Physics Letters B 772 (2017) 174–178
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Physics Letters B
www.elsevier.com/locate/physletb
Integration of trace anomaly in 6D
Fabricio M. Ferreira
a,b
, Ilya L. Shapiro
a,c,d,∗
a
Departamento de Física, ICE, Universidade Federal de Juiz de Fora, Campus Universitário – Juiz de Fora, 36036-330, MG, Brazil
b
Instituto Federal de Educação, Ciência e Tecnologia do Sudeste de Minas Gerais, IF Sudeste MG – Juiz de Fora, 36080-001, MG, Brazil
c
Tomsk State Pedagogical University, Tomsk, 634041, Russia
d
National Research Tomsk State University, Tomsk, 634050, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
14 May 2017
Received
in revised form 27 May 2017
Accepted
1 June 2017
Available
online 15 June 2017
Editor: M.
Cveti
ˇ
c
Keywords:
Conformal
anomaly
Effective
action
Conformal
operators
Topological
terms
The trace anomaly in six-dimensional space is given by the local terms which have six derivatives of
the metric. We find the effective action which is responsible for the anomaly. The result is presented in
non-local covariant form and also in the local covariant form with two auxiliary scalar fields.
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Within the modern approach the effective action (EA) is a cen-
tral
object in quantum theory of fields. In particular, evaluation of
vacuum EA is one of the main targets in the semiclassical approach
to quantum gravity.
Usually
complete derivation of EA is impossible, and the main
purpose is to develop an approximation which is controllable in
the sense we can see what is the physical situation when the given
approximation can be applied. The most important examples are
the anomaly induced EA, which are derived by integrating trace
anomalies and therefore hold full information about the vacuum
quantum effects in UV. As far as the original theory possess lo-
cal
conformal symmetry, the notion of UV can be usually extended
to most of the physically relevant domains. In some physical sit-
uations
one can regard masses as small perturbations, extending
the area of application of conformal anomaly. Last but not least is
that the anomaly induced EA can be derived in a closed, compact
and useful form. For this reason the conformal anomaly and the
anomaly-induced actions are the main instruments used in cos-
mology
and black hole physics to deal with the quantum effects of
vacuum.
*
Corresponding author.
E-mail
addresses: fabricio.ferreira@ifsudestemg.edu.br (F.M. Ferreira),
shapiro@fisica.ufjf.br (I.L. Shapiro).
In the dimension D = 2the integration of anomaly yields the
Polyakov action [1], which proved important for the development
of string theory. Since the D = 2result is exact, it also plays
the role of a reference for other dimensions, where one has to
use approximations. In the D = 4case the analog of Polyakov EA
has been obtained by Riegert [2], Fradkin and Tseytlin [3], just
three years after the two-dimensional analog. This result proved
to be an extremely useful tool for numerous applications [4,5]. The
anomaly-induced EA in D = 4is essentially more complicated than
the Polyakov action, mainly because in D = 4there is an “integra-
tion
constant”, an arbitrary conformal functional S
c
, which can not
be determined from anomaly. The problems with Ward identities
of the RFT action [2,3] which were discussed in a number of works
starting from [6,7] are related to the fact that this conformal func-
tional
remains unknown. At the same time, since S
c
is not related
to the UV sector of the theory, in all known applications this un-
certainty
proves to be irrelevant (see the discussion in [5]).
Along
with the conformal functional, in D = 4there is also
one more complication. Compared to D = 2, the anomaly includes,
along with the topological term, also conformal and surface terms.
Furthermore, the coefficients of the most relevant topological and
conformal terms possess a mysterious universality of signs, which
do not depend on the Grassmann parity, but only on the number
of derivatives in the action of quantum fields. There is a very in-
teresting
and fruitful statement that this universality of signs and
related property of the renormalization group flows holds beyond
http://dx.doi.org/10.1016/j.physletb.2017.06.014
0370-2693/
© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.