ersidad de Zaragoza, 50009 Zaragoza, Spain. 1 orica, Univ e sica T to de F b er 1994. E-mail: [email protected] View metadata,citationandsimilarpapersatcore.ac.uk [email protected] tso
t address: Departamen HEP-TH-9412061 Presen Address june-septem E-mail: o din 3 1 2 E-mail: [email protected] 1 HUPD-9413 provided byCERNDocumentServer y of the nite tum ably simple (al- en to consist of v brought toyouby ws is discussed as y of Barcelona y remark ersit arbara, 17300 Blanes yp e is pro CORE en b y of Barcelona, h are the generalized coupling ta B omsk, Russia, er t w omsk, Russia omsk, Russia, ersit , Higashi-Hiroshima 724, Japan 3 ysics, Univ , Higashi-Hiroshima 724, Japan 2 y y de San o = 4 theory corresp onding to a dimen- ed spacetime is explored. The classical v . The niteness of the conformal theory d ersit ersit yofPh ysics, Univ theory for quan cksenaev t functions, whic solutions y of renormalization group o acult Abstract I.L. Shapir yofPh E. Elizalde AE, F S.D. Odintso A.G. Ja and acult e obtain explicitly , CEAB, CSIC, Cam edagogical Institute, 634041 T edagogical Institute, 634041 T edagogical Institute, 634041 T explicit e calculate the one-lo op b eta functions and then consider the ysics, Hiroshima Univ ysics, Hiroshima Univ The set of exact solutions of p o W t ECM, F tains 12 indep enden e scalar eld mo del in curv . ersion of a renormalizable omsk P t ECM and IF Diagonal 647, 08028 Barcelona, Catalonia, Spain Diagonal 647, 08028 Barcelona, Catalonia, Spain omsk P omsk P tofPh ativ anced Study tofPh with conformal and nonconformal T T ysical applications. and T y estigated and the p ossibilit Departmen v vit eral ph ter for Adv trivial) functions that w ts of the theory and Departmen A four-dimensional and Departmen Cen gra and Departmen The most general v ell as sev w indicates the absence ofsolutions a is conformal anomaly in in the nite sector. The stabilit constan conditions for niteness. precisely three conformal and threeb eit nonconformal non solutions, giv action of the theory con sionless higher-deriv
1 Intro duction
The considerable achievements that have b een obtained in the eld of two-dimensional quan-
tum gravityhave inspired di erent attempts to use it as a pattern for the construction of
the more realistic theory of quantum gravity in four dimensions. Unfortunately the direct
analogies of the two cases do not work here, for rather evident reasons. First of all, the
quantum metric in d = 4 has more degrees of freedom, which include the physical degrees
of freedom of spin two, what is quite di erent from the d = 2 case. Second, the Feynman
integrals in d =4 haveworst convergence prop erties as compared with the d = 2 case, from
what follows that higher-derivative terms have to b e included in order to ensure renormal-
2
izability. An example of this sort is given by quantum R -gravity (for a review and a list
of references see [1]), whichismultiplicatively renormalizable [2] (not so is Einstein's grav-
ity) and also asymptotically free. However the presence of higher derivatives leads to the
problem of massive spin-two ghosts, which violate the unitarityoftheS-matrix. It has b een
2
conjectured, nevertheless, that the problem of non-unitarityinR -gravity might p erhaps b e
solved in a non-p erturbative approach.
The alternative approach is based on the assumption that gravity is the induced interac-
tion and the equations for the gravitational eld arise as e ective ones in some more general
theory, as the theory of (sup er)strings [3]. It is also interesting to notice that higher-derivative
gravitational theories (like string-inspired mo dels) often admit singularity-free solutions (for
a recent discussion and a list of references, see [4, 5 ]). In string theory, higher-derivative
actions also arise in quite a natural way.For instance, if one wants to study the massive
higher-spin mo des of the theory one has to mo dify the standard -mo del action by adding to
it an in nite numb er of terms, which contain all p ossible derivatives. On the other hand, the
e ective action of gravity, which follows from string theory, contains higher-derivative terms,
and the higher p owers in derivatives corresp ond to the next order of string p erturbation the-
ory. One can exp ect that the unitarity of the theory will b e restored when all the excitations
are taken into account. Therefore, it is quite natural to consider fourth-order gravity as some
kind of e ective theory, whichisvalid as an approximation to a more fundamental theory,
still unknown.
String-inspired mo dels of gravity contain, at least, two indep endent elds, which are the
metric and the scalar dilaton eld. Hence, the aforementioned e ective theory has to dep end
on the dilaton eld as well. The more general action (1) for a renormalizable theory of this
typ e has b een recently formulated in [6]. Since this mo del is rather complicated, even the
one-lo op calculations are very tedious. At the same time it is p ossible to make quite a
considerable simpli cation: since b oth the metric and the dilaton are dimensionless, higher-
derivative elds, the structure of divergences is essentially the same even if the metric is
taken as a purely classical background. Indeed, the renormalization constants are di erent,
if compared with the complete theory, but their general structures have to b e similar.
Let us recall that the theory of a quantum dilaton eld has b een recently prop osed for the
description of infrared quantum gravity [7] (see also [8] and [9]). Furthermore it has turned
out that the quantum dilaton theory enables one to estimate the back reaction of the vacuum
to the matter elds [10]. It is very remarkable that the e ect of the quantum dilaton is quali-
tatively the same as the e ect of the quantum metric, evaluated earlier in [11]. In a previous
article [6] wehave considered the one lo op renormalization and asymptotic b ehaviour of the
sp ecial constrained version of the dilaton theory. In fact the action of this sp ecial mo del 2
is the direct extension of the action for induced gravity [12, 13 , 14 , 7]. In particular, we
have found that this constrained mo del has induced gravity as the renormalization-group
xed-p oint, and that it also exhibits asymptotical conformal invariance.
