The higher derivative fermionic operator and trace anomaly

16 August 2001

Physics Letters B 514 (2001) 377–384 www.elsevier.com/locate/npe

The higher derivative fermionic operator and trace anomaly Guilherme de Berredo-Peixoto a,b , Ilya L. Shapiro c,d,1 a Instituto de Física, Universidade de Brasilia, 70910-900, Brasilia, Brazil b Centro Brasileiro de Pesquisas Físicas-CBPF-CNPq, Rio de Janeiro, Brazil c Departamento de Física Teorica, Universidad de Zaragoza 50009, Zaragoza, Spain d Departamento de Física, Universidade Federal de Juiz de Fora, Brazil

Received 29 January 2001; received in revised form 21 June 2001; accepted 22 June 2001 Editor: P.V. Landshoff

Abstract We construct a new example of the higher derivative four-dimensional conformal operator. This operator acts on fermions, and its contribution to the trace anomaly has opposite sign, when compared with the conventional scalars, spinors and vectors. Possible generalizations and applications are discussed.  2001 Published by Elsevier Science B.V.

1. Introduction The local conformal symmetry plays an important role in gravitational physics. In some cases the local conformal symmetry together with the diffeomorphism invariance is referred to as the Weyl invariance. In this Letter we understand the local conformal symmetry as a symmetry under the simultaneous transformation of the metric gµν → gµν exp[2σ (x)] and φi → φi exp[dφi σ (x)], where φi is some matter field with the conformal weight dφi . Since we do not consider other sorts of conformal transformations or conformal symmetries, in what follows we will not write the word “local”, and simply use expressions like “conformal symmetry” or “conformal action”. In every case it is supposed that the corresponding action possesses local conformal symmetry and general covariance. Therefore, we consider the conformal invariance as being equivalent to the Weyl invariance. E-mail addresses: [email protected] (G. de Berredo-Peixoto), [email protected] (I.L. Shapiro). 1 On leave from Tomsk Pedagogical University, Tomsk, Russia.

On quantum level the Weyl invariance is violated by the anomaly. The origin of the anomaly is the renormalization of the ultraviolet (UV) divergences [1,2]. If one chooses a regularization in which the diffeomorphism invariance is preserved, then the (trace) anomaly breaks the conformal symmetry. The anomaly is relevant for the applications of quantum field theory in curved spacetime [3,4]. In particular, it is important for the inflationary cosmological model of Starobinsky [5] and the semiclassical approach to the derivation of the Hawking radiation [6]. Recently, there has been a considerable interest in the general properties of the anomaly and the anomaly-induced effective action [7,8] (see also [9] for the consequent discussions). In particular, the anomaly-induced effective action has been used to obtain systematic classification of the black hole vacuum states [10] and for the more detailed analysis of the Starobinsky model [11]. One should note that the anomaly-induced action is defined with accuracy to an arbitrary conformal invariant functional [12]. Since there is no regular way to derive this functional, it is useful to have various versions of the conformal invariant actions, which can be ap-

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plied to mimic the unknown conformal invariant part of the effective action [10]. We remark, that the form of some conformal invariants constructed from curvature tensor has been already discussed by mathematicians (see, e.g., [13]). The anomaly induced gravitational action can be considered as a four-dimensional (4d) analog of the two-dimensional (2d) Polyakov action. As the quantum analysis of the Polyakov theory has led to numerous interesting investigations (starting from [14]), it is natural to use the 4d induced action of [7,8] to construct the quantum theory of gravity [15,16] and to generalize the c-theorem from 2d for the 4d spacetime [17]. The contributions of the matter fields to the β-functions of the parameters of the vacuum action possess a remarkable universality. The vacuum action which is sufficient for the renormalizability of the conformal theory has the form
 √  Svacuum = d 4 x −g a1 C 2 + a2 E + a3 ✷R , (1) where 1 2 2 C 2 = Cµναβ C µναβ = Rµναβ − 2Rµν + R2 , 3 2 2 E = Rµναβ − 4Rµν + R2

