Renormalization group techniques are used in order to obtain the improved
non-local gravitational effective action corresponding to any asymptotically
free GUT, up to invariants which are quadratic on the curvature. The
corresponding non-local gravitational equations are written down, both for the
case of asymptotically free GUTs and also for quantum R$^2$-gravity. The
implications of the results when obtaining the flux of vacuum radiation through
the future null infinity are briefly discussed.Comment: LaTeX file, 11 pages, no figure

Elizalde, E.

Odintsov, S. D.

English

arXiv

E. ELIZALDE

S.D. ODINTSOV

Crossref

A RENORMALIZATION GROUP IMPROVED NONLOCAL GRAVITATIONAL EFFECTIVE LAGRANGIAN

Renormalization group techniques are used in order to obtain the improved non-local gravitational effective action corresponding to any asymptotically free GUT, up to invariants which are quadratic on the curvature. The corresponding non-local gravitational equations are written down, both for the case of asymptotically free GUTs and also for quantum R^2-gravity. The implications of the results when obtaining the flux of vacuum radiation through the future null infinity are briefly discussed

Elizalde, E

Odintsov, S D

CERN Document Server

UB-ECM-PF 95/
April 1995
A renormalization group improved non-local
gravitational eective Lagrangian
E. Elizalde
1
Center for Advanced Studies CEAB, CSIC, Cam de Santa Barbara, 17300 Blanes,
and Department ECM and IFAE, Faculty of Physics, University of Barcelona,
Diagonal 647, 08028 Barcelona, Catalonia, Spain
and
S.D. Odintsov
2
Department ECM, Faculty of Physics, University of Barcelona,
Diagonal 647, 08028 Barcelona, Catalonia, Spain
Abstract
Renormalization group techniques are used in order to obtain the improved non-local
gravitational eective action corresponding to any asymptotically free GUT, up to invariants
which are quadratic on the curvature. The corresponding non-local gravitational equations
are written down, both for the case of asymptotically free GUTs and also for quantum R
2
-
gravity. The implications of the results when obtaining the ux of vacuum radiation through
the future null innity are briey discussed.
1
E-mail: eli@zeta.ecm.ub.es
2
E-mail: odintsov@ecm.ub.es. On leave from: Tomsk Pedagogical Institute, 634041 Tomsk, Russia.
1
The eective action has turned out to be a quite important subject in the study of
dierent aspects of quantum eld theory. Among the phenomena to which it has been applied
successfully, we can mention symmetry breaking/restoration eects, phase transitions in
general, models of quantum corrected eld equations, etc.
Most of the studies of the eective action have been limited to a quasi-local approach
(for a general introduction see [1, 2]), that is, they deal with almost constant background
elds, as is the case of quantum gravity on a De Sitter background [3] |which is important
in inationary universes. Recently, some interest has arisen ([4]-[7], see [6] for an extense
account) for the case of weak but very quickly varying background elds, which typically
lead to non-local eective actions. In the present note, by using simple renormalization
group (RG) methods |implemented by means of a Wilsonian procedure [8]| we are going
to show how one can obtain in fact an improved non-local eective gravitational action for
a big class of theories.
The starting point for our considerations will be a massless, multiplicatively renormaliz-
able theory including scalar, spinor and vector elds on a classical gravitational background.
The corresponding Euclidean Lagrangian has the following form
L = L
m
+ L
ext
;
L
m
= L
Y M
+
1
2
(r

')
2
+
1
2
R'
2
+
1
4!
f'
4
+ i (

r

  h') ;
L
ext
= a
1
R
2
+ a
2
C
2

+ a
3
G+ a
4
2R: (1)
By choosing a specic gauge group, we can assume that some multiplets of the scalar, ',
and spinor,  , elds are given in some concrete representation of the gauge group.
We will assume that our theory (1) is a typical asymptotically free GUT in curved
spacetime (for a general introduction, see [2]). In principle one could equally well consider
other types of GUTs, what would not change qualitatively the conclusions of our study
below.
The running coupling constants corresponding to the asymptotically free couplings of the
theory (1) have the form [9, 10]
g
2
(t) = g
2
"
1 +
B
2
g
2
t
(4)
2
#
 1
; g
2
(0) = g
2
;
h
2
(t) = 
1
g
2
(t); f(t) = 
2
g
2
(t); (2)
where t is the RG parameter while 
1
and 
2
are numerical couplings dened by the specic
features of the theory under consideration. We know of many examples of such theories,
2
with gauge groups SU(N), O(N), E
6
, etc. [9, 10]. Asymptotic freedom (g
2
(t) ! 0, t! 1)
is realized for all running couplings: gauge, Yukawa and scalar ones, as is easy to see from
(2).
The study of asymptotically free GUTs in curved spacetime was started in Ref. [11] (for
a review and detailed list of references see [2]). In the theories with one scalar multiplet, for
the running scalar-graviton coupling constant one gets
(t) =
1
6
+

