30
3 Scalar-Tensor Gravities
(1)
= iTr ln(i∂ / − m − b /γ 5 + ω
/).
(3.33)
Just the same approach was used in [48] for the Lorentz-breaking extension of QED.
In the weak gravity case, we can use the approximation e μa η μa +
1
2
h μa . This trace
of the logarithm, however, can be calculated both in the weak field case and in the
full-fledged gravity case, with use of the Feynman diagrams or of the proper-time
method.
It is interesting that, similarly to the CFJ term, the 4D gravitational CS term is
ambiguous, i.e. the results for it depend on the calculation scheme. So, within all
these approaches, the linearized gravitational CS term
S C S = C
d
4 xh μν
μρκλ b κ ∂ λ
h ρ
ν
− ∂
ν
∂
σ h ρσ
,
(3.34)
or its full-fledged analogue (3.8) multiplied by 2C, was shown to arise, with the
constant C depends on the method of computation. So, in [51], where the calculations were carried out in the weak gravity case with use of the Feynman diagrams
constructed for the action (3.32), it was found that C =
1
192π 2 . Further, in [52], this
scheme has been realized for the finite temperature case where the zero component
of the internal momentum is supposed to be discrete, k 0 = (2n + 1)πT , so that the
result is
S C S =
d
4 x h μν
1
192π 2
ρμκλ b κ ∂ λ
h ρ
ν
− ∂
ν
∂
σ h ρσ
(3.35)
+
T
2
12
b 0
ρμκλ u κ ∂ λ
∂ 0 ∂
ν
− u
ν
∂ 0 ∂
σ
− u
σ
h ρσ
,
i.e. it looks like a sum of the zero-temperature result (3.34) and the additive term
proportional to T
2 .
In [53], where the proper time method has been used for the full-fledged gravity, the result was found in the form (3.8), with C =
1
128π 2 . Finally, in [54] it has
been argued that due to the arbitrariness in defining of conserved currents within the
functional integral approach, the constant C is actually completely ambiguous. The
similar situation occurs in QED [55]. However, the ambiguity of results is known to
be highly controversial, and in gravity it is even more controversial than in electrodynamics. For example, in [56] it was claimed that, if one suggests that the b μ is the
vacuum expectation value (v.e.v.) of a some dynamical field, the correct result for the
4D gravitational CS term is zero, as is also required by the gauge invariance of the
Lagrangian (and not only the action). Nevertheless, the question whether the requirements of [56] are indeed so necessary is still open, as the presence of ambiguities in
generic Lorentz-breaking theories is a strongly polemical problem.
However, there are also other interesting scalar-tensor gravity models which we
will consider now.
3 Scalar-Tensor Gravities
(1)
= iTr ln(i∂ / − m − b /γ 5 + ω
/).
(3.33)
Just the same approach was used in [48] for the Lorentz-breaking extension of QED.
In the weak gravity case, we can use the approximation e μa η μa +
1
2
h μa . This trace
of the logarithm, however, can be calculated both in the weak field case and in the
full-fledged gravity case, with use of the Feynman diagrams or of the proper-time
method.
It is interesting that, similarly to the CFJ term, the 4D gravitational CS term is
ambiguous, i.e. the results for it depend on the calculation scheme. So, within all
these approaches, the linearized gravitational CS term
S C S = C
d
4 xh μν
μρκλ b κ ∂ λ
h ρ
ν
− ∂
ν
∂
σ h ρσ
,
(3.34)
or its full-fledged analogue (3.8) multiplied by 2C, was shown to arise, with the
constant C depends on the method of computation. So, in [51], where the calculations were carried out in the weak gravity case with use of the Feynman diagrams
constructed for the action (3.32), it was found that C =
1
192π 2 . Further, in [52], this
scheme has been realized for the finite temperature case where the zero component
of the internal momentum is supposed to be discrete, k 0 = (2n + 1)πT , so that the
result is
S C S =
d
4 x h μν
1
192π 2
ρμκλ b κ ∂ λ
h ρ
ν
− ∂
ν
∂
σ h ρσ
(3.35)
+
T
2
12
b 0
ρμκλ u κ ∂ λ
∂ 0 ∂
ν
− u
ν
∂ 0 ∂
σ
− u
σ
h ρσ
,
i.e. it looks like a sum of the zero-temperature result (3.34) and the additive term
proportional to T
2 .
In [53], where the proper time method has been used for the full-fledged gravity, the result was found in the form (3.8), with C =
1
128π 2 . Finally, in [54] it has
been argued that due to the arbitrariness in defining of conserved currents within the
functional integral approach, the constant C is actually completely ambiguous. The
similar situation occurs in QED [55]. However, the ambiguity of results is known to
be highly controversial, and in gravity it is even more controversial than in electrodynamics. For example, in [56] it was claimed that, if one suggests that the b μ is the
vacuum expectation value (v.e.v.) of a some dynamical field, the correct result for the
4D gravitational CS term is zero, as is also required by the gauge invariance of the
Lagrangian (and not only the action). Nevertheless, the question whether the requirements of [56] are indeed so necessary is still open, as the presence of ambiguities in
generic Lorentz-breaking theories is a strongly polemical problem.
However, there are also other interesting scalar-tensor gravity models which we
will consider now.