IL :NUOVO CIMENTO VOL. 46 B, N. 1 11 Luglio 1978

Quantum Gravity with Higher Derivative Terms.

g. JULVE

Ins t i tu t e de Estructuxa de la Materia, C S I C - Madr id , Spai,~

M. To ~Ii~-

Is t i tu to di F i s ica deU' Universi th - Padova, I ta l ia

Is t i tu to Nazionale eli l" is ica Nucleate - S e z i o n e di Padova, Ttalia

(ricevuto il 13 Marzo 1978)

S u m m a r y . - - We consider a renormMizable (but perturbatively non- unitary) model of quantum gravity which includes quadratic terms in the curvature tensors. One-loop counterterms are calculated for this enlarged action by truing the background field method in the context of the dimensional regularization scheme and the renormalization group equations are derived. Finally the possibility is discussed that the Weyl ghosts which are characteristic of this model and would in principle expoil unitarity do not actually occur. 2r difficulty with the cosmological term is pointed out.

l . - I n t r o d u c t i o n .

Covariant quant izat ion of generM re la t iv i ty can be per formed by the same method which has been successful in non-Abelian gauge theories. The recipe

is to add a gauge-fixing t e rm and the corresponding ]~'addeev-Popov one to

the invar iant act ion and to quantize by using for instance the pa th integral

method. I n the case of gravi ty , the only complicat ion is the occurrence of

noncanonicM t e rm coming f rom the invar iant functional space measure which

is not trivial. The specific fo rm of this t e rm has been obtained recently (1).

Moreover, i t can be avoided if dimensional regularizat ion (DR) is used.

Actual ly, the difficult point in quan t m n grav i ty lies in its renormMization.

I n fact , the s tandard Einste in-Hi]ber t actio~ gives rise to divergences which

(1) E. S. FI~ADKIN and G. A. VILKOVISKY: preprint CERN, TtI. 2332, June, and references therein.

137

138 ~. zuLv~ and ~. TONI~

are not renormalizable (~'~). Of course, the reason is tha t the gravi tat ional constant has dimensions of (length) ~ or equivalent ly tha t the gravi tat ional field has canonical dimension equal to one in mass. Therefore, the gravity- ma t t e r interact ion te rm and the Einstein action itself have canonical dimensions

greater than four. A way out of this difficulty is to modify the s tandard action by adding terms

quadi'atic in thc curvatare tensor (~). ~ o w the gravitat ional field becomes dimensionless and all the terms have canonical dimensions ~4 , so tha t the resulting theory fuliils the renormalizabil i ty requirements ('~,~). Tha t this theory is indeed rcnormalizable has been proved recent ly by STEELE (~). Besides the gravitat ional coupling constant G -- 1/u ~, in this model two new adimen- sional couplings a and b appe,~r as coefficients of the quadrat ic terms.

The trouble one meets, however, is the occurrence of massive Weyl ghosts which cannot be eliminated by the usual cancellation mechanism based on g~uge invariance. I f they are present, they will cause uni tar i ty to fail. This is the reason why this theory has received little at tent ion.

In tree approximat ion the masses of these ghosts are simple rat ional func- tions of x ~, a and b. However, it is conceivable tha t higher-order corrections

push these masses to infinity. I f this happens, one can conjecture tha t the theory is un i ta ry after all. For this reason we think that , in spite of the ghost

difficulty, this theory is worth being investigated. In sect. 2, we introduce l~he model and outline the quantizat ion procedure.

In parbicular, unconvent ional (~) gauge-fixing terms are used which are specially suitable in our case and the corresponding Faddeev-Popov terms are obtained. In sect. 3 the one-loop counter terms are computed in the context of D]~ by using the background field method (~.~,s). ]n sect. 4 we discuss the renormal- ization group equat ion and speculate about the possible mechanism which hopefully could rule out the ghosts. We suggest tha t the ghosts are absent if the theory reaches a non-Gaussian ~fltraviolet stable fixed point and if, at this fixed point, the mass parameter which gives the ghost mass at the tree level

has an anomalous dimension ~ < - 1. Finally, a difficulty which concerns the cosmological te rm is pointed out.

