D. No-go result for in

The solution to the `homogeneous' equation (119) can be represented as

where

and is a nonvanishing, local current.

According to the general result expressed by (75) in both antighost and pure ghost numbers equal to zero, equation (217) implies

where are listed in (75). Solution (219) is assumed to provide a cross-coupling Lagrangian. Therefore, since is the most general gauge-invariant quantity depending on the field , it follows that each interaction vertex from is required to be at least linear in and to depend at least on a BF field. But contains two spacetime derivatives, so the emerging interacting field equations would exhibit at least two spacetime derivatives acting on the BF field(s) from the interaction vertices. Nevertheless, this contradicts the general assumption on the preservation of the differential order of each field equation with respect to the free theory (see assumption ii) from the beginning of section 4), so we must set

Next, we solve equation (218). In view of this, we decompose with respect to the number of derivatives acting on the fields as

where each contains precisely spacetime derivatives. Of course, each is required to mix the BF and field sectors in order to produce cross-interactions. In agreement with (221), equation (218) is equivalent to

Using definitions (45)-(47) and an integration by parts it is possible to show that

From (225) we observe that is solution to (222) if and only if the following conditions are satisfied simultaneously

Because is derivative-free, the solutions to equations (226)-(227) read as

where , , , , , and are some real, constant tensors. In addition, display the same mixed symmetry properties like the tensor field and , , and are completely antisymmetric. Because there are no such constant tensors in , we conclude that (226 )-(227) possess only the trivial solution, which further implies that

Related to equation (223), we use again definitions (45)-(
47) and integrate twice by parts, obtaining

Inspecting (231), we observe that satisfies equation (223) if and only if the following relations take place simultaneously

The solutions to equations (232)-(233) are expressed by

where the quantities , , , , , , and are some tensors depending at most on the undifferentiated fields from (2). In addition, they display the symmetry/antisymmetry properties

and , , and are completely antisymmetric. Because both tensors and are derivative-free, their are related through

Using successively properties (237)-(239) and formula (240), it can be shown that is completely antisymmetric. This last property together with (239) leads to

which replaced in the latter equality from (234) produces

This means that the entire dependence of on is trivial (reduces to a full divergence), and therefore can at most describe self-interactions in the BF sector. Since there is no nontrivial solution to (223) that mixes the BF and field sectors, we can safely take

In the end of this section we analyze equation (224). Taking one more time into account definitions (45)-(47), it is easy to see that (224) implies that the EL derivatives of are subject to the equations

Because (and also its EL derivatives) contains two spacetime derivatives, the solution to both equations from (242) is of the type

where depends only on the undifferentiated fields and exhibits the mixed symmetry . This means that is simultaneously antisymmetric in its first three and respectively last two indices and satisfies the identity . The solutions to the remaining equations, (243) and (247), can be represented as

where the functions , , and are completely antisymmetric and contain a single spacetime derivative.

Let be a derivation in the algebra of the fields
,
, ,
,
,
, and of their derivatives, which counts the powers of these fields and of
their derivatives

We emphasize that does not `see' either the scalar field or its spacetime derivatives. It is easy to check that for every nonintegrated density we have

If is a homogeneous polynomial of degree in the fields , , , , , and their derivatives (such a polynomial may depend also on and its spacetime derivatives, but the homogeneity does not take them into consideration since is allowed to be a series in ), then

Based on results (245)-(247), we can write

We decompose along the degree as

where ( in ( 251) because , and hence every , is assumed to describe cross-interactions between the BF model and the tensor field with the mixed symmetry ), and find that

Comparing (252) with (250), it follows that decomposition ( 251) induces a similar one with respect to each function , , , , , and

Inserting (253) and (254) in (250) and comparing the resulting expression with (252), we get

Replacing the last result, (255), into (251), we further obtain

where

So far, we showed that the solution to (224) can be put in the form ( 256). By means of definitions (36)-(37), we can bring (256) to the expression

The -exact modulo terms in the right-hand side of (259) produce purely trivial interactions, which can be eliminated via field redefinitions. This is due to the isomorphism in all positive values of the ghost number and respectively of the pure ghost number [42], which at allows one to state that any solution of equation ( 224) that is -exact modulo is in fact a trivial cocycle from . In conclusion, the only nontrivial solution to (224) can be written as

where displays the mixed symmetry , is derivative-free, and is required to depend at least on one field from the BF sector. But already contains two spacetime derivatives, so such a disagrees with the hypothesis on the differential order of the interacting field equations (see also the discussion following formula (219)), which means that we must set

Substituting results (230), (241), and (261) into decomposition (221), we obtain

which combined with (220) proves that indeed there is no nontrivial solution to the `homogeneous' equation (119) that complies with all the working hypotheses

2018-03-26