ¼ Aba ; ðB4Þ Lorentzian or fermionic indices reduces to the uncontracted I ∂πϕb ∂πϕb combination of them. Here, Eq. (A8) implies μρ ν ðσ εÞαβðεσ¯ ρ Þγ_ δ_ is symmetric under the exchange of μ in the last equalities. Thus, Eqs. (B2) and (B3) tell us that and ν and traceless, no dependence of Fα;I on the time derivative of the fermions means that we have 4N primary constraints for μρ ðσ εÞαβðεσ¯ ρμÞγ_ δ_ ¼ 0; ðA12Þ the fermions. Thus, Eqs. (B2) and (B3) suggest that the maximally degeneracy conditions, Eqs. (13) and (14), are through (A4). The matrices and their complex conjugates not only the sufficient condition for the existence of 4N are related through primary constraints for fermions but also the necessary condition for that. σμ σμ σμνε εσ¯ μν ð αβ_Þ ¼ βα_; ðð ÞαβÞ ¼ð Þβ_ α_: ðA13Þ

APPENDIX C: SCALAR-FERMION APPENDIX B: MAXIMALLY DEGENERATE THEORIES QUADRATIC IN FIRST CONDITIONS AND PRIMARY CONSTRAINTS DERIVATIVES OF MULTIPLE WEYL FIELDS As discussed in the Sec. II, we need 4N constraints In this Appendix, we extend our analysis in Sec. III to the making the number of the d.o.f. of fermions half to avoid case of multiple Weyl fields and construct the most general negative norm states in a n-scalar and N-fermion scalar-fermion theory of which the Lagrangian contains up Lagrangian with up to first derivatives. For the purpose, to quadratic in first derivatives of scalar and fermionic we have adopted the maximally degenerate conditions, fields. As in Sec. III, we first construct Lorentz-invariant Eqs. (13) and (14), having 4N primary constraints for the scalar quantities with no derivatives, which only contain ϕa ψ α ψ¯ α_ fermions, Eqs. (17) and (18). Here, we show that, if we scalar fields and fermionic fields I , I . Since the ε ε σμ have 4N primary constraints for the fermions, the max- fermionic indices can be contracted with αβ, α_ β_, αα_, μν μν imally degenerate conditions are automatically satisfied. ðσ εÞαβ, and ðεσ¯ Þα_ β_, we have five possibilities, From the definition of the canonical momenta of fermions, we generally have Ψ ψ αψ Ψ¯ ψ¯ ψ¯ α_ μ ψ¯ α_σμ ψ α IJ ¼ I J;α; IJ ¼ I;α_ J;JIJ ¼ I αα_ J; μν β μν β_ β β β_ β_ β β_ α μν ¯ α_ μν a a _ K ¼ ψ ðσ εÞαβψ ; K ¼ ψ¯ ðεσ¯ Þ _ψ¯ ; ðC1Þ πψα ¼ Fα ðϕ ; πϕa ; ∂ ϕ ; ψ ; ∂ ψ ; ψ¯ ; ∂ ψ¯ ; ψ_ ; ψ¯ Þ; IJ I J IJ I α_ β J I ;I i J i J J i J J J ðB1Þ which satisfy the following properties,