  
1
6

"
1 +
B
2
g
2
t
(4)
2
#
b
; (3)
where (0) =  and where for the dierent GUTs the constant b can be either positive,
negative or zero (see Ref. [2]).
The gravitational running couplings are dened by the following dierential equations
(we shall consider the gravitational equations in the Euclidean region)
da
1
(t)
dt
=
1
(4)
2

(t)  
1
6

2
N
s
2
;
da
2
(t)
dt
=
1
120(4)
2
(N
s
+ 6N
f
+ 12N
A
) ;
da
3
(t)
dt
=  
1
360(4)
2
(N
s
+ 11N
f
+ 62N
A
) ; (4)
where N
s
; N
f
and N
A
are the number of real scalars, Dirac spinors and vectors, respectively
(notice that the running of a
4
(t) will not be meaningful for us, as we shall see below).
Owing to the fact that the theory under discussion is multiplicatively renormalizable, the
eective Lagrangian satises the RG equation
 

@
@
+ 
i
@
@
i
  
i

i
@
@
i
!
L
eff
(; g

; 
i
; 
i
) = 0; (5)
where  is the mass parameter, 
i
= (g
2
; h
2
; f; ; a
1
; a
2
; a
3
; a
4
) is the set of all coupling
constants, the 
i
are the corresponding -functions and 
i
= (A

; ;  ). The solution of
Eqs. (5) by the method of the characteristics gives (for all quantum elds we consider a zero
background eld, 
i
= 0):
L
eff
(; g

; 
i
) = L
eff
( e
t
; g

; 
i
(t)); (6)
where
d
i
(t)
dt
= 
i
(
i
(t)) ; 
i
(0) = 
i
: (7)
3
Observe that for some of the coupling constants, the corresponding Eq. (7) has been written
above explicitly (Eq. (4)), while for a subset of them Eqs. (7) have been actually solved (see
Eqs. (2) and (3)).
Actually, the idea itself of a RG improvement procedure was suggested many years ago
[12]. What we do here is to make use once more of this interesting concept. Physically, the
meaning of expression (6) is the following: L
eff
(called sometimes the Wilsonian eective
action [8]) is obtained through the above equations provided its functional form for some
value of t is known (usually it is the classical Lagrangian that serves as boundary condition at
t = 0). Another diculty is related with the choice of t, which cannot be given a unique de-
nition due to the presence, in general, of several dierent efective masses (see the discussions
in Refs. [13, 14] concerning that point, for curved and for at spaces, respectively).
There are dierent approaches to the gravitational eective action (for a general intro-
duction, see [1, 2]). In the literature, mainly the case of a local eective action has been
discussed (i.e., the situation where the gravitational eld is slowly varying). One-loop non-
local eective actions have been considered in Refs. [4]-[7] (see also the references therein),
in dierent contexts, but almost exclusively the case of a free scalar eld theory has been
taken into account.
We will be interested in the situation where the gravitational eld is weak, but rapidly
varying, e.g.
rrR >> R
2
: (8)
The non-local one-loop eective action for a free scalar eld theory in this case has been
calculated in Ref. [5] (see also [6, 7, 15]), up to the second order on curvature invariants.
Such a calculation is quite tedious, moreover, its extension to other elds (especially, to
interacting elds) is anything but trivial (see [6] for a discussion and list of references).
We will make use of this RG improvement technique in our calculation, what is going
to yield a correspondingly more precise result than the one that has been obtained till
now by means of previous approaches to the problem. First, all those calculations have
been carried out in the one-loop approximation, while ours here will yield the RG improved
eective Lagrangian to leading-log order (through summation of all possible logarithms) of
perturbation theory, i.e., clearly beyond one-loop. Secondly, the theory under discussion had
been usually restricted to scalar elds, while the considerations here will be applicable to
any renormalizable theory on a curved background, including the ordinary renormalizable
models of quantum gravity, as R
2
-gravity (see [2] for a review). In particular we will present
4
results for an arbitrary asymptotically free GUT in curved spacetime (see [2, 11]).
To begin, using the general expression (6) we can write explicitly the RG improved
eective Lagrangian for the theory (1), employing the classical Lagrangian as boundary
condition:
L
eff
= a
1
(t)R
2
+ a
2
(t)C
2

