EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

MASSIVE VECTOR PARTICLES IN THE
ANTISYMMETRIC TENSOR MODEL
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This chapter presents a detailed analysis of the antisymmetric tensor
model for three massive vector particles including the interaction terms
with coupling constants of mass unit eV 1 and eV 0 . First, all available
Lorentz structures relevant to three- and four-field interactions are derived.
Consequently, the U(1) invariance is employed to reduce the
number of free coupling constants. Next, a constraint analysis is used to
impose self-consistency on the coupling constants. Finally, conditions
for the renormalizability of the theory are calculated.
6.1 available lorentz structures of the interaction
terms
From the Lagrangian density of the free tensor model in equation (4.7)
on page 30 with mass unit eV 4 one derives 1 that the antisymmetric tensor
field W µν has mass unit eV. Using the fact that all Lorentz indices in the
interaction terms must be completely contracted, interaction terms with
coupling constants possessing mass unit eV 1−n , where n = 0, 1, 2, . . . ,
can be constructed. In the following, all available Lorentz structures for
the cases n = 0, i.e. three fields, and n = 1, i.e. four fields, are motivated.
Note that the derivative operator ∂ µ —with mass unit eV—needs not to
be taken into account since possible terms
(1) must contain an even number of derivatives,
(2) must not represent a total divergence, and
(3) must not be equal to the kinetic term.
The Case n = 0: Three Fields
A general interaction term has the form of a rank-six tensor
W a α 1β 1
W b α 2β 2
W c α 3β 3
, (6.1)
1 The parameters a and b in equation (4.7) are dimensionless in order to be consistent
with the fundamental commutation relations in canonical quantization.
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