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Ψ Ψ μν − μν μν − νμ – IJ ¼ JI;KIJ ¼ KJI;KIJ ¼ KIJ; where we have used (A4) (A9). In general, any quantities † ¯ μ † μ μν † μν without derivatives and Weyl indices (Weyl indices are ðΨ Þ ¼ Ψ ; ðJ Þ ¼ J ; ðK Þ ¼ K¯ ; ðC2Þ IJ JI IJ JI IJ JI appropriately contracted) are written as and μ μ μ μ μ μ μ μν μν A 1 2 n A 1 2 n η ϕa Ψ Ψ¯ ¯ ¼ ð μν; ; IJ; IJ;JIJ;KIJ; KIJÞ; 1 μ ν − ημνΨ¯ Ψ − 2 ¯ λν λμ Ψ¯ μν − Ψ ¯ μν ðC12Þ JIJJKL ¼ 2 IK JL KIKKJL þ IKKJL JLKIK; ðC3Þ where n is zero or a natural number. In addition, from the above identities, we know that all scalar quantities without μ μ η ν −2Ψ¯ Ψ a ¯ JIJJμ KL ¼ μνJIJJKL ¼ IK JL; ðC4Þ derivatives can be expressed only by ϕ , ΨIJ, and ΨIJ.We also see that the space-time index of vector quantities is 1 μ μν μ μ described by J , and those of second-rank tensors are K Jν KL ¼ ðJ ΨJL − J ΨILÞ; ðC5Þ IJ IJ 2 KI KJ ημν μν ¯ μν μλ ¯ ν described by , KIJ, KIJ, and KIJKλ KL, i.e., μν ¯ K Jν Jμ ¼ −Ψ ðΨ Ψ − Ψ Ψ Þ; ðC6Þ IJ KL MN KM IN JL JN IL F F ϕa Ψ Ψ¯ Gμ GIJ μ ¼ ð ; IJ; IJÞ; ¼ ð1ÞJIJ; 1 ¯ μν μ ¯ μ ¯ μν μν IJ μν IJ ¯ μν IJKL μλ ¯ ν K Jν −J Ψ J Ψ ; C7 H ¼ H 1 η þ H K þ H K þ H K Kλ ; IJ KL ¼ 2 ð IL JK þ JL IKÞ ð Þ ð Þ ð2Þ IJ ð3Þ IJ ð4Þ IJ KL ðC13Þ ¯ μν −Ψ Ψ¯ Ψ¯ − Ψ¯ Ψ¯ KIJJν KLJμ MN ¼ NLð MJ KI MI KJÞ; ðC8Þ where F, Gμ, and Hμν are an arbitrary scalar, vector, and μν λ 1 μλ 1 μλ η −Ψ Ψ Ψ Ψ Ψ second-rank tensor, respectively. G 1 , H 1 , H 2 , and so on KIJKν KL ¼ 4 ð IL JK þ IK JLÞþ4 ðKIL JK ð Þ ð Þ ð Þ are scalar quantities just like F. Though further inves- − μλ Ψ − μλ Ψ μλ Ψ KIK JL KJL IK þ KJK ILÞ; ðC9Þ tigation is needed for the construction of arbitrary functions with Weyl indices, we can formally write down the most 1 μν ¯ λ μ λ − μ λ − μ λ μ λ general action with arbitrary functions, and this is given by KIJKν KL ¼ 8 ðJKIJLJ JLIJKJ JKJJLI þ JLJJKIÞ Z λν ¯ μ ¼ KIJKν KL; ðC10Þ 4 S ¼ d xðL0 þ L1 þ L2Þ; ðC14Þ μν ¯ 1 μ μ K Kνμ ¼ ðJ Jμ − J Jμ Þ¼0; ðC11Þ IJ KL 4 KI LJ LI KJ where

L0 ¼ P0; L ð1Þμ∂ ϕa ð2Þμ;I∂ ψ α ∂ ψ¯ α_ ð2Þμ;I † 1 ¼ P a μ þ P α μ I þ μ I ðP α Þ 1 μν a b μν;I a α a α_ μν;I † L2 V ∂μϕ ∂νϕ S α ∂μϕ ∂νψ ∂μϕ ∂νψ¯ S α ; ¼ 2 ab þ a I þ I ð a Þ 1 1 _ μν;IJ∂ ψ α∂ ψ β ∂ ψ¯ β∂ ψ¯ α_ μν;IJ † ∂ ψ¯ α_ μν;IJ∂ ψ α þ 2 Wαβ μ I ν J þ 2 ν J μ I ðWαβ Þ þ μ I Qαα_ ν J; ðC15Þ