(~) :B. S. I)EWI'.~T: :Phys. Rev., 162, 1195, 1239 (1967). (3) G. 'T IIooFT and M. V]~L~'.~LX~: A~n. [,nst. H. 1)oinearg, 20, 69 (1974). (i) a) S. Dv, s~,~ a1~d P. V~N ~I~UW~'rIuIZEN : Phys. Rev. D, 10, 401 (1974); b) S. DESV, R, P. VAX ~I):UWEh'IIUIZE~ and II. S. TSAO: :phys. Rev. D, 10, 3337 (1974); c) S. D~s~n: Qu~nlum Gravity, edited by C. J. ISLAM, R. 1)E-~aOS~, aI~d D. W. ScIA~ (Oxford, 1975), p. 136; Con]erence o~,~ Gauge Theories and i~iodern Field Theory, edited by R. A~NOWITT and P. NA'r~ (1975), p. 230. (~) R. UTIY~A and B. S. D~:W~: Journ. Math. Phys., 3, 608 (1962). (~) A. GAV~ELU)~S, T. K. KuO and S. Y. L~E: Phys. Rev. D, 12, 1829 (1975). (v) K. S. S~):LL)~': I)hys. J~ev. D, 16, 953 (1977). (s) J. HoN~r~K~e: Nucl. Phys., 48 B, 269 (1972).

QUANTUhI GI~.AVITY W I T I I ]:[IGILER DERIVATIV}; TEI/3IS 139

2. - The complete action and its quantization.

General re la t ivi ty can be regarded as a classical field theory invariant under general co-ordinate transformations

x ' . = x . - / . ( x ) ,

i .e . a non-Abelian group of t ransformations depending oll the four local para- meters ] d x ) . As fundamenta l fields one (;an take the components of the sym- metr ic rank-two tensor g , , ( x ) , the metric tensor, which under infinitesimal

general co-ordinate t ransformations have the var ia t ions

~g..(x) - g.(x) ~l~),(x) + g.~,(x) ~/~,~(x) + ~f (x) ~ g.,(x) .

T h e only difference between grav i ty and non-Abelian gauge theories is tha t the former undergoes local t ransformations involving also the space-time variables.

The affine connection coefficients F~, play the role of the Yang-Mills fields and the curvature tensor

= - r ~ , / ' L - r ' 1 ~

corresponds to the field strength tensor. 3Ioreover, the condition tha t the co- var iant derivatives of the metric tensor vanish,

D ~ g,, = 0 ,

fixes the affine connections in terms of g~,~:

F ~ , , ~ l - ) .o i ~ I , , 2 Y tYo~,y T ,Jov,g~- g~v.o) �9

Up to second order in the curvature tensors, the invariants are

where

g : : - - det(g,v),

R,~ - R~:~,

R = g ~ ' R l ~ .

Moreover, the ident i ty (~)

(1) R, .pT~R~r 4 R ~ v R "~ -t- R" =- total eovariant derivative

140 J. JULVE and M. TONIN

holds in four-dimensional space-time. Then the more general action up to second order in the curvature tensors is

(2) ~f=fd'x~[a(R,~R.~--3R2 ) +bR'--~R + ).],

where a and b are adimensional constants, G----1/z ~ is the lqewton constant

and ). is the cosmological one. Le t us write as usual

g,,.(x) = w,, § h,,,(x) ,

where ~/~. is the diagonal Lorentz metric tensor ( 1 , - 1 , - ] , - - 1 ) and set 2 - - 0 in eq. (2) for the moment . The quadrat ic par t in eq. (2) is

(3) ~ Lfo =--~ ([:]hz~)(r~h~'~) - - F~,~F~'~-- (~eAe) ~ 4w 3 + 1 (8~Ao ~q~)o. +

+ -a-[h~,, ,~h,, + A e X o + ( A o - 8~cf) (Ao-- 8%0)]

= h ~ b/a. In eq. (3), where ~ = h ~ , A~ 8,, ,,, F~ ,~=8 .A~- -O~A. and op------ the indices arc raised by means of the metric tensor ,1~ instead of gu~. The same convention will be used up to the end of this section with the exception of eq. (10). Of course, the differential operator which appears in the quadrat ic form eq. (3) is not invertible due to the general covariance of the original action

eq. (2). In order to quantize this theory, it is necessary to add a gauge-fixing term.

Since the general gauge transformations depend on four arb i t rary functions, we must impose four gauge conditions. The conventional choice is to take some linear combination of A , and ~ 0 as gauge functions. An al ternat ive pos- sibility is to choose the four-dimensional curl and the four-dimensional di-

vergence of this linear combination. Thus we add the following gauge-fixing

term to ~o :

= a ~] 2

This term breaks the gauge invariance and gives rise to well-dcfined prop- agators for h~,~. However, to ensure the gauge independence of the theory, it is necessary to add also a compensating Faddeev-Popov term, which is

(5) cs e : - - - ( ~zU~ - 8 ~ ) ~ ~o(D~ u ~ + D~u p) -- ~2 [~"~(D~u, D~Uo)--~D~u~] - 2(~ ( ~ ~o) +

QUANTU:~I GRAVITY ]VITII ttIGIIEF~ D}]RIVATIY:E TERMS 1 4 1

u, and ~ are ant icommuting vector Faddeev-Popov fields, and D ouo = ~euo-- --l~'~u~. depends on h~,, through F ~ , Iu fact, the total action

