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Ghost-Free Scalar-Fermion Interactions PHYSICAL REVIEW D 98, 044043 (2018) Ghost-free scalar-fermion 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 fermions 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 field 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 matter, 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 effective field theory, which allows a ghost ing, which needs interactions between an inflaton and as long as it appears above the scale we are interested in; standard model particles, 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 particle 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 metric signature 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, 044043-2 GHOST-FREE SCALAR-FERMION INTERACTIONS PHYS. REV. D 98, 044043 (2018) 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
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