+ a
3
(t)G+ a
4
(t)2R; (9)
where the choice of RG parameter t will be described below. From the explicit one-loop
calculation [10, 11], the RG parameter is found to be
t 
1
2
ln
 2+ c
1
R

2
; (10)
where the constant c
1
is dierent in the dierent sectors (scalar, spinor and vector). By
looking at (8) one can see that in order to get the dominant contribution we may just keep
the rst term in (10), i.e. t ' (1=2) ln( 2=
2
). From the explicit study of the non-local
eective action [5]-[7] it follows that the thing one has to understand is the way non-local
form factors act, as formal operators obeying the variational rules of nite matrices (in the
Lorentzian region). Note also that the terms a
4
(t)2R and a
3
(t)G are still total derivatives
after the RG improvement (compare with the other regime in [13] where these terms become
important). Notice that a dierent way of understanding the appearance of the  2 under
the logarithm is to resort to RG considerations in curved space [2], where we know that a
scale transformations of the metric, g

! e
 2t
g

, ought to be performed. Since, under
this transformation, R
2
! e
4t
R
2
and 2! e
2t
2, the logarithm corresponding to those terms
becomes relevant in the high-energy limit t!1.
Finally, the RG improved non-local gravitational eective Lagrangian takes the form
L
eff
= R
8
<
:
a
1
 
(   1=6)
2
N
s
2B
2
g
2
(2b+ 1)
2
4
 
1 +
B
2
g
2
ln( 2=
2
)
2(4)
2
!
2b+1
  1
3
5
9
=
;
R
+C

"
a
2
+
ln( 2=
2
)
240(4)
2
(N
s
+ 6N
f
+ 12N
A
)
#
C

; (11)
where a
1
and a
2
are intital values for the corresponding eective couplings. Notice that with
the above form factors the solutions fulll the requeriment of asymptotic ateness [15]. As
it has been discussed in Refs. [15, 16], the coecients of the terms linear in ln( 2) give a
measure of the energy radiation through the future null innity.
Here we have obtained an eective Lagrangian, L
eff
, which sums all the logarithms of
perturbation theory, up to second order terms on curvature invariants on the background,
5
of weak but quickly varying curvature. The theory under consideration is an asymptotically
free GUT but, in principle, we can consider in the same way any other kind of renormalizable
quantum eld theory.
Notice, however, that the price one has to pay for the universality of the approach (i.e.,
for the possibility to write (11) for a variety of theories beyond the one-loop approximation)
is the fact that we cannot proceed to higher orders in the curvature. The reason is that
the terms as R
3
, R
4
; : : : are ultraviolet nite. At the same time, the ordinary technique to
one-loop order [6, 7] gives the possibility, in principle, to calculate the non-local eective
action up to any desired order in the curvature |although it is quite complicated, already
in the case of the scalar theory. It turns out, therefore, that the two approaches complement
each other quite well.
Using L
eff
one can obtain the eective gravitational equations. Adding the quantum
matter-induced eective Lagrangian (11) to the classical Einstein Lagrangian (without the
cosmological constant, for simplicity), one gets the eective gravitational equations in close
analogy with Refs. [6, 7, 15]. Before doing this, it is convenient to rewrite
C
2

= G + 2R
2

 
2
3
R
2
; (12)
and to substitute it into Eq. (11). Then, one nds the following Euclidean eective gravita-
tional equations
 
1
8G

R

 
1
2
g

R

+
8
<
:
a
1
+
(   1=6)
2
N
s
2B
2
g
2
(2b+ 1)
2
4
 
1 +
B
2
g
2
ln( 2=
2
)
2(4)
2
!
2b+1
  1
3
5
 
2
3
a
2
 
ln( 2=
2
)
360(4)
2
(N
s
+ 6N
f
+ 12N
A
)
)
h
4r

r

R  4g

2R +O(R
2
)
i
(13)
+2
"
a
2
+
ln( 2=
2
)
240(4)
2
(N
s
+ 6N
f
+ 12N
A
)
#
h
2r

r

R   g

2R   22R

+O(R
2
)
i
= 0:
Observe that in order to obtain the eective gravitational equations it is not necessary to
take into account the g