ð1Þμ ð2Þμ;I μν μν;I μν;IJ μν;IJ and P0, P a, P α , Vab, Saα , Wαβ , and Qαα_ are appropriately contracted arbitrary functions of Lorentz scalar quantities, Pauli matrices, and fermionic fields. The properties of these coefficients are

† ð1Þμ † ð1Þμ μν † μν μν;IJ † νμ;JI ðP0Þ ¼ P0; ðP aÞ ¼ P a; ðVabÞ ¼ Vab; ðQαβ_ Þ ¼ Qβα_ ; μν νμ μν;IJ − νμ;JI Vab ¼ Vba;Wαβ ¼ Wβα : ðC16Þ

The maximally degenerate conditions, Eqs. (13) and (14), are

− C baB 00;IJ 00;I 00 −1ba 00;J 0 Dαβ;IJ αb;IA aβ;J ¼ Wαβ þ Sbα ðV Þ Saβ ¼ ; ðC17Þ

− C baB − 00;JI − 00;I 00 −1ba 00;J † 0 Dαβ_;IJ αb;IA aβ_;J ¼ Qαβ_ Sbα ðV Þ ðSaβ Þ ¼ : ðC18Þ

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We can write down the momenta as

π ð1Þ0 0ν ∂ ϕb 0ν;I∂ ψ α ∂ ψ¯ α_ 0ν;I † ϕa ¼ Pa þ Vab ν þ Saα ν I þ ν I ðSaα Þ ; ðC19Þ

πI − ð2Þ0;I − μ0;I∂ ϕa 0ν;IJ∂ ψ β − ∂ ψ¯ β_ μ0;JI ψα ¼ Pα Saα μ þ Wαβ ν J μ Qαβ_ ; ðC20Þ

πI − πI † ψ¯ α_ ¼ ð ψα Þ : ðC21Þ

00 ð0Þ ≠ 0 ϕ_ a 4 As in Sec. III, we assume det ðVabÞ . Substituting (C19) into (C20) for eliminating , we obtain the N primary constraints,

ΦI πI ð2Þ0;I − 00;I 00 −1 ab ð1Þ0 00;I 00 −1 abπ − 00;I 00 −1 ab 0i − i0;I ∂ ϕc ψ α ¼ ψ α þ Pα Saα ðV Þ Pb þ Saα ðV Þ ϕb ½Saα ðV Þ Vbc Scα i _ − 00;I 00 −1 ab 0i;J 0i;IJ ∂ ψ β ∂ ψ¯ β 00;I 00 −1 ab 0i;J † i0;JI ½Saα ðV Þ Sbβ þ Wαβ i J þ i J½Saα ðV Þ ðSbβ Þ þ Qαβ_ ; ΦI − ΦI † ψ¯ α_ ¼ ð ψα Þ : ðC22Þ