The present pap er is devoted to the study of the quantum prop erties of the most general
higher-derivative scalar theory in curved spacetime. The pap er is organized as follows.
Section 2 contains a brief description of the mo del. In section 3 we calculate the one-lo op
divergences with the use of the standard Schwinger-DeWitt technique, which is mo di ed a
little, in accordance to the needs of our higher-derivative dilaton theory. Sections 4 and 5
are devoted to the search for all the one-lo op nite solutions (of a sp eci c p ower-liketyp e)
of the renormalization group (RG) equations. First of all, we consider the conformal version
of dilaton gravity (this mo del is an extension of the one formulated in [15, 12 ]) and thus
construct three di erent examples of anomaly-free dilaton mo dels. Then the more general
nonconformal version is explored. In section 6 we present some analysis of the asymptotic
b ehaviour of the theory, together with a numb er of mathematical to ols which are useful in
this eld. Section 7 contains the discussion of our results, including the p ossible role of the
e ects of the quantum metric.
2 Description of the mo del
We start with an action of -mo del typ e which is renormalizable in a generalized sense.
A basic assumption will b e that the scalar ' b e dimensionless in four-dimensional curved
spacetime, namely that [']= 0. We will also admit that there is just one fundamental
dimensional constant, which has dimension of mass squared. The only eld, aside from the
scalar, which will b e present in the theory is the gravitational eld g .
Then, dimensional considerations lead us to the following general action of sigma-mo del
typ e
Z
p
4 2 2
S = d x g fb (')(') + b (')(r ')(r ')' + b (')[(r ')(r ')]
1 2 3
+b (')(r ')(r ')+b (')+c (')R(r ')(r ')+c (')R (r ')(r ')
4 5 1 2
2 2 2
+ c (')R' + a (')R + a (')R + a (')R + a (')Rg + (s.t.); (1)
3 1 2 3 4
where s:t: means `surface terms'. All generalized coupling constants are dimensionless, except
for b , b and a , for whichwehave: [b (')] = 2, [b (')] = 4, [a (')] = 2. All other p ossible
4 5 4 4 5 4
terms that can app ear in dimension 4 in the ab ove mo del can b e obtained from (1) by simple
integration by parts, and thus di er from these structures of the ab ove action by some surface
terms (s.t.) only. One can easily verify the following reduction formulas
2 0
R(r ') c R(')+(s:t:) c (r R)(r ')= c
4 4
4
2 0 00
R(')+(s:t:) R(r ') + c c (R)=c
5
5 5
1 1
0 0 2
c R (r r ')= c R (r ')(r ')+ c R(r ') + c R(')+(s:t:)
6 6
6 6
2 2
0 2 2
b (r ')(r ')= b (r ') (') b (') +b R (r ')(r ')+(s:t:)
6 6 6
6
1 3
2 00 0 4 2 2
b (r r ') = b b (r ') + (r ') (')+b (') b R (r ')(r ')+(s:t:)
7 7 7
7 7
2 2 3
1
0 4 2
b (r ')(r ')(r r ')= ( )[b (r ') + b (r ') (')]+(s:t:)
8 8
8
2
00 2 0 2 0
b (r r ')=b (r ') (')+b (') b R (r ')(r ')+(s:t:)
9
9 9 9
2 00 2 0 2
b ( ')=b (r ') (')+b (') +(s:t:)
10
10 10
0 2 2
b (r ')(r ')= b (r ') (') b (') +(s:t:)
11 11
11
Here c = c (');b = b (') are some (arbitrary) functions. We shall extensively
4;5;6 4;5;6 6;:::;11 6;:::;11
use these formulas b elow. Notice that, for constant ', this theory represents at the classical
2
level the standard R gravity.
Theory (1) is renormalizable in a generalized sense, i.e., assuming that the form of the
scalar functions b (');:::;a (') is allowed to change under renormalization. As we see,
1 4
also some terms corresp onding to a new typ e of the non-minimal scalar-gravityinteraction
app ear, with the generalized non-minimal couplings c (');c (') and c (').
1 2 3
It is interesting to notice that, at the classical level and for some particular choices of
the generalized couplings, the action (1) may b e viewed, in principle, as a sup erstring theory
e ective action |the only background elds b eing the gravitational eld and the dilaton, see
[3]. It has b een known for some time that string-inspired e ective theories with a massless
dilaton lead to interesting physical consequences, as a cosmological variation of the ne
structure constant and of the gauge couplings [3], a violation of the weak equivalence principle
[16], etc. It could seem that all these e ects are in con ict with existing exp erimental data.
However, some indications have b een given [17] that non-p erturbative lo op e ects might
op en a window for the existence of the dilaton, b eeing p erfectly compatible with the known
exp erimental data. This gives go o d reasons for the study of higher-derivative generalizations
of theories of the Brans-Dicketyp e [18 ] and, in particular, of their quantum structure.