(2)

are the square of the Weyl tensor and the integrand of the Gauss–Bonnet topological term. The conformally coupled fields: scalars  
√ 1 µν 1 g ∂µ φ∂ν φ + Rφ 2 S0 = d 4 x −g (3) 2 12 spinors
 √  i γ µ Ψ  γ µ ∇µ Ψ − ∇µ Ψ S1/2 = d 4 x −g Ψ 2
√  γ µ ∇µ Ψ + surface term = i d 4 x −g Ψ (4) and vectors  
√ 1 S1 = d 4 x −g − Fµν F µν 4

(5)

give positive contributions to the β-function of the parameter a1 and negative contributions to the β-function of the parameter a2 (see formulas (36)). In relation to the β-function of the a2 parameter the situation

remains the same in the framework of a supergravity theory. Besides (3), (4) and (5) conformal supergravity includes Weyl gravity (spin-2) and corresponding spin-3/2 fields. This universality has one important consequence: one can not cancel the anomaly by choosing the appropriate number of the fields of different spins. Therefore, at this point there is a difference between 2d and 4d cases. Indeed, in 2d the anomaly cancellation is possible, providing the existence of the critical dimension in string theory. Taking into account the interest to the higher (than 2d) dimensional conformal field theories, one can formulate two relevant questions: (i) whether one can construct the conformal invariant theories distinct from the theories (3), (4) and (5)? (ii) if this is possible, what would be the contribution of these fields to the anomaly? In principle, it may occur that including some special quantity of new fields into the definition of the integration measure one could cancel the anomaly. In this case these new conformal fields would violate the universality of the renormalization group flow. Then, the second question can be formulated as follows: whether it is possible to maintain the conformal symmetry at quantum level by introducing some new fields? Regarding the second problem, it was found that the cancellation is impossible for the conformal supergravity theory [18]. Also, as an example of the search for the new conformal theories, one can indicate Ref. [19], where the conformal tensor operators have been constructed. On the other hand, the higher derivative conformal operator acting on scalars has been found long ago [7,8,20] 1 2  = ✷2 + 2R µν ∇µ ∇ν − R✷ + (∇µ R)∇ µ . 3 3

(6)

Moreover, the contribution of the corresponding free scalar
√ S4 = d 4 x −g ϕϕ (7) to the trace anomaly has opposite sign [7], when compared to the usual fields (3), (4) and (5). 2 2 In [7] the contribution of (6) to the anomaly was taken with negative sign, because it was the compensation of the integration over the auxiliary field.

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Why is the existence of two different conformal scalars (3) and (7) possible? The remarkable difference between two conformal scalars is the transformation law for the fields

Using dimensional arguments, one can fix the possible form of the operator Dµ as

= gµν e2σ , gµν → gµν

where k1 and k2 are some unknown coefficients. Integrating by parts and omitting the surface terms, we get
√ µ ψ + · · · , ¯ µD S = i d 4 x −g ψγ (13)

φ → φ = φe−σ ,

ϕ → ϕ = ϕ.

(8)

The main purpose of the present article is to construct the spinor analog of the action (7). For the conventional spinor case (4) the conformal transformation has the form Ψ → Ψ = ψe−3σ/2 ,

→Ψ  = ψe ¯ −3σ/2 . Ψ

(9)

To construct the higher derivative conformal action, one has to start from the Hermitian action
 √  µ i ¯ µ ψ , (10) ¯ Dµ ψ − Dµ ψγ S3 = d 4 x −g ψγ 2 where Dµ is some third derivative covariant operator, and postulate the following transformation law for the spinor ψ: ψ → ψ = ψe−σ/2 ,

¯ −σ/2 . ψ¯ → ψ¯ = ψe

(11)

These conditions guarantee the global conformal invariance with σ = const [21]. However, the possibility to have local conformal invariance is not obvious and we are going to consider it in detail. The article is organized as follows. In Section 2 we present the construction of the conformal operator Dµ , and in Section 3 the calculation of its contribution to the trace anomaly. In the last section we draw our conclusions and discuss some mathematical conjecture and some speculations about possible physical applications of the conformal operators.

