A. No-go result for $ I=5$ in $ a^{\mathrm {int}}$

In agreement with (86), the general solution to the equation $ sa^{
\mathrm{int}}=\partial ^{\mu }m_{\mu }^{\mathrm{int}}$ can be chosen to stop at antighost number $ I=5$

$\displaystyle a^{\mathrm{int}}=a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{int }}+a_{3}^{\mathrm{int}}+a_{4}^{\mathrm{int}}+a_{5}^{\mathrm{int}},$ (220)

where the components on the right-hand side of (181) are subject to the equations (68) and (66)-(67) for $ I=5$.

The piece $ a_{5}^{\mathrm{int}}$ as solution to equation (68) for $ I=5$ has the general form expressed by (75) for $ I=5$, with $ \alpha
_{5}$ from $ H_{5}^{
\mathrm{inv}}(\delta \vert d)$. According to (81) at antighost number five, it follows that $ H_{5}^{
\mathrm{inv}}(\delta \vert d)$ is spanned by the generic representatives (82). Since $ a_{5}^{\mathrm{int}}$ should effectively mix the BF and the $ (2,1)$ tensor field sectors in order to produce cross-couplings and (82) involves only BF generators, it follows that one should retain from the basis elements $ e^{5}\left( \eta ^{\bar{\Upsilon}}\right) $ only the objects containing at least one ghost from the $ (2,1)$ tensor field sector, namely $ D_{\mu \nu
\rho }$ or $ S_{\mu }$. Recalling that we work precisely in $ D=5$, we obtain that the general solution to (68) for $ I=5$ reduces to

$\displaystyle a_{5}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{3!}\left( \left( \tilde{U}_{1}\right)
C+\left( \tilde{U...
...alpha }\tilde{D}^{\alpha \beta }\tilde{D}_{\beta \nu }\sigma ^{\mu \nu }
\notag$ (221)
    $\displaystyle +\tfrac{1}{2}\left( \left( \tilde{U}_{3}\right) \eta S^{\mu }-\le...
...D}_{\nu \alpha }\tilde{D}_{\rho
\beta }\sigma ^{\alpha \beta }\right) S_{\mu }.$ (222)

Each tilde object from the right-hand side of (182) means the Hodge dual of the corresponding non-tilde element, defined in general by formula ( 92). The elements $ \tilde{U}$ are dual to $ \left( U\right) _{\mu
_{1}\ldots \mu _{5}}$ as in (82), with $ W\left( \varphi \right) $ respectively replaced by the smooth function $ U\left( \varphi \right) $ depending only on the undifferentiated scalar field $ \varphi $.

Introducing (182) in equation (66) for $ I=5$ and recalling definitions (35)-(52), we obtain

$\displaystyle a_{4}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{6}\tilde{D}_{\mu \alpha }\tilde{D}
^{\alpha \beta }\si...
...a \nu }+\tfrac{3}{2}C\tilde{F}
_{\beta \nu \vert\lambda }\right) \right. \notag$ (223)
    $\displaystyle \left. -\left( \tilde{U}_{2}\right) ^{\lambda }\left( \tfrac{1}{5...
...ft( V_{\lambda }S_{\mu }+\eta \mathcal{C}_{\lambda
\mu }\right) S^{\mu } \notag$ (224)
    $\displaystyle -\tfrac{1}{4}\left( \tilde{U}_{4}\right) ^{\lambda }\left[ D^{\mu...
...{\rho \beta
\vert\lambda }\sigma ^{\alpha \beta }S_{\mu }\right) \right. \notag$ (225)
    $\displaystyle \left. -F^{\mu \nu \rho \vert\gamma }\tilde{D}_{\nu \alpha }\tild...
... ^{\alpha \beta }\sigma _{\gamma \lambda }\right] +\bar{
a}_{4}^{\mathrm{int}}.$ (226)