APPENDIX D: A SYSTEMATIC WAY TO FIND Thus, our goal in this Appendix is to find the most general INDIVIDUAL FUNCTIONS form of (D2). Let us begin with focusing on the quantities without the Weyl fields as well as the scalar fields, given by We show a systematic way to construct the concrete expression for the coefficients of the derivatives in (C15), no space-time index∶ f1; εαβ; ε _g; ðD3Þ while in the case of N ¼ 1, we have constructed them one by α_ β one. Our prescription proposes a minimal set of functions ∶ σμ categorized with respect to each tensorial type, but we do one space-time index f αα_g; ðD4Þ not, at this moment, consider the elimination of the redun- ∶ ημν σμνε εσ¯ μν dant functions of which the origin is in the invariance of a two space-time indices f ; ð Þαβ; ð Þα_ β_; Lagrangian under the addition of total derivatives when ρμ ρν ðσ εÞαβðεσ¯ Þγ_ δ_g: ðD5Þ the proposed functions are used as the coefficients in the Lagrangian. In Sec. III A 2, we concretely reduced the Here, we considered tensors with up to two space-time redundant functions coming from the total derivatives in indices, which are needed to construct (D2), and classified 1 the N ¼ case. We also leave the introduction of the them according to the number of the space-time indices. symmetries (C16) here. Let us first introduce a new notation. Other tensors can be obtained by combining and contracting Since any coefficients, in general, have space-time indices μν elements of (D4) and (D5),e.g.,σν••ðσ εÞ••.Itis,however, and Weyl indices, we label each with shown by using (A4)–(A9) that any combination of tensors ðindices for space-timejindices for Weyl componentsÞ; in (D4) and (D5) with any contraction of space-time indices is reduced to the linear combination of (D4) and (D5) with μν ðD1Þ coefficients of (D3), just as σν••ðσ εÞ•• is a linear combi- ε σμ according to the tensorial type. We do not include the labels nation of •• •• according to (A7). What we should consider αβ α_ β_ of scalar fields, a; b; , and fermions, I;J ,inthe next is the contractions of Weyl indices with ε and ε for notation since they are not related to any so far each combination of elements in (D3)–(D5).Ifthecontrac- and are supposed to be arbitrarily contracted with functions tion is taken for an index of (D3), it does not produce new μ αα a ¯ εαβσα • ε 1 of ϕ , ΨIJ,andΨIJ. With this notation, we can classify the types of tensors; e.g., 1 × is just reduced to −δα1 σμ candidates for the coefficients with respect to the tensorial β α1•. The contraction of two Weyl indices of each ημνψ σμνε ψ β ⊂ μν α μν;I type, e.g., f α;I; ð Þαβ I g ð j Þ.Then,Saα element in (D5) vanishes as they are symmetric. Therefore, should be composed of ðμνjαÞ tensors, and similar state- in order to find the most general form of (D2), Weyl indices ments also apply to other coefficients as of each combination of elements in (D3)–(D5) should not be contracted with epsilon tensors but with the Weyl fields ψ α α_ P0 ∈ ðjÞ; and ψ¯ . ð1Þμ ð2Þμ;I We begin with the quantities with no space-time index P a ∈ ðμjÞ;Pα ∈ ðμjαÞ; such as ðjαÞ, ðjα_Þ, ðjαβÞ, ðjαβ_Þ, and ðjα_ β_Þ. These are simply μν ∈ μν μν;I ∈ μν α μν;IJ ∈ μν αβ ψ α Vab ð jÞ;Saα ð j Þ;Wαβ ð j Þ; obtained by multiplying the building block (D3) by and α_ μν;IJ ψ¯ . All the possible combinations for each are listed in Q ∈ ðμνjαβ_Þ: ðD2Þ αβ_ Table I. [Column 1 is composed of an element in (D3) and

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TABLE I. For the sequence with no space-time index. TABLE II. For the sequence with one space-time index.

Type Column 1 Column 2 Type Column 1 Column 2 α ψ μ μ ðj Þ α;I ð jÞ JIJ α_ ψ¯ μ ðj Þ α_;I μ α μ β_ ψ ð j Þ σ ψ¯ JIJ αK αβ ε ψ ψ αβ_ I ðj Þ αβ α;I β;J μ β μ ðμjα_Þ σ ψ J ψ¯ α_ _ ψ ψ¯ βα_ I IJ K ðjαβÞ α;I β_;J _ ε ψ¯ ψ¯ ðjα_ βÞ α_ β_ α_;I β_;J

TABLE III. For the sequence with two space-time indices.