2. The derivation of the conformal operator The covariant derivative of the spinor is defined in a usual way i ∇µ ψ = ∂µ ψ + ωab µ Σab ψ, 2 i ¯ ab , ∇µ ψ¯ = ∂µ ψ¯ − ωab µ ψΣ 2 where Σab = 2i (γa γb − γb γa ) and ωab µ is the spinor connection. Consequently, µν ψ = 1 γ α γ β Rαβµν ψ. [∇µ , ∇ν ]ψ = R 4

Dµ = ∇µ ✷ + k1 Rµν ∇ ν + k2 R∇µ ,

(12)

where µ = 1 (∇µ ✷ + ✷∇µ ) + k1 Rµν ∇ ν + k2 R∇µ D 2  k1 k2 + + (∇µ R). 4 2

(14)

Making commutations of the covariant derivatives 1 1 (∇µ ✷ + ✷∇µ ) = ∇µ ✷ + [✷, ∇µ ] 2 2 one can calculate

(15)

[✷, ∇µ ]ψ = [∇ρ , ∇µ ]∇ ρ ψ + ∇ ρ [∇ρ , ∇µ ]ψ 1 = − γ α γ β Rαβµρ ∇ ρ ψ + Rµρ ∇ ρ ψ 2

 1 − γ α γ β ∇ ρ Rαβµρ ψ. (16) 4 Substituting (16) into (14), we obtain the useful form of the operator: µ = ∇µ ✷ + a1 Rµρ ∇ ρ + a2 R∇µ + a3 (∇µ R), D

(17)

where a1 = k1 , a2 = k2 and a3 = a41 + a22 − 18 — the last is a condition of Hermiticity. Our purpose will be to find values of a1,2,3 which provide both Hermiticity and conformal invariance of the action (10). For the one-parameter Lie group, one can consider just the infinitesimal transformation

gµν → gµν = (1 + 2σ )gµν ,

ψ → ψ = (1 − σ/2)ψ, ¯ ψ¯ → ψ¯ = (1 − σ/2)ψ. Then, disregarding higher orders in σ , after some long algebra we arrive at the following transformations

µ  ¯ ∇µ ✷ψ

ψγ

 

µ ¯ µ ✷ψ + ∇µ ψγ ¯ ✷ψ

= − ∇µ ψγ ¯ µ ✷ψ ¯ µ ✷ψ − ∇µ σ ψγ = −(1 − 4σ )∇µ ψγ

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¯ µ ∇ν σ ∇ ν ψ + i∇µ ψγ ¯ µ Σρν ∇ ν σ ∇ ρ ψ − ∇µ ψγ

µ  1 ¯ ✷ψ , ¯ µ ✷σ ψ + ∇µ ψγ + ∇µ ψγ (18) 2

µ  ¯ Rµρ ∇ ρ ψ

ψγ i ¯ µ Rµ ν Σνρ ∇ ρ σ ψ ¯ µ Rµρ ∇ ρ ψ − ψγ = (1 − 4σ )ψγ 2 1 ¯ µ ∇µ ∇ν σ ∇ ν ψ ¯ µ Rµν ∇ ν σ ψ − 2ψγ − ψγ 2 ¯ µ ✷σ ∇µ ψ, − ψγ (19) 

µ ¯ R∇µ ψ ψγ ¯ µ R∇µ ψ − 6ψγ ¯ µ ✷σ ∇µ ψ = (1 − 4σ )ψγ ¯ µ R∇µ σ ψ, + ψγ

µ  ¯ ∇µ Rψ

ψγ ¯ µ ∇µ Rψ − 2ψγ ¯ µ R∇µ σ ψ = (1 − 4σ )ψγ ¯ µ ∇µ ✷σ ψ. − 6ψγ

(20)

(21)

Substituting these formulas into (13) with (17), we find that the conformal invariance

√  √ µ ψ = −g ψγ µ ψ ¯ µD ¯ µD (22) −g ψγ holds for the unique choice of the (Hermitian) parameters a1 = 1,

a2 = −

5 , 12

a3 = −

1 . 12

(23)

3. One-loop divergences and anomaly The one-loop effective action, for the free theory of field ψ can be presented in the form

 

µ = − i Tr ln γ µ D µ γ ν D ν . Γ (1) = −i Tr ln γ µ D 2 (24) It proves useful to rewrite the last expression as the minimal six derivative operator µ γ ν D ν = H  = 1ˆ ✷3 + V µν ∇µ ∇ν ✷ γ µD µν ∇µ ∇ν µνα ∇µ ∇ν ∇α + U +Q µ ∇µ + P . +N

(25)