In (183) $ \left( \tilde{U}\right) ^{\lambda }$ are dual to (83 ), with $ W\left( \varphi \right) \rightarrow U\left( \varphi \right) $. In addition, $ \mathcal{C}_{\mu \rho }$ is implicitly defined by formula (74) so it is a ghost field of pure ghost number one without definite symmetry/antisymmetry property, $ \mathcal{C}^{\ast \nu \lambda }$ is its associated antifield, defined such that the antibracket $ \left( \mathcal{C}
_{\mu \rho },\mathcal{C}^{\ast \nu \lambda }\right) $ is equal to the `unit' $ \delta _{\mu }^{\nu }\delta _{\rho }^{\lambda }$

$\displaystyle \mathcal{C}^{\ast \nu \lambda }\equiv 3S^{\ast \nu \lambda }+A^{\ast \nu \lambda }.$ (227)

The nonintegrated density $ \bar{a}_{4}^{\mathrm{int}}$ stands for the solution to the homogeneous equation (68) for $ I=4$, showing that $ \bar{a}_{4}^{\mathrm{int}}$ can be taken as a nontrivial element of $ H\left( \gamma \right) $ in pure ghost number equal to four.

At this stage it is useful to decompose $ \bar{a}_{4}^{\mathrm{int}}$ as a sum between two components

$\displaystyle \bar{a}_{4}^{\mathrm{int}}=\hat{a}_{4}^{\mathrm{int}}+\check{a}_{4}^{\mathrm{ int}},$ (228)

where $ \hat{a}_{4}^{\mathrm{int}}$ is the solution to (68) for $ I=4$ which is explicitly required by the consistency of $ a_{4}^{\mathrm{int}}$ in antighost number three (ensures that (67) possesses solutions for $ i=4
$ with respect to the terms from (183) containing the functions of the type $ U$) and $ \check{a}_{4}^{\mathrm{int}}$ signifies the part of the solution to (68) for $ I=4$ that is independently consistent in antighost number three

$\displaystyle \delta \check{a}_{4}^{\mathrm{int}}=-\gamma \check{c}_{3}+\partial _{\mu } \check{m}_{3}^{\mu }.$ (229)

Using definitions (35)-(52) and decomposition (185 ), by direct computation we obtain that
$\displaystyle \delta a_{4}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \delta \left[ \hat{a}_{4}^{\mathrm{int}}-
\tfrac{1}{2}S^{\alpha }...
...u \nu \rho \lambda }\eta _{\mu \nu \rho \lambda }^{\ast }\right. \right.
\notag$ (230)
    $\displaystyle \left. \left. +\tfrac{1}{60}\left( \tilde{U}_{3}\right) ^{\mu \nu...
... }^{\ast }\right) \right]
+\gamma c_{3}+\partial _{\mu }j_{3}^{\mu }+\chi _{3},$ (231)

where we made the notations
$\displaystyle c_{3}$ $\displaystyle =$ $\displaystyle -\check{c}_{3}+\tfrac{1}{12}\left( \tilde{U}_{1}\right) ^{\lambda...
... \vert\lambda
}^{\rho \alpha }\tilde{F}_{\alpha \nu \vert\sigma }\right] \notag$ (232)
    $\displaystyle -\tfrac{1}{240}\left( \tilde{U}_{2}\right) ^{\lambda \sigma }\til...
... \vert\lambda }^{\rho \alpha }\tilde{F}_{\alpha \nu \vert\sigma }\right]
\notag$ (233)
    $\displaystyle -\tfrac{1}{12}\left( \tilde{U}_{3}\right) ^{\lambda \sigma }\left...
...al{C}_{\lambda \rho }\mathcal{C}_{\sigma
\mu }\sigma ^{\rho \mu }\right] \notag$ (234)
    $\displaystyle -\tfrac{1}{2}\left( \left( \tilde{U}_{3}\right) ^{\mu \nu \sigma ...
...nu \rho \lambda }^{\ast }\right)
S^{\alpha }\mathcal{C}_{\sigma \alpha } \notag$ (235)
    $\displaystyle -\tfrac{1}{24}\left( \tilde{U}_{4}\right) ^{\lambda \sigma }\sigm...
...lpha }\tilde{D}_{\rho \beta }t_{\lambda \sigma \vert\mu }\right) \right.
\notag$ (236)
    $\displaystyle \left. +3F_{\qquad \vert\lambda }^{\mu \nu \rho }\tilde{D}_{\nu \...
...{C}_{\sigma \mu }-2\tilde{F}_{\rho \beta
\vert\sigma }S_{\mu }\right) \right] ,$ (237)

