Third Order Lagrangians, Weyl Invariants and Classical Trace Anomaly in Six Dimensions

Abstract

We have proceeded the analogy (represented in our previous works) of the Einstein tensor and the alternative form of the Einstein field equations for the generic coefficients of the eight terms in the third order of the Lovelock Lagrangian. We have found the constraint between the coefficients into two forms, an independent and a dimensional dependent versions. Each form has three degrees of freedom, and not only the exact coefficients of the third order Lovelock Lagrangian do satisfy the two forms of the constraints, but also the two independent cubic of the Weyl tensor satisfy the independent constraint in six dimensions and yield the dimensional dependent version identically independent of the dimension. Then, we have introduced the most general effective expression for a total third order type Lagrangian with the homogeneity degree number three which includes the previous eight terms plus the new three ones among the all seventeen independent terms. We have proceeded the analogy for this combination, and have achieved the relevant constraint. We have shown that the expressions given in the literature as the third Weyl-invariant combination in six dimensions do satisfy this constraint. Thus, we suggest that these constraint relations to be considered as the necessary consistency conditions on the numerical coefficients that a Weyl-invariant in six dimensions should satisfy. Finally, we have calculated the “classical” trace anomaly (an approach that was presented in our previous works) for the introduced total third order type Lagrangian and have achieved a general expression with four degrees of freedom in more than six dimensions (three degrees in six dimensions). Then, we have demonstrated that the resulted expression contains exactly the relevant coefficient of the Schwinger–DeWitt proper time method (that linked with the relevant heat kernel coefficient) in six dimensions, as a particular case. Of course, this result is a necessary consistency check, nevertheless our approach can be regarded as an alternative (perhaps simpler, and classical) derivation of the trace anomaly which also gives a general expression with the relevant degrees of freedom.