3 Calculation of the counterterms
In this section we shall present the details of the calculation of the one-lo op counterterms of
the theory for the dilaton in an external gravitational eld. For the purp ose of calculation
of the divergences we will apply the background eld metho d and the Schwinger-De Witt
technique. The features of higher-derivative theories do not allow for the use of the last
metho d in its original form. At the same time, a few examples of calculations in higher-
derivative gravity theory are known [19]{[24] (see also [1] for a review and more complete
list of references) which p ossess a more complicated structure than (1), b ecause of the extra
di eomorphism symmetry. Let us start with the usual splitting of the eld into background
' and quantum parts, according to
0
' ! ' = ' + : (2)
The one-lo op e ective action is given by the standard general expression
i
= TrlnH; (3)
2
where H is the bilinear form of the action (1). Substituting (2) into (1), and taking into
account the bilinear part of the action only, after making the necessary integrations by parts 4
(the surface terms give no contribution to ), we obtain the following self-adjoint bilinear
form:
2
H =2b ( +L r r r + V r r + N r + U ); (4)
1
where the L have the sp ecially simple structure
1
0
L r r r = [4b (r ')r ]=L r :
1
2b
1
The quantities V ;N and U are de ned according to
1
0 0 00 2 2
V r r = f[(c c )R +(3b 2b )(')+(b 2b )(r ') b m ]
1 2 3 4
3 1 1
b
1
0
+[ c R +(2b 4b )(r ')(r ')+2b (r r ')]r r g
2 3 2
2
1 1
0 00 0 0
N r = c c )(r R)r +(2b c )R (r ')r f(c c )R(r ')r +(c
2 1 2
2 3 1 3
b 2
1
00 00 0 2 0
+2(b 2b )(')(r ')r +2(b 3b )(r ') (r ')r +4(b 2b )(r ')(r r ')r
3 3
1 2 3 2
0 0 2
+2b (r ')r b m (r ')r g;
1 4
1 1 1 1
0 0 000 00 2 00 0 00
U = c c )(r R)(r ')+( c c )R(r ') f(c c )R(')+(c
2 1 3 1 3 1 3
b 2 2 2
1
1 3
0 00 000 0 2 00 0 2
+(b c )R (r ')(r ')+(b 2b )(r ') (')+( b b )(') +
2 2 1 3 1 2
2 2
1 3
000 00 4 00 0
+( b b )(r ') +(2b 4b )(r ')(r ')(r r ')
2 3 2 3
2 2
0 0 2
c R (r r ')+b ( ')
2 1
1 1
00 0 2 0 00 2 2
+2b (r ')(r ')+b (r r ') + c (R) b m (r ')
1 2 3 4
2 2
1 1 1 1 1
0 2 00 2 00 2 00 2 00 2 00 4
b m (')+ a R + a R + a R + a m R + b m g: (5)
4 1 2 3 4 5
2 2 2 2 2
The next problem is to separate the divergent part of the trace (3). First of all, let us
note that (4) is just a particular case of the general fourth-order op erator which has b een
considered in [25]. However, direct use of the general results in [25] leads to very cumb ersome
calculations and we use a di erent pro cedure, already employed in [20]. Let us rewrite the
trace (5) under the form
2
TrlnH=Tr ln(2b )+Tr ln( +L r + V r r + N r + U ); (6)
1
and notice that the rst term do es not give contribution to the divergences. Let us explore
the second term. From standard considerations based on p ower counting and covariance, it
follows that the p ossible divergences have the form
2
Tr ln( +L r + V r r + N r + U )j
div
N +k L r V k L r V =Tr fk U +k L
3 4 1 2
k VL L k V L L + k r L r L + k r L r L
5 6 7 8 5
2 2
+k L L r L + k L L L L + k R + k R
9 10 11 12
2
+ k RV + k R V + k V + k V V g + (s.t.); (7)
13 14 15 16
where k are some (unknown) divergent co ecients.
1:::16
The questions is now to nd their explicit values in the one-lo op approximation. It is easy
to classify the terms in (7) into several groups. The rst group is formed by the structures
the structures with numerical factors k |those are the ones which do not dep end on
1;11;:::;16
L . The divergences of this typ e are just the same as for the op erator
2
+ V r r + N r + U; (8)
and we can use the well-known values from [20]. To the second group b elong the structures
with k . Here we will use the following metho d. Since these structures do not contain
7:::10
V; N and U , it is clear that k will b e just the same as for (7) with V = N = U = 0. Hence
7:::10
we can simply put V = N = U = 0. Then, taking into account that L r r r = L r ,
we can write
2
Tr ln( +L r )= Tr ln() + Tr ln(+L r ): (9)
The rst term gives contribution to the k only, whichwehave already taken into account.
11;12
The second term has a standard structure, and its contribution has a well-known form (see,
for example, [1]).
The third group is just the mixed sector with co ecients k . Here we use the following
2:::6
metho d [23]. Performing the transformation
2
Trln(+Lr +V r r + N r + U )
1 2 2 2 2
=Tr ln(1 + L r + V r r + N r + U )+Tr ln( ); (10)
we can easily nd that the second term contributes only to k . Then we can expand the
11;12
logarithm in the rst term into a p ower series (see [1] for details) and use the universal traces
of [25]. After a little algebra, we obtain the nal result in the form:
2i 1 1 1
TrlnH= Trf U + L N + L r V L r V
" 4 6 6
1 1 1 1 1
2 2
VL L V L L + P + S S + R
24 12 2 12 30
1 1 1 1 1
2 2
+ R + RV R V + V + V V g;
60 12 6 48 24
where
1 1 1
P = R r L L L ;V =V ;
6 2 4
1 1
S = (r L r L )+ (L L L L ): (11)
2 4
Finally, substituting (5) into (11) and after a very tedious algebra which uses the reduc-
tion formulas (2), we arrive at the following result:
Z
p
2
(1 loop)
4 2 2 2 2