-dependence of the form factors. As was discussed in Ref. [15], the
eective gravitational equations can be used in order to study the problem of collapse.
To be remarked is the fact that the above approach works well for renormalizable models
of quantum gravity too. In order to exemplify this, let us consider R
2
-gravity under the form
L =
1


R

 
1
3
R
2

 
!
3
R
2
: (14)
Such a theory is multiplicatively renormalizable, being non-unitary in the perturbative ap-
proach (for a general review and a list of references, see [2]). The RG improved non-local
6
eective Lagrangian corresponding to this theory, with the same gravitational background
(8), can be easily constructed. The eective gravitational equations are (for simplicity, only
leading-log terms have been kept)
"
1

+
133
20(4)
2
ln( 2=
2
)
#
h
r

r

R   g

2R   22R

+O(R
2
)
i
+2
"
 
1
3
 
!
3
+

10
9
!
2
+
5
3
! +
5
36

ln( 2=
2
)
2(4)
2
#

h
4r

r

R  4g

2R +O(R
2
)
i
= 0; (15)
where  and ! are the initial values for the corresponding eective couplings.
Following now Refs. [15, 16] (it is explained there in which form non-local eective
action can be relevant for black-hole physics), we can discuss the implications that the
above non-local gravitational action has concerning the ux of the vacuum radiation in an
asymptotically free GUT. Working with the asymptotically at (Lorentzian) solution of Eqs.
(13) one may consider the congruence u(x) = const. of the light rays that can reach the
future null innity F
+
. We shall denote, as in [15], by r the luminosity distance along rays
and by M(u) the Bondi mass at F
+
(see [17]). Then, the nal expression for the radiation
corresponding to the vacuum energy in a spherically symmetric state has been found to be
the following [15]:
dM(u)
du
=  
1
4
(w
1
+ 2w
2
)
d
2
d
2
u
Z
F
+
F
 
dr r R+O(R
2
); (16)
where w
1
and w
2
are the coecients of terms linear in ln( 2) of (11), that is
L
eff
=
(
R

"
a
1
 
2
3
a
2
 
w
1
2(4)
2
ln
 
 
2

2
!#
R

+R
"
2a
2
 
w
2
2(4)
2
ln
 
 
2

2
!#
R
)
; (17)
where a
1
and a
2
can be taken to be zero and where a
1
(t) has been expanded up to terms
linear on ln( 2). Taking into account the overall change of sign of L
eff
in the Lorentzian
region, from (4) we obtain
w
1
=
1
60
(N
s
+ 6N
f
+ 12N
A
) ; w
2
=  
1
180
(N
s
+ 6N
f
+ 12N
A
) +
N
s
2

  
1
6

: (18)
In this way we can calculate the rate of the vacuum energy radiation through the future null
innity, taking into account corrections to the GUT under consideration. To be remarked is
the fact that the choice of  can inuence this rate of radiation signicantly (18). Radiation
disappears when the null surface u = const. comes very close to the horizon [15, 16]. Then,
7
in order to nd the Hawking radiation [18] one has to calculate the next-to-leading correction
in (17), namely the O(R
2
)-terms.
To summarize, using rather simple RG considerations, we have constructed a RG im-
proved non-local gravitational Lagrangian corresponding to a general asymptotically free
GUT and also to R
2
-quantum gravity. The corresponding eective gravitational equations
have been written down as well. It would be now of interest to study the applications of
these equations to black hole physics in more detail, since they certainly modify a number
of results obtained previously.
Acknowledgments
We would like to thank Roberto Percacci and Sergio Zerbini for their interest in this work.
SDO is grateful with the members of the Department ECM, Barcelona University, for warm
hospitality. This work has been supported by DGICYT (Spain), project Nos. PB93-0035
and SAB93-0024, by CIRIT (Generalitat de Catalunya), and by RFFR, project 94-020324.
8
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10


A renormalization group improved non-local gravitational effective Lagrangian

A renormalization group improved non-local gravitational effective
  Lagrangian

A renormalization group improved non-local gravitational effective Lagrangian

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