View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server HUPD-9413 A four-dimensional theory for quantum gravity with conformal and nonconformal explicit solutions E. Elizalde Center for Advanced Study, CEAB, CSIC, Cam de Santa Barbara, 17300 Blanes and Department ECM and IFAE, FacultyofPhysics, University of Barcelona Diagonal 647, 08028 Barcelona, Catalonia, Spain 1 and DepartmentofPhysics, Hiroshima University, Higashi-Hiroshima 724, Japan A.G. Jacksenaev Tomsk Pedagogical Institute, 634041 Tomsk, Russia 2 S.D. Odintsov Department ECM, FacultyofPhysics, University of Barcelona, Diagonal 647, 08028 Barcelona, Catalonia, Spain and Tomsk Pedagogical Institute, 634041 Tomsk, Russia, 3 and I.L. Shapiro Tomsk Pedagogical Institute, 634041 Tomsk, Russia, and DepartmentofPhysics, Hiroshima University, Higashi-Hiroshima 724, Japan Abstract The most general version of a renormalizable d = 4 theory corresp onding to a dimen- sionless higher-derivative scalar eld mo del in curved spacetime is explored. The classical action of the theory contains 12 indep endent functions, which are the generalized coupling constants of the theory. We calculate the one-lo op b eta functions and then consider the conditions for niteness. The set of exact solutions of p ower typ e is proven to consist of precisely three conformal and three nonconformal solutions, given by remarkably simple (al- b eit nontrivial) functions that we obtain explicitly. The niteness of the conformal theory indicates the absence of a conformal anomaly in the nite sector. The stability of the nite solutions is investigated and the p ossibility of renormalization group ows is discussed as well as several physical applications. HEP-TH-9412061 1 Address june-septemb er 1994. E-mail: [email protected] 2 E-mail: o [email protected] 3 Present address: Departamento de Fsica Teorica, Universidad de Zaragoza, 50009 Zaragoza, Spain. E-mail: [email protected] 1 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.