d x g [A R = + A R + A R + A Rm
1 2 3 4
div
" 6
2
+C R(r ') + C R (r ')(r ')+C R(')
1 2 3
2 2 4 2 2 4
+B (') +B (r ') (')+B (r ') +B m (r ') +B m ]; (12)
1 2 3 4 5
where
1 1
00
A = a
1
1
90 2b
1
1 1 c 1 c
2 2
00 2
A = a + ) + (
2
2
90 2b 24 b 6b
1 1 1
1 1 1 1 1
00 0 0 2 0 0
A = a + (c c + [(4c 4c c ) +4(c c )(2c 2c c )] c )
3 1 1 2 1 1 2 2
3 3 3 3 3
2
36 2b 48b 6b 2
1 1
1
1 1 1
00 0
a b b (4c 4c c ) A =
4 4 1 2 4
4 3
2
2b 4b 6b
1 1
1
00
b 1
1
0 2 0 2
B = + (8(b ) 10b b +5b )
1 2
1 1 2
2
2b 4b
1
1
00
1 1 b
2
00 0 0 0 0 0 2 2 0
+ (b b +4b b b b 10b b +10b b ) (2(b ) b + b b ) B =
2 2 3 2 3 2 2
1 1 2 2 1 1 2 1
2 3
2b 2b 2b
1
1 1
00
b 1 3
3
0 2 0 2 0 0 0 00
(b ) 5b b +15b +5b b +b b ) B = + (5b b +
3 2 3 3
2 2 3 1 3 3 1
2
2b 3b 4
1
1
2 0 2
1 b (b )
2 1
0 2 0 00 2 0 0
(20(b ) b +4b b b +b b +2b b b )+
3 2 3 2
1 1 1 2 1 2
3 4
6b 2b
1 1
00
b 1 1
4
00 0 0 0 0 0 0 2
B = + (4b b 4b b +6b b 5b b 3b b )+ (3b b b 5(b ) b )
4 4 4 3 4 2 2 4 4
1 2 1 4 4 1 1
2 3
2b 2b b
1
1 1
00
b b 1
4
5
2
( B = + )
5
2b 2 b
1 1
00
1 1 1 1 2 c 2b 1 1
3
1
0 0 0 00 0 00 0
b c 3c b + b c c b +3c b + b c c b + c b C = + (
3 1 1 1 3 2 2 2 3 1
2 3 3 1 2 1 2
2
2b 3b b 2 2 2 6 6 3
1 1
1
1 1 1 1
0 0 0 0 0 0 0 0 2 0 2
+c b + b b c b + c b )+ (b b c 2(b ) c 6(b ) c )
2 2 2 2 2 1
1 1 1 2 2 1 1 1 1
3
6 12 12 6b
1
0 00
b b c c 2b 1
2 2 3
1 2
0 0 0 2
(5b c +2c b +c b b ) C = + +
2 3 2 2
1 2 2 2
2 3
2b 3b 6b 3b
1 1
1 1
00
c 1 1
3
0 0 0 0 0 0
(12b c 9b c 9b c +9c b 2b c +2c b ): (13) C = + (b b )+
2 1 1 2 2 2 2 3 2
1 3 3 1 1 1
2
2b 3b 6b
1 1
1
Let us now brie y analyze the ab ove expression. First of all, notice that the divergences
(12), (13) have just the same general structure as the classical action (1). This fact indicates
that the theory under consideration is renormalizable, what is in full accord with the more
direct analysis based on p ower counting. All the divergences can b e removed by a renor-
malization transformation of the functions a(');b(');c('), in analogy with two-dimensional
sigma mo dels. We do not include the renormalization of the quantum eld ', since in the
case of arbitrary b it leads to unavoidable diculties. Let us nowsay some words ab out
1
the p ossible role of the matter elds. Supp ose that the dilaton mo del under consideration 7
1
is coupled to a set of free massless matter elds of spin 0; ; 1. Then the matter elds con-
2
tributions to the divergences of vacuum typ e lead to the following change of the functions
A (') (see, for example, [20]).
2;3
1
A ! A = A + ; N +6N +12N
2 2 2 0 1
1=2
60
2
1 1 1
A ! A = A + N +6N +12N N ; (14)
3 3 3 0 1 0
1=2
180 2 6
where N ;N and 12N are the numb ers of elds with the corresp onding spin, and is the
0 1
1=2
parameter of the non-minimal interaction in the scalar eld sector. Here wehave omitted
the top ological Gauss-Bonnet term for simplicity.Thus, we see that even in the presence of
the matter elds all the divergences can b e removed by the renormalization of the functions
a(');b(');c('). (In the case massive scalars and spinors a matter contribution to B and A
5 4
will also app ear). Below it will b e shown that the ab ovechange of A (')doesnot a ect
1;2;3
our results seriously.
It is imp ortant to notice that renormalization of the generalized couplings a(');b(');c(')
explicitly manifests the prop erties which are usual for any quantum eld theory in an external
gravitational eld [1]. All these functions can b e easily separated into three groups, with
a di erent renormalization rule. The rst group is constituted by the b(') functions. The
renormalization of these functions is indep endent of the other functions, a(');c('), and is
4
similar to the renormalization of matter elds couplings in usual mo dels (like the coupling
constant in the case of an ordinary scalar eld). The second group are the c(') functions,
2
which renormalize in a manner similar to that for the nonminimal constant of the R
interaction [1]. This means that their renormalization transformations are indep endenton
a('), but strongly dep end on b('). The third group of couplings is comp osed by the a('),
and they are similar to the parameters of the action of the vacuum for ordinary matter
elds. Furthermore, the renormalization of the dimensionless functions do es not dep end on
that of the dimensional ones, a ;b ;b , what is in go o d accord with a well-known general
4 4 5
theorem [26]. Thus, the theory under consideration p ossesses all the standard prop erties of
the mo dels on a curved classical background. The only distinctive feature of the present one
is that the couplings in our theory are arbitrary functions of the eld '. This fact can b e
interpreted as p ointing out to the presence of an in nite numb er of coupling constants.
Since the theory is renormalizable, one can formulate the renormalization group equations
for the e ective action and couplings and then explore its asymptotic b ehaviour. The renor-
malization group equations for the e ective action have the standard form, since the number
( nite or in nite) of coupling constants is not essential for the corresp onding formalism [1].