Type Column 1 Column 2 Column 3 μν ημν μν ¯ μν μρ ¯ ν ð jÞ , KIJ, KIJ, KIJ Kρ KL μν β μρ β ν μν μν ¯ μν μρ ¯ ν ðμνjαÞ σ ε ψ σ ε ψ ¯ η ψ α , K ψ α , K ψ α , K Kρ ψ α ð Þαβ I , ð Þαβ I Kρ JK I IJ K IJ K IJ KL M μν μν μρ μν α_ μν β_ μρ ν β_ μν ¯ ¯ ν ð j Þ εσ¯ ψ¯ εσ¯ ψ¯ η ψ¯ α_I, K ψ¯ α_K, K ψ¯ α_K, K Kρ ψ¯ α_M ð Þα_ β_ I , KIJ ð ρ Þα_ β_ K IJ IJ IJ KL μν αβ σμνε σμρε ¯ ν σμνε ψ γψ σμρε ψ γ ¯ ν ψ ημνψ ψ μνψ ψ ¯ μνψ ψ ð j Þ ð Þαβ, ð ÞαβKρ IJ ð Þαγ I βJ, ð Þαγ I Kρ JK βL, αI βJ, KIJ αK βL, KIJ αK βL, σμνε ψ γψ σμρε ψ γ ¯ ν ψ μρ ¯ ν ψ ψ ημνε μνε ð Þβγ I αJ, ð Þβγ I Kρ JK αL KIJ Kρ KL αM βN, αβ;KIJ αβ, ¯ μνε μρ ¯ ν ε KIJ αβ, KIJ Kρ KL αβ μρ γ ν γ_ μν γ μρ γ ¯ ν μν μν μν αβ_ σ ε ψ εσ¯ ψ¯ σ ε ψ ψ¯ _ σ ε ψ Kρ ψ¯ _ η ψ α ψ¯ _ , K ψ α ψ¯ _ , ð j Þ ð Þαγ I ð ρ Þβ_ γ_ J ð Þαγ I βJ, ð Þαγ I JK βL, I βJ IJ K βL μν γ_ μρ ν γ_ ¯ μν μρ ¯ ν εσ¯ ψ¯ ψ εσ¯ ψ¯ ψ K ψ α ψ¯ _ , K Kρ ψ α ψ¯ _ ð Þβ_ γ_ I αJ, KIJ ð ρ Þβ_ γ_ K αL IJ K βL IJ KL M βN _ εσ¯ μν μρ εσ¯ ν μν γ_ μρ ν γ_ ημνψ¯ ψ¯ μνψ¯ ψ¯ ðμνjα_ βÞ ð Þα_ β_, KIJ ð ρ Þα_ β_ ðεσ¯ Þα_ γ_ψ¯ ψ¯ β_ , K ðεσ¯ ρ Þα_ γ_ψ¯ ψ¯ β_ , α_I β_J, KIJ α_K β_L, I J IJ K L μν μρ μν γ_ μρ ν γ_ ¯ ¯ ν εσ¯ ψ¯ ψ¯ εσ¯ ψ¯ ψ¯ K ψ¯ α_Kψ¯ β_ , K Kρ ψ¯ α_Mψ¯ β_ , ð Þβ_ γ_ I α_J, KIJ ð ρ Þβ_ γ_ K α_L IJ L IJ KL N ημνε μνε ¯ μνε μρ ¯ ν ε α_ β_, KIJ α_ β_, KIJ α_ β_, KIJ Kρ KL α_ β_ the Weyl fields, and a cell in column 2 is made up of the obtained from the building block (D5) as the list of the combination of the upper left elements.] The quantities quantities with two space-time indices. [A cell in column 3 only with one space-time index such as ðμjÞ, ðμjαÞ, and is composed of the combination of the upper left elements ðμjα_Þ can be similarly obtained by multiplying the building with no Weyl indices and ðjαβÞ, ðjαβ_Þ, ðjα_ β_Þ.] block (D4) by the Weyl fields, shown in Table II. (A cell in We can write down all the coefficients in (C15) with the column 2 is composed of the combination of the upper left linear combinations of the elements listed in Tables I–III element and ðjαÞ, ðjα_Þ.) In a similar way, Table III is according to each tensorial type.

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