The derivation of the divergences can be performed using the generalized Schwinger–DeWitt technique developed in [22]. Since all the steps for this calculation are quite similar to the ones presented in [22] for the

four-derivative operator, we will not present the details here. The general formula for the divergences has the form i  − Tr ln H 2 div µn−4 (4π)2 (n − 4)


7 2 1 n √ R × d x −g + R2 120 µναβ 15 µν

=−

1 2 − R 2 − ✷R 6 5   1 µν  1 µν 1 Rµν − Rgµν + tr V Rµν + V 2 6 2

 1 µν gµν 2 µν − 1 V µν − 1 V µν V + g µν U 4 24 48 

 1 1 µν gµν µν − ✷V , − ∇µ ∇ν V (26) 3 12 µ and P  — terms, µνα , N which does not depend on Q as one would expect from dimensional arguments. Now we are in a position to calculate the divergent part of the one-loop effective action using (26). For the sake of generality we shall perform the calculations for arbitrary values of a1,2,3 and substitute (23) afterwards. After some algebra, disregarding the nonessential terms with more than four derivatives of the external metric, we arrive at the relations µ , D ν ] γ µ γ ν [D 1 = γ α γ β γ µ γ ν Rαβρλ Rµν ρλ ✷ 8 − 2γ µ γ ν Rνρ ∇µ ∇ ρ ✷   1 1 − ✷R✷ − R✷2 + γ µ , γ ν Rνρ ∇µ ∇ ρ ✷ 2 2  + a1 γ µ γ ν γ α γ β −Rαβσρ R σ [ν ∇µ] ∇ ρ 1 σ + Rαβ[µ Rν]σ ✷ 2 + a2 R 2 ✷ − 2a2 γ µ γ ν RRνρ ∇µ ∇ ρ + · · · ,

(27)

µ D µ = ✷3 + (2a1 + 1)Rµν ∇ µ ∇ ν ✷ D 1 − γ α γ β γ λ γ σ Rαβµρ Rλσ µρ ✷ 16 + a1 ✷R µν ∇µ ∇ν + 2a1 ∇µ ∇ρ R µσ ∇ ρ ∇σ

G. de Berredo-Peixoto, I.L. Shapiro / Physics Letters B 514 (2001) 377–384

 + a1 + a12 R µσ Rσρ ∇µ ∇ ρ + (a2 + a3 )✷R✷ + 2(a2 + a3 )∇µ ∇ν R∇ µ ∇ ν + (a2 + 2a1 a2 )RRµν ∇ µ ∇ ν + 2a2 R✷2 + a22 R 2 ✷ + · · · .

(28)

Then, the relevant blocks of (25) are  1 µν µν  V = 2a1 R + 2a2 − (29) Rg µν , 4 µν = a1 ✷R µν + 2a1 ∇ρ ∇ µ R ρν U  1 µν g ✷R + a2 + a3 − 4

 + a12 + a1 R µ ρ R ρν + 2(a2 + a3 )∇ µ ∇ ν R  a2 µν 2 + (2a1 a2 + a2 )RR µν + a12 − g R 2 1 − a1 γ [µ γ ρ] γ α γ β Rαβ σ ν Rσρ 2 a1 2 µν − Rαβ g − a2 γ µ γ ρ Rρ ν R. (30) 2 Using the formula (3), for the generic operator (17) we arrive at the following divergences
√ 1 (1) Γdiv = − d 4 x −g ε  2  2 × αRµναβ + βRµν + γ R 2 + δ✷R , (31) where 1 7 2 1 , β = a12 − a1 + , 120 3 3 15 1 2 1 4 1 2 γ = − a1 − 4a2 − 2a1 a2 − a1 − a2 − , 3 6 3 8 2 11 δ = a2 + 6a3 − . (32) 3 15 Replacing the coefficients (23) corresponding to the Hermitian conformal operator, and using (2), we arrive at the final result

α=

µ(n−4) ε  
√ 68 1 3 × d n x −g − C 2 + E − ✷R , 60 40 45

(1) Γdiv =−

(33)

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which is conformal invariant up to total derivatives. √ The cancellation of the non-conformal −g R 2 — term confirms the correctness of our calculations of both conformal operator and divergences. The conformal anomaly is directly related to the divergences [2], so that we have, using (33):  µ  T µ =−