The general solution of this equation has the form
2t
[e g ;a ;b ;c ;]= [g ;a (t);b (t);c (t);]; (15)
i j k i j k
where is the renormalization parameter and the e ective couplings satisfy renormalization
group equations of the form
da (t)
i
= ; a = a (0);
a i i
i
dt
db (t)
i
= ; b = b (0);
b i i
i
dt 8
dc (t)
i
= ; c = c (0): (16)
c i i
i
dt
Note that we do not takeinto account the dimensions of the functions a ;b ;b . In fact we
4 4 5
consider here these quantities as dimensionless and supp ose that the dimension of the corre-
sp onding terms in the action is provided by some fundamental nonrenormalizable constant.
The b eta-functions are de ned in the usual manner. For instance,
db
1
= lim : (17)
b
1
n!4
d
The derivation of the -functions is pretty the same as in theories with nite number of
couplings, and we easily get
2 2 2
= (4 ) A ; = (4 ) B ; = (4 ) C : (18)
a i b i c i
i i i
In the next sections we shall present the analysis of the renormalization group equations
(16),(18). In accordance with the considerations ab ove, one can rst explore the equations
for the e ective couplings b , then for c and nally for the \vacuum" ones a . All
1;::;5 1;2;3 1;::;4
that analysis lo oks much more simple for the conformal version of the theory.
4 The conformally-invariant theory and some explicit
solutions
Let us now consider the most general conformally-invariantversion of the theory (1):
Z
n o
p
2
4 4 2
S = d x g f (')'r ' + q (')C + p(')[(r ')(r ')] : (19)
c
2 1
4 2
Here f (');q(') and p(') are arbitrary functions, r = +2R r r R + (r R)r is
3 3
a fourth-order conformally invariant op erator, and we should recall that due to the fact that
['] = 0 the conformal transformation of our dilaton is trivial:
2
g ! e g ; ' ! ': (20)
Now, using expressions (2), one can integrate by parts the rhs in (19) and present the result
as a particular case of the theory (1), with
1
a (')= q('); a (')= 2q('); a (')= q(');
1 2 3
3
0 00 0 0
b (')=f (')'+f('); b (')= f (')' +2f (')=b ('); b (')=p(');
1 2 3
1
2 2 1
0 0
c (')= f (')'+ f(')= c ; c (')= 2f (')' 2f('): (21)
1 2 2
3 3 3
The rest of the generalized couplings a ;b ;b ;c are equal to zero. So, the general action
4 4 5 3
(1) is invariant under the conformal transformation (20) when the functions a ;b ;c ob ey
i j k
the constraints (21). 9
Substituting the relations (21) into the general expression for the divergences of the
e ective action, we get the divergences of the conformal theory in the form (12), where
instead of (13) wehave
00 0 2
b 3(b ) 2
1 1 0
B = + ; B = B ; C = B ; C = 2B ; B = B =0;
1 2 1 1 2 1 4 5
1
2
2b 4b 3
1
1
0 00 4
h i
1 3 (b b 1 )
1 3
0 2 00 0 2 00 2 2 0 0
+ 8(b ) b + b (b ) + B = (b ) +15b +6b b ;
3 3
1 1 1 1 3 1 3
2 3 4
2b 3b 4 2b 2b
1
1 1 1
1 1 1
00
a ; A = 2A ; A = A = A ; A = A =0: (22)
2 1 3 1 1 4 5
1
90 2b 3
1
As one can see from the last expressions, the divergences of the conformally-invariant
theory (19) ap eear also in a conformally invariant form (up to the total divergence), as
it should b e. The functions A(');B(');C(') ob ey the same conformal constraints (21)
with some F (');Q(');P(') instead of f (');q(');p('). Below, we shall use the functions
a ;b ;c , taking into account the restrictions (21), b ecause in this way calculations b ecome
i j k
more compact. Hence, wehave shown that the conformal invariant, higher-derivative scalar
theory considered here is renormalizable at the one-lo op level in a conformally invariantway,
and therefore it is multiplicatively renormalizable at one-lo op. One can supp ose that the
general pro of of one-lo op conformal renormalizability in an external gravitational eld, given
in [27] (see also [1]), is valid for the higher-derivative dimensionless scalar eld as well. Note
that taking into account the matter eld cotributions do es not lead, according to (14), to
1
the violation of conformal invariance. Indeed the conformal value of = must b e cho osen.
6
From a technical p oint of view, the cancellation of non-conformal divergences gives a very
e ective to ol for the veri cation of the calculations. It moreover enables us to hop e that the
general dilaton mo del (1) might b e asymptotically conformal invariant [28, 1], just as the
sp ecial case considered in [6].
Asabypro duct, the ab ove expression also gives us the conformal anomaly of the confor-
mal invariant theory (19): T is equal to the integrand of (12), (22) (up to total derivatives,
that wehave dropp ed). Thus, if one nds the form of the functions f (');q(');p(') which
provide the one-lo op niteness in the theory (19), the last will b e free from the conformal
anomaly. Actually, the one-lo op e ective action ob eys the equation
(1)
2
(1)
p g = T ; (23)
g g
(1)
where T is the one-lo op part of the anomaly trace of the energy-momentum tensor. Eq.
(1)
(23) allows one to de ne with accuracy up to some conformally invariant functional.
Hence if we nd the solution of the equations
A (f (');q(');p(')) = B (f (');q(');p(')) = C (f (');q(');p(')) = 0; (24)
i j k
taking into account the constraints (21), the right-hand side of Eq. (23) will b e zero (up to
(1)
surface terms), and will b e a conformally invariant (but probably nonlo cal) functional.
