# Primary operators in $d=3$ (bosonic free) conformal field theory

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?

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

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