 1 2 ωC + bE + c✷R 2 (4π)

(34)

with 1 3 68 (35) , b=+ , c=− . 60 40 45 Consider the cancellation of anomaly. In the fourdimensional space (D = 4), if one has only conventional scalar, spinor and vector fields, the cancellation of anomaly is impossible due to the fact that all these fields contribute to the coefficients ω and b (35) with the same signs [23]. But, the situation changes if we have higher derivative fields. For instance, some examples of finite and anomaly-free theory has been given in [24], where the infrared (IR) and UV conformal fixed points of the renormalization group flow were established. Now we shall see that including the new conformal fields one can achieve the anomaly cancellation in a different way. Let us consider a theory with N0 real conformally coupled scalars (3), N1/2 Dirac spinors (4) and N1 massless vectors (5). In addition, the theory includes n3 copies of the higher derivative spinor (10) and n4 copies of the higher derivative scalar (7). Then the total expression for the anomaly is (34), with the following total coefficients:  1 1 1 N0 + N1/2 + N1 ωt = 120 20 10  1 1 n4 + n3 , − 15 60  1 11 31 N0 + N1/2 + N1 bt = − 360 360 180  7 3 n4 + n3 , + 90 40 N1/2 N1 N0 68 2 ct = (36) + − − n4 − n3 . 180 30 10 45 45 In order to arrive at the conditions of anomaly cancellation, it is sufficient to consider only the coefficients ωt and bt , because ct can be always canceled ω=−

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by adding the local finite counterterm [2]
√ ct d 4 x −g R 2 . Sc = 12(4π)2

(37)

Then from (36), one arrives at the following solutions for the amount of the higher derivative fields 1 1 n3 = N1 − N1/2 − N0 , 2 8 5 7 5 n4 = N1 + N1/2 + N0 . (38) 4 8 32 Indeed, both n3 and n4 must be integers. One can see that the cancellation of trace anomaly is in principle possible, but it puts some restrictions on the field composition N0,1/2,1 of the matter theory. Namely, we need N0 + 4N1/2
8N1 , N0 to be multiple of 32, N1/2 to be multiple of 8 and N1 to be multiple of 4. The last conditions can be easily satisfied for some gauge groups. Then in some theory with extended supersymmetry, if the one-loop anomaly is exact, the proper choice of the numbers n3 and n4 can provide the complete cancellation of anomaly. In particular, this concerns the special values of the w and b coefficients (a and c in usual notations, see, e.g., [26]) which were derived in [26] using the AdS/CFT correspondence (see also further references therein). Indeed, this cancellation is achieved by the expense of an explicit violation of the supersymmetry. Another important question is the origin of the negative sign in the contribution to anomaly from the higher derivative operators ∆ and (17), (23). One can make a hypothesis that this “sign anomaly” is due to the presence of the higher derivative ghosts [27–29]. Unfortunately, we do not know any method to check this idea, and the only thing that can be investigated is whether the ghosts are really present in the two higher derivative conformal operators. For this purpose it is sufficient to consider the flat metric, because the pole structure of the propagator in curved spacetime can be investigated using the normal coordinates and local momentum representation [30]. Therefore we meet higher derivative theories with the Lagrangians L0 = µ ∂ )3 ψ. The content ¯ 1/2 (✷ϕ)2 and L1/2 = −ψ(iγ µ of the first theory has been extensively discussed in the literature starting from [28]. The general answer concerning the particle content of the theory can be obtained using the Hamiltonian formulation. In case of the higher derivative theory one needs the

Ostrogradsky method (see, e.g., [31]), which can be useful even for the theory with constraints (see, e.g., [32,33]). In our case we can use simplified Ostrogradsky method suggested in [29]. For the scalar theory L0 this method immediately demonstrates the existence of the unphysical ghost, that is the state with negative kinetic energy or negative norm in the vector space. This result agrees with the previous one of [28]. Let us concentrate our attention on the spinor case. For the higher derivative scalars, the simplified Ostrogradsky procedure of [29] is equivalent to the simple introduction of auxiliary fields. One has to remember that the choice of these fields does not influence the physical content of the theory (see corresponding theorem in [33]). It proves useful to introduce the following notations: Πψ = iγ µ ∂µ ψ,

¯ Π∗ ψ¯ = iγ µ ∂µ ψ.