So the solution gives us the conformal invariant theory (19) that is free from the anomaly
(at least on the one-lo op level). Moreover, according to the structure of the conformal Ward
identities [27], [1] it is clear, that the two-lo op divergences of the corresp onding theory will
b e conformally invariantaswell. 10
The conditions (24) are nothing but a set of nonlinear (and rather complicated) ordinary
di erential equations. Fortunately, one can use the results of the qualitative analysis of the
previous section and divide the equations into three groups B =0;C = 0 and A =0,
j k i
resp ectively. It turns out that the only nontrivial problem is to explore the equations of the
rst group. Note that, due to the conformal constraints (21), the equation for b (') can
3
b e factorized out and we just have to deal with the ones for b (') and b (') rst. Since
1 2
the variable b (') is not indep endent, we end up with only one equation for b (') that can
2 1
b e solved, in principle (actually just a very reduced numb er of explicit solutions could b e
obtained, see b elow). The only three solutions of p ower-like form are the following:
b = k; k = const.; b =0; b =0; (25)
1 2 3
3k
0
=0; b = b = k; b = b ; ' = const.; (26)
3 1 2 1 0
2
5(' ' )
0
and
2 2 2
2k k k
0
; b = b = ; b = : (27) b =
2 1 3 1
2 3 4
(' ' ) (' ' ) (' ' )
0 0 0
We should observe that the second solution (26) is a particular p oint of a whole surface of
conformal xed p oints (i.e., conformal solutions) which can b e expressed as
1
b = k; b =0; b =F (' ' ); (28)
1 2 3 0
10
00 2
where the function F (p)=x is the solution of the di erential equation p p = 0, and
k
is given by the quadrature:
Z
dp
q
= x; (29)
20
3
p + c
1
3k
with c an arbitrary constant.
1
Since within the conformal theory the functions c are not indep endent, the corre-
1;2;3
sp onding equations are satis ed automatically. The equations for a have the following
1;2;3
corresp onding solutions. For (25) and (26), the common one
2
(')
a (')= + a ' + a ; (30)
1 11 12
90
and for (27),
1
a (')= ln j' ' j + a ' + a ; (31)
1 0 11 12
45
where a and a are integration constants and a (') and a (') are b oth de ned via the
11 12 2 3
conformal constraints (21). Let us notice that the ab ove nite solutions (with evidentnu-
merical mo di cations) is stable under the contributions of the matter elds, that directly
follows from (14).
In this waywehave constructed three explicit examples of one-lo op nite, anomaly
free, conformal theories. The mo del (25) is essentially the same theory which had b een
investigated in previous articles [6]. It is closely related with the theory of induced conformal
factor [7, 12 , 13 , 9]. Since the only nontrivial interactions here are of \nonminimal" and
\vacuum" typ e, it is renormalized in a manner similar to the one for the theory of a free
(ordinary) scalar eld in an external metric eld. That is why the niteness of this mo del is 11
rather trivial. Not so are the solutions (26), (27) and (28). These mo dels contain nontrivial
interaction sectors and their niteness do es not lo ok trivial at all. Moreover, the form of
solution (27) probably indicates that some extra symmetry is present. Notice also that b oth
nontrivial solutions dep end on the arbitrary value ' and are singular in the vicinityof
0
this value. One could argue that this fact hints towards the existence of some di erent,
nonsingular parametrization of the eld variable. This conformally invariant nite mo del
might b e quite interesting in connection with some attempts to generalize the C -theorem
[29] to four dimensions [30].
5 Explicit non-conformal solutions
Wenow turn to the search of nite solutions of the general mo del (1), (13), free of the
0
must conformal constraints. Since we are lo oking for non-conformally invariant solutions, b
1
b e di erent from b (otherwise we get back to the conformally invariant case). From the
2
mathematical p oint of view, to obtain solutions of the general system (24), (13) is a rather
dicult problem, b ecause in this case the equation B = 0 is not factorized out. So, already
3
at a rst stage, we are faced up with a set of the nonlinear, higher-dimensional di erential
equations. Fortunately, these equations exhibit some homogeneity prop erty, and hence it is
natural to lo ok for solutions of the exp onential form b (')=k exp [(' ' ) ], and of the
j j 0 j
j
power-like form b (')=k (' ' ) , where k and are some constants.
j j 0 j j
Accurate analysis shows that all the are necessary equal to zero in the exp onential
j
case. Quite on the contrary, the search for solutions of p ower typ e yields the following three
non-conformal xed p oints:
k 2k 16k
b = ; b = ; b = ; (32)
1 2 3
5=3 8=3 11=3
(' ' ) (' ' ) 15(' ' )
0 0 0
k 4k 4k
b = ; b = ; b = ; (33)
1 2 3
5=3 8=3 11=3
(' ' ) 3(' ' ) 9(' ' )
0 0 0
and
k
; b =0; b =0: (34) b =
2 3 1
1=3
(' ' )
0
These are in fact the only solutions of p ower typ e. The solution of the equations for a (')
i
and c (') is then straightforward (but involved). We shall present only the results of this
k
analysis. For all three solutions (32){(34), the a (') are given by the integrals
i
" #
' '
Z Z
1
00
2b (') A (')+ a = a (') : (35)
1 i i
i
2b (')
1
' '
i1 i2
Notice that in the last expression the integrands do not dep end on a (') while ' and '
i i1 i2
are arbitrary constants. In the case of the theory coupled to matter elds the values of A
1;2;3
have to b e substituted according to (14). The solutions for b and c are written b elow.
4;5 1;2;3
For the case (34), these solutions have the form
4
9
c = r x + r
2 0 1 12
1 4 2
2 1
9 3 3
c = + r x r x r + r x
1 3 0 1 2
7 3
1 2k 9r 7 45 13 9 5 2
3
9 3 3 3
+ + r + r x c = r x r x r x x
4 5 3 0 1 2
364 12 10 10
p p
4 4+ 22 22
3 3
+ r x b = r x
7 4 6
9 14
3
b = r + r x + r r x
5 8 9 6 7
44
2 2
p p
2 2
r r
6 7
22 22 5+ 5
3 3
+ + ; (36) x x
p p p p
2 2 2 2
k 4+ 22 5+ 22 k 4 22 5 22
3 3 3 3
where, for the sake of brevity,wehave denoted x ' ' and intro duced the set of
0
integration constants r ; :::; r . For the cases when the b are given by (32) and (33), the
0 9 i
solutions for c , b and b are still easily found in a closed form, but we will not b other the
i 4 5
reader with such lengthy espressions here. Thus wehave constructed the nite nonconformal
versions of the theory (1). The functions a ('), b (') and c (')above corresp ond to the
i j k
nite theory.
