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Consider the free bosonic conformal field theory (CFT) in spacetime dimension $d=3$. I would like to explicitly construct a primary operator of spin $l=4$, with four scalar fields $\phi$ and five derivatives. The operator should also have odd (spacetime) parity, namely it should have an epsilon symbol $\epsilon^{\mu\nu\rho}$.

Since in free theory the dimension of $\phi$ is $\Delta_\phi=\frac{d-2}2=\frac12$ and $\partial^\mu$ has dimension 1, the primary we are looking for has dimension $4\frac12+5=7$ (I know from the character table that such primary exists). Here is my attempt to compute it.

We want an operator $O^{\mu\alpha\beta\gamma}$ which is symmetric in all indices and traceless. If I did not do anything wrong, the building blocks should be $$ O^{\mu\alpha\beta\gamma}_1 =\epsilon^{\mu\nu\rho} \quad\partial_\nu \partial_\alpha \partial_\beta \partial_\gamma \phi \quad \partial_\rho \phi \quad \phi \quad \phi, \\ O^{\mu\alpha\beta\gamma}_2 =\epsilon^{\mu\nu\rho} \quad\partial_\nu \partial_\alpha \partial_\beta \phi \quad \partial_\rho \partial_\gamma \phi \quad \phi \quad \phi, \\ O^{\mu\alpha\beta\gamma}_3 =\epsilon^{\mu\nu\rho} \quad\partial_\nu \partial_\alpha \partial_\beta \phi \quad \partial_\rho \phi \quad \partial_\gamma \phi \quad \phi, \\ O^{\mu\alpha\beta\gamma}_4 =\epsilon^{\mu\nu\rho} \quad\partial_\nu \partial_\alpha \phi \quad \partial_\rho \phi \quad \partial_\beta \partial_\gamma \phi \quad \phi, \\ O^{\mu\alpha\beta\gamma}_5 =\epsilon^{\mu\nu\rho} \quad\partial_\nu \partial_\alpha \phi \quad \partial_\rho \phi \quad \partial_\beta \phi \quad \partial_\gamma\phi. $$ I put some spaces for clarity. All other things vanish because of the antisymmetry of $\epsilon^{\mu\nu\rho}$. Also, you cannot contract all indices of $\epsilon^{\mu\nu\rho}$, otherwise you will need seven derivatives and you would increase the dimension. The goal would now be to take a combination $$ O^{\mu\alpha\beta\gamma} =\sum_{i=1}^5 c_i O^{\mu\alpha\beta\gamma}_i$$ where $c_i$ are real numbers fixed by the requirment that $O^{\mu\alpha\beta\gamma}$ is a primary, namely $$ K^\xi O^{\mu\alpha\beta\gamma} =0, $$ where $K^\xi$ is the generator of special conformal transformation. Since a primary is defined up to a numerical constant, we can fix $c_1=1$ and solve for four coefficients. In practice one needs to move $K^\xi$ to the right using the commutation relations. For the conformal algebra, I am using the conventions of https://arxiv.org/abs/1907.05147, see section 1.3. Of course $\phi$ itself is a primary, i.e. $ K^\xi \phi =0.$ I have been trying to implement all this in Mathematica and solve for the coefficients $c_2,\dots,c_5$, but could not find any solution. Does anyone know how could I proceed?

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    $\begingroup$ You couldn't find a solution for the coefficients? Or you couldn't find a solution to the problem of avoiding a tedious calculation? $\endgroup$ Jul 6, 2023 at 20:52
  • $\begingroup$ The first one, unfortunately. $\endgroup$
    – Rubilax96
    Jul 6, 2023 at 21:45

1 Answer 1

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I've got an answer: $$ c_1=0, \quad c_2=1, \quad c_3=-3, \quad c_4=5, \quad c_5=0.$$

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