Then, introducing the auxiliary spinor fields χ, χ¯ and η, η¯ according to χ = Πψ,

¯ χ¯ = Π∗ ψ,

η = Πχ,

η¯ = Π∗ χ, ¯

we can write the action in the following Hermitian form
S1/2 = d 4 x L1/2
1 ¯ = − Π∗ ηψ ¯ − χΠχ ¯ + Π∗ χχ). ¯ d 4 x (ψΠη 4 Introducing new variables u, v, w ψ = 2(u + w),

η = 2(u − w),

χ = 2v,

and making some partial integrations one can cast the action in the final form
S1/2 = d 4 x (uΠu ¯ − Π∗ uu) ¯
¯ − Π∗ vv) ¯ − d 4 x (vΠv
¯ − Π∗ ww). ¯ − d 4 x (wΠw (39) Then, depending on the overall sign of the action, the theory has one or two ghosts with, correspondingly, two or one positively defined constituents. In the same way one can treat the similar scalar theory ϕ✷3 ϕ [29].

G. de Berredo-Peixoto, I.L. Shapiro / Physics Letters B 514 (2001) 377–384

In order to check the validity of the simplified Ostrogradsky procedure, one has to check that the constraints between the new variables u, v, w do not lead to the reduction in the dimension of the phase space. This has been done (for the flat metric case) in the standard (but tedious) way, completely similar to the one described in [31]. We will not present the details here. The next observation concerns the physical interpretation of (39). This problem has been discussed by Pais and Uhlenbeck in [28] in 1950. The conclusion of [28] is that a fermionic theory of this kind does not have problems with ghosts, because the “appropriate definition of vacuum” can transform ghosts into physical degrees of freedom. We will not discuss this issue further, but only mention that the relation between the existence of the states with negative energy and the contribution to anomaly might be an interesting subject for further studies. For a fermionic theory the contributions to anomaly do not depend on the interpretation of the ghost states. The last remark is that in curved spacetime the introduction of the auxiliary fields (in both scalar and spinor cases) can not be done without breaking conformal invariance or locality. At the same time the original formulation of the Ostrogradsky method can be, in principle, applied and it is obvious that the dimension of the phase space will not be altered. Here we have used the weak-field approximation and the existence of the local momentum representation in order to establish the particle content of the theory in curved spacetime. The general investigation of the problem of the ghost states and unitarity in the higher derivative fermionic theories looks difficult but interesting and deserves serious independent study.

383

odd (for spinors) number of derivatives in D = 4. The scalars and spinors corresponding to these operators transform according to their classical dimension. The generalization to the D = 4 is also possible (see [25] for the ∆ operator). To check this conjecture would be an interesting mathematical problem. The signs of the contributions of a new third derivative spinor and of the fourth derivative scalar (7) to the trace anomaly are opposite to the ones of the usual scalars, spinors and vectors. One can guess that this sign distribution is related to the emergence and dominating contributions of the higher derivative ghosts, which are usually present in the spectrum of the higher derivative operators. As an extension of our conjecture, one can suppose that the signs of the contributions of the (yet unknown) higher order conformal operators to the trace anomaly coefficients ωt and bt will alter, as in (36). Therefore, if there exists for example, a supersymmetric gauge theory for which the one-loop anomaly is exact, one can define the integration measure in curved spacetime in such a way that the D = 4 conformal symmetry is exact. This definition of the measure must include a proper number of functional determinants of the operators µ . like ∆ and γ µ D

Acknowledgements Authors are grateful to I.L. Buchbinder and I.V. Tyutin for useful discussions, to K.E. Hibbard for reading the manuscript and to the CNPq (Brazil) for support. I.Sh. is also grateful to the RFFI (Russia) for the support of the theoretical physics group at Tomsk Pedagogical University through the project 9902-16617.

4. Conclusions and speculations We have constructed the 3-derivative Hermitian conformal spinor action. Our solution is a generalization of the flat-space operators [21] with global scale invariance. The relation between the corresponding spinor and conventional massless Dirac spinor is similar to the one between the fourth derivative scalar (7) and the usual conformal scalar (3). One can formulate the following mathematical conjecture: those are only first representatives of the infinite family of the conformal invariant operators with even (for scalars) and

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