6 Renormalization group and stability analysis
Here we apply a metho d of analysis based on the renormalization group for the investiga-
tion of the general mo del (1). If we do not imp ose the conditions (24) on the interaction
functions, then the theory is not nite (of course, it is p ossible that there exist some other
nonconformal nite solutions) but renormalizable. As it was already p ointed out ab ove, the
renormalization group -functions are de ned in a unique way (18), and we arrive at the
following renormalization group equations for a ('), b (') and c ('):
i j k
da db dc
i j i
= A ; = B ; = C ; (37)
i j i
0 0 0
dt dt dt
0 2
where t =(4) t, and t is the parameter of the rescaling of the background metric (13).
The renormalization group equations (37) have a complicated structure. In fact the
e ective couplings a; b; c dep end not only on t, but also on ' and, therefore, (37) is nothing
but a set of nonlinear, higher-order di erential equations in terms of partial derivatives. For
this reason, to obtain the complete solution of these equations do es not seem to b e p ossible.
At the same time, we already know the values of a; b; c which corresp ond to vanishing -
functions. From the renormalization group p oint of view these values are the xed p oints of
the theory.Thus, we can explore the stability of the xed p oints (37) and then formulate
some conjectures concerning the asymptotic b ehaviour of the theory.
Wethus face the problem of the stability analysis of a system with an in nite number of
variables. A p ossible way to attack it consists in combining the standard Lyapunov metho d
and harmonic Fourier analysis. Let us rst illustrate the metho d on the most simple example
of the conformal xed p oint (26). The advantage of this solution is that the equations for b
1
and b do not dep end on each other. One can start with the equation for b , and put k =1
3 1
0
for the sake of simplicity. Moreover, we shall write t instead of t . According to the Lyapunov
metho d we write b =1+y(x), where x = ' ' and y is the in nitesimal variation of b .
1 0 1 13
0
Hence we preserve the conformal constraint b =1+y(x), where the derivative is taken with
2
resp ect to x. Substituting the ab ove expressions into the renormalization group equation,
we get
dy 1 3
00 0 2
= 2y (1 + y ) (y ) : (38)
xx x
dt 4(1 + y ) 4
Since we are only interested in the b ehaviour at the vicinity of the xed p oint, the nonlinear
terms of the last equation can b e safely omitted, and we obtain
dy 1
00
= y : (39)
xx
dt 2
This equation lo oks very simple but it dep ends still on twovariables. However (39) can b e
easily reduced to a set of ordinary di erential equations. One can expand y (x)inFourier
series with t dep endent co ecients:
1
X
y(t)
0
e
y (x; t)= + y (t) cos nx + y (t) sin nx: (40)
n n
2
n=1
Substituting (40) into (39) we obtain
2 2
e
dy n dy n dy
n n 0
e
=0; = y ; = y : (41)
n n
dt dt 2 dt 2
From (41) it follows that all the co ecients except for y vanish in the limit t ! +1. Since
0
the in nitesimal variation cannot contain a zero mo de, one can put y = 0 and hence the
0
xed value b = 1 is stable in the mentioned limit. Notice that if we do not input the
1
0
conformal constraints, that is, if we take b 6=(b ) , then the values b =1;b = 0 givea
2 1 1 2
saddle p oint of the theory.
Then one can start with b , what is a bit more complicated. If one intro duces the
3
5
in nitesimal variation z as b = + z and omits all the nonlinear terms, the remaining
3
3(' ' )
0
equation is
2
1 50 z dz
00
= z : (42)
''
dt 2 3 ' '
0
If we consider the b ehaviour of z in a region far from the value of ' , then the factor
0
1
(' ' ) is slowly varying and one can regard it as a constant x . After expanding z into
0 0
aFourier series, we get
! !
e
dz dz 50 z 1 50 dz 1 50
n 0 0 n
2 2
e
; = = n z ; = n z ; (43)
n n
2 2 2
dt 3 x dt 2 3x dt 2 3x
0 0 0
what reveals the stable nature of this conformal xed p oint. The exploration of the b ehaviour
of the c is not necessary, b ecause they are related with b by the conformal constraints,
1;2;3 1
and thus their b ehaviour is completely determined.
If one takes the values of the in nitesimal corrections which violate the conformal con-
straints then this xed p oint is a saddle one. The last claim is actually trivial, since we
already knew this from the b ehaviour of b . Stability analysis p erformed on the last of the
2
non-conformal solutions (34) shows that it is a saddle p oint of the non-conformally invariant
theory. It is clear already from the b ehaviour of b ;b ;b and hence further investigation is
1 2 3
not necessary. 14
So we can see that among the xed p oints of the theory there are some which are com-
pletely stable in UV limit and others which are partially stable, namely saddle p oints of the
renormalization group dynamics. One can conjecture that the b ehaviour of the functions
a(');b(');c(') essentially dep ends on the choice of the initial data (with resp ect to the
renormalization group parameter), whichhave to b e p ostulated at some given energy.In
particular, it is natural to exp ect that for some conformal mo dels at high energies, asymp-
totic niteness manifestly app ears. Simultaneously, there is a cancellation of the conformal
anomaly in this limit. In suchway, the theory (1) predicts the existence of renormalization
group ows from arbitrary values of a; b; c to the one which provides niteness and conformal
invariance of the theory.
