Eur. Phys. J. C (2019) 79 :50 Page 3 of 15 50
where e
a
μ
(x) are the vielbeins; g
μν
(x) = e
a
μ
(x)η
ab
e
b
ν
(x);
η
ab
is the Minkowski metric of the tangent space at a
space-time point with coordinates x
μ
; the generalised spin-
connection is:
ω
abμ
= ω
abμ
+ K
abμ
; K
abc
=
1
2
(H
cab
−
H
abc
− H
bca
) =−
1
2
H
abc
; ω
abμ
is the standard torsion-
free spin connection.
4
Our convention is that Latin letters
denote tangent-space indices, while Greek letters refer to
space-time indices. In (10), we used standard properties of
the γ -matrices. For a Robertson-Walker metric g
μν
back-
ground, of relevance to us here, F
μ
= 0, and thus we can
write the action (10)intheform:
S
Di r a c
=
d
4
x
√
−g
¯
ψ
ıγ
μ
∂
μ
− m
ψ
+
d
4
x
√
−gB
μ
¯
ψγ
5
γ
μ
ψ
≡ S
Free
Di r a c
−
d
4
x
√
−gB
μ
J
5μ
, (11)
thus yielding a minimal coupling of the H
μνρ
field to the
fermion axial current.
In four space-time dimensions, the KR three-form H can
be expressed in terms of its dual pseudoscalar b(x) (KR
“axion” ) field [10,17]
∂
μ
b =−
1
4
e
−2φ
ε
μ
abc
H
abc
, (12)
where ε
0123
=+1,ε
0123
=−1, etc. is the gravitationally
covariant totally antisymmetric Levi-Civita tensor. From the
definition of B
μ
in (10), we deduce that
B
μ
= ∂
μ
b(x). (13)
The full effective action S
ef f
is given by
S
ef f
= S
B
+ S
Di r a c
. (14)
Anewformoftheeffective action, in terms of the KR axion
field, can be obtained as follows [17]:
• First, we formulate the path integral, which involves a
functional integration over the KR field strength H .
• We insist on the preservation of the Bianchi identity
(8)ataquantum level, via the addition of appropriate
counterterms (in a renormalisation group sense) order by
order in perturbation theory. This guarantees the conser-
vation of the “H-torsion charge ”Q =
d
3
x ε
ijk
H
ijk
,
which is implemented in the path-integral by adding a
δ-function constraint in the form δ
κ
2
ε
μνρσ
∂
μ
H
νρσ
,
4
The spin connection is given by
ω
μ
ab
≡ e
a
ν
∂
μ
e
νb
+
ν
μσ
e
σ b
,
with
λ
μν
=
λ
νμ
the standard Christoffel symbol.
and expressing the latter in terms of a (dimensionless)
Lagrange multiplier field b(x), which eventually will cor-
respond to the dual KR axion field:
δ(κ
2
ε
μνρσ
∂
μ
H
νρσ
) =
D b exp
i κ
−2
×
d
4
x
√
−gb(x)ε
μνρσ
∂
μ
H
νρσ
=
D b exp
−i κ
−2
×
d
4
x
√
−g ∂
μ
b(x)ε
μνρσ
H
νρσ
,
(15)
where the second equality has been obtained by partial
integration, upon assuming that the KR field strength dies
out at spatial infinity.
• Integrating out the H -field in the path integral with the
action (14), we obtain a path integral over the Lagrange
multiplier field b(x),
Z =
Dg Dψ D
¯
ψ Db exp[ı
˜
S
ef f
],
˜
S
ef f
=
1
2κ
2
d
4
x
√
−g
R +
8
3
∂
σ
b ∂
σ
b −
+ S
Free
Di r a c
−
d
4
x
√
−g∂
μ
bJ
5μ
−
3κ
2
16
×
d
4
x
√
−gJ
5
μ
J
5μ
. (16)
In realistic situations there are many fermion species ψ
i
,
i = 1, 2,...N with masses m
i
. Then the axial current is a
sum over species
J
5
μ
=
i
ψ
i
γ
μ
γ
5
ψ
i
. (17)
The reader should notice the appearance, in the effec-
tive action S
ef f
(16), of a four fermion axial-current-current
term, which is a repulsive four-fermion term, yielding de-
Sitter type (positive) contributions to the vacuum energy,
as standard in Einstein-Cartan theories of quantum torsion,
where the latter can be integrated exactly in a path integral
[13].
1.2 Choice of background
Upon splitting the quantum field into a background,
¯
b(x),
and fluctuations,
˜
b(x),
b(x) =
¯
b(x) +
˜
b(x) (18)
we then have for S
ef f
:
123