$\displaystyle \chi _{3}$ $\displaystyle =$ $\displaystyle -\tfrac{1}{4}\left( \left( \tilde{U}_{1}\right) ^{\lambda
\sigma ...
...ha }\tilde{D}^{\alpha \beta }
\tilde{R}_{\beta \nu \vert\lambda \sigma } \notag$ (238)
    $\displaystyle +\tfrac{1}{6}\left( \tilde{U}_{3}\right) ^{\mu \nu }\eta S^{\rho ...
...\nu \rho }\tilde{D}
_{\nu \alpha }\tilde{D}_{\rho \beta }S_{\mu }\right. \notag$ (239)
    $\displaystyle \left. +D^{\mu \nu \rho }\tilde{D}_{\nu \alpha }\left( \tilde{D}_...
...gma \mu }-6\tilde{R}_{\rho \beta \vert\lambda \sigma
}S_{\mu }\right) \right] ,$ (240)

and $ j_{3}^{\mu }$ are some local currents. In (187)-(189) $ \left( \tilde{U}\right) ^{\mu \nu }$ and $ \left( \tilde{U}\right) ^{\mu \nu
\rho }$ denote the duals of (84) and (85) with $ W\left( \varphi \right) \rightarrow U\left( \varphi \right) $. In addition, $ \left(
\tilde{U}\right) ^{\mu \nu \rho \lambda }$ represents the dual of $ \left(
U\right) _{\mu }=\frac{dU}{d\varphi }H_{\mu }^{\ast }$ and $ \left( \tilde{U}
\right) ^{\mu \nu \rho \lambda \sigma }$ the dual of $ U\left( \varphi \right) $. Inspecting (187), it follows that the consistency of $ a_{4}^{\mathrm{int}}$ in antighost number three, namely the existence of $ a_{3}^{\mathrm{int}}$ as solution to (67) for $ i=4
$, requires the conditions

$\displaystyle \chi _{3}=\gamma \hat{c}_{3}+\partial _{\mu }\hat{\jmath}_{3}^{\mu }$ (241)

$\displaystyle \hat{a}_{4}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}S^{\alpha }S_{\alpha }\left(
\left( \tilde{U}_{3}\rig...
...tilde{U}_{3}\right) ^{\mu \nu \rho }\eta _{\mu \nu \rho }^{\ast
}\right. \notag$ (242)
    $\displaystyle \left. +\tfrac{1}{12}\left( \tilde{U}_{3}\right) ^{\mu \nu \rho \...
... \nu \rho \lambda \sigma }\eta _{\mu \nu \rho \lambda
\sigma }^{\ast }\right) ,$ (243)

where we made the notations $ \hat{c}_{3}=-\left( a_{3}^{\mathrm{int}
}+c_{3}\right) $ and $ \hat{\jmath}_{3}^{\mu }=\overset{(3)}{m}^{\mathrm{int~}
\mu }-j_{3}^{\mu }$. Nevertheless, from (189) it is obvious that $ \chi _{3}$ is a nontrivial element from $ H\left( \gamma \right) $ in pure ghost number four, which does not reduce to a full divergence, and therefore (190) requires that $ \chi _{3}=0$, which further imply that all the functions of the type $ U$ must be some real constants

$\displaystyle U_{1}\left( \varphi \right) =u_{1},\qquad U_{2}\left( \varphi \ri...
...d U_{3}\left( \varphi \right) =u_{3},\qquad U_{4}\left( \varphi \right) =u_{4}.$ (244)

Based on (192), it is clear that $ a_{5}^{\mathrm{int}}$ given by ( 182) vanishes, and hence we can assume, without loss of nontrivial terms, that

$\displaystyle a_{5}^{\mathrm{int}}=0$ (245)

in (181).

Ashkbiz Danehkar