7 Discussion
Wehaveinvestigated the renormalization group b ehaviour of the general dilaton mo del (1)
on the background of a classical metric. The theory under consideration p ossesses interesting
nontrivial features, as nite xed p oints and plausible renormalization group ows b etween
these p oints. This fact has imp ortantphysical applications, if we make use of the hyp othesis
in [7] and regard the dilaton theory as an approximation to some more fundamental theory
of quantum gravity (like the theory of strings) at low energies. The action of gravity, induced
0
by string lo op e ects, has the form of a series in the string lo op parameter , and at second
order it contains the terms with fourth derivatives of the target space metric and the dilaton
[32]. Thus, within some accuracy, the e ective action of the string is a particular case of our
dilaton mo del (the well known arbitrariness in the second order e ective action for the string
do es not a ect our sp eculations here). This particular case is not a xed p oint of our mo del
(mayb e only at one lo op). One can supp ose that our theory of the dilaton is valid at scales
between the Planck energy M and some energy M , where the e ects of quantum gravity are
p l
weak and only matter elds can b e regarded as quantum ones. It is rather remarkable that
the action for the dilaton |generated by quantum e ects of the matter elds| is an IR xed
p oint of our general dilaton mo del. Hence our dilaton mo del can describ e the transition from
string induced dilaton gravity at the M scale to matter induced gravity at the M scale.
p l
We can also say some words ab out the exp ected e ects of the quantum metric. In spite
of the fact that the theory (1) is rather involved, one can calculate the one-lo op divergences
with the use of the metho d prop osed in [14]. Moreover, some conjectures concerning the
renormalization of the theory of quantum gravity based on (1) can b e made even without
carring out calculations to the end explicitly. As has b een already p ointed out ab ove, the
general structure of the expressions for the counterterms will b e similar to (13). This means
that all the structures (but not necessarily the numerical co ecients, of course) will b e
actually the same. However, the structure of the renormalization mightbemuch more
complicated. In particular, for the theory of quantum gravity the hierarchy of the couplings
is lacking and all the functions b; c; a have to b e renormalized simultaneously, what is rather
more cumb ersome as compared with the dilaton theory describ ed ab ove. However, the
general structure of the counterterms in the case of the quantum metric must b e the same as
for our dilaton mo del. In particular, the functions A ;B ;C are exp ected to b e homogeneous
i j k
just as in the case considered ab ove. Hence one can hop e to get similar nite solutions in
the general theory. 15
The nal p oint of our discussion is related with the conformal invariance prop erties at the
quantum level. Some features of the theory of quantum gravity based on (19) (with p(')=0)
have b een recently discussed in [15 ]. It was shown there that, generally, the theory leads
to a conformal anomaly. This anomaly app ears already in the one lo op counterterms and
prevents the theory from b eing renormalizable. For this reason, we cannot exp ect from a
theory of quantum gravity based on (1) to have a conformal xed p oint. However, it should
b e p ossible to obtain conformal invariance at the quantum level within the general mo del
(1), byintro ducing the lo op expansion parameter in an explicit way.
It would b e of interest to study the cosmological consequences that arise from the family
of nite mo dels (1), as the p ossible existence of solutions of black hole typ e and their in uence
on the evolution of the early universe. The dilaton in the starting theory is massive, owing
to the nontrivial dimensions of the functions b ( );b ( ). For the nite conformal versions
4 5
of the theory it is not more so. However one can get massive parameters as a result of some
symmetry breaking and for this purp oses it is necessary, for instance, to derive the e ective
p otential and to explore the p ossibility of a phase transition (see [31, 11 ] for the discussion
of that approach). Indeed, the e ective p otential in dilatonic gravity under discussion has
the form (in the linear curvature approximation)
" #
1 (') 1 (')
V = b (')+ B (')ln + R a (')+ A (')ln ;
5 5 4 4
2 2
2 2
where (') is some combination of the dimensional functions a and b . Its explicit form
4 4
plays no role in this qualitative discussion. It is clearly seen that the one-lo op level e ective
action of our theory at low energies represents the standard Einstein theory with '-dep endent
cosmological and gravitational constants. Hence, our theory leads to the induction of general
relativityatlow energies, what serves as an additional physical motivation for its detailed
study. Notice also that one can intro duce massive terms even in the conformal case, what is
something like soft breaking of the conformal invariance. It is p ossible to provide niteness
even in this case (as well as in nonconformal versions of the theory, of course). We exp ect
to return to such questions elsewhere.
Summing up, wehave explored some features of the general dilaton mo del (1) which can
b e regarded as a toy mo del for the same theory with a quantum metric. Some sp ecial versions
of the mo del are nite at one lo op and, moreover, some of them are conformally invariant
b oth at the classical and at the quantum level. The lack of conformal anomaly holds even if
the matter eld contributions are taken into account. The last prop erty is likely to survive
for the more general mo del with a quantum metric. In this resp ect the theory discussed
ab ove is the rst example of such kind. Furthermore wehaveinvestigated its stabilityof
found several xed p oints (developing by the way new mathematical to ol for this purp oses).
This enables us to draw some conclusions on the p ossibility of renormalization group ows
between the di erentversions of the theory. In particular, one can hop e to apply our mo del
to obtain the connection b etween the string induced gravity action at the Planck energy
scale and the matter eld induced action at some lower scale, what is certainly valuable for
phenomenology purp oses.
Acknowledgments
EE and ILS are grateful to T. Muta and to the whole DepartmentofPhysics, Hiroshima
University, for warm hospitality. SDO would liketoacknowledge the kind hospitalityof 16
the memb ers of the Department ECM, Barcelona University. We thank also the referee
for relevant comments to our previous version of the pap er. This work has b een supp orted
by the SEP Program, by DGICYT (Spain), by CIRIT (Generalitat de Catalunya), by the
RFFR (Russia), pro ject no. 94-02-03234 , and by ISF (Russia), grant RI1000. 17
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