# D. No-go result for in

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

 (307)

where
 (308) (309)

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

 (310)

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

 (311)

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

 (312)

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
 (313) (314) (315)

Using definitions (45)-(47) and an integration by parts it is possible to show that
 (316) (317) (318)

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

 (319) (320)

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

 (321) (322)

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

 (323)

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

 (324) (325) (326)

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

 (327) (328)

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

 (329) (330) (331)

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

 (332) (333) (334)

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

 (335)

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

 (336)

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

 (337) (338) (339)

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

 (340)

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

 (341) (342)

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

 (343) (344) (345)

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
 (346) (347)

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
 (348) (349)

We decompose along the degree as

 (350)

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

 (351)

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

 (352) (353)

Inserting (253) and (254) in (250) and comparing the resulting expression with (252), we get
 (354) (355)

Replacing the last result, (255), into (251), we further obtain
 (356) (357)

where

 (358) (359)

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
 (360) (361)

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

 (362)

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

 (363)

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

 (364)

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

 (365)

Ashkbiz Danehkar
2018-03-26