(6) ~qf = ~fo + Lfz d- c~fse

is inv~riant under the following Becchi-Rouet-Stora (BRS) supersymmetrie

transformations (9):

The inv~riance of eq. (6) can be easily checked. Indeed one can verify tha t

~(D,u~ d- 1)~us) = 0,

so tha t the vari~tion of c~f~, r just compensates tha t of L f . The gauge-fixing term eq. (4) can be generalized as follows:

where $~.$.,. and/or Ss are different from zero. For later convenience we notice tha t a.nother class of gauge-fixing terms

which generalizes eq. (4) is

= ~ {~ s,~l':'~ + ~s~},

where

as~ being an arbi trary external field. The corresponding Faddeev-Popov term is

.SfFp----[(~7/~--8~i~) ~F~ ~2 ~B] ~ h ~ ' ~ ( ~ ) ~-~-~o] (O~u~ -t- O ~ @ .

(9) C. B~ce~r, A. ROW:T and R. STORA: Comm. Math. Phys., 42, 127 (1975).

142 .r. JULVn and ~. ~roxz~

In this case, the ]~IES t ransformation eq. (7c) must be subst i tuted by

(9) 3~/~, ;~ k$~ ~ - - ,

which is nonloeM.

As discussed in the introduction, a fur ther t e rm should be added which

comes from the invariant measure. However, it is proport ional to ~(0) and, therefore, it does not contr ibute if, as we shall do in the following, DR is used. In the DIE scheme the action should also contain the term, which is vanishing in four dimensions,

(10) f d,x (R@x~R~'~x~-- 4R~Rt'~ + R2),

where n = 4 + z is the dimension of space-time. This te rm could give finite contributions (at z = 0) at higher loop orders. Since we are interested in the

polar one-loop contribution, it will be neglected in the following. ~ o w the theory can be quant ized in the s tandard way. Let us repor t the

explicit form of the h~ propagator , choosing eq. (8) as gauge-fixing te rm a.nd taking ~ : 0.

Following ref. (~) we shall define the projectors for the spin-two, spin-one ~nd the two spin-zero parts of the field in momentum space as

(11)

1 P(~') = ,~ (0~o 0,o + 0~,~ 0~) - -a- 0~,~ 0~ , ktv;Oa

19(I)

,uv;Qa

~(o) izv;Oa : ( .Olt~O)~a ,

where

__ q~ q~ q~ q~ ( 1 2 ) Ol,v : ~]/'~ q2 ' (Ore - - q2

Then the propagator in the momentum space is

2 1 1 1 _ t ) ( 2 ) / 9 ( o ) (~ 3) A~,;oo(q) a q'~(q~-- x2/a) • 2oJa q2(q.2 _ _ x2/2(.oa) ~';~a ~-

The spin-one component a.nd the spin-zero one projected by p(1) and p(0)

can be gauged away. The gauge-independent par t of Az,.,o~(q) describes massless gravitons togeth-

er with a spin-two particle with square mass x2/a and a scalar particle with

QUANTU3[ GRAVITY WITH IIIGIIER DERIVATIVE TER~S 143

square mass -- u:/b. The massive tensor particle is a tachion or a ghost depending on whether a < 0 or a > 0. The scalar massive particle is a physical one if b < 0 and a tachion if b > O.

3. - O n e - l o o p e o u n t e r t e r m s .

In this section we calculate the one-loop contributions divergent at ~ -- 0 , to be used as counterterms for the Lagrangian equation (2). We shall use the background field method (BFM). I t consists of a splitting of the fundamental fields in a classical background part and a quantum one, which propagates only as internal lines, and then of computing only one-loop graphs with the classical background fields as external lines. A g,~uge-fixing term can be chosen, so that the total action looks invariant under general co-ordinate transforma- tions which involve only the classical background fields. The adv-mtage of ~his method is that only invariant counterterms (polar in v) appear.

We rewrite eq. (2) in the form

b ~ , , _ z . '-~ -',- ;~],

where now we have renamed with ~,~ the fundamental fiehts and witlz ~, 1~,,, h ,~ and/~ the correspollding determinant, affinc connections and curva-

ture tensors, respectively. We shall leave the symbols without the hat for the classical background quantities. Thus let us define

#~,~ = gl,, -4- h~,, ,

where g,~ is the background field and h~ the internal quantum field and expand in series of h/,~. The term of zeroth order is irrelevant. Terms linear in h~,,,

can be removed by using the motion equations. Terms of order higher than two contribute only to graphs with two or more loops. T h e n for our purpose only quadratic terms in h/,~ are relewmt. A straightforward calculation gives

a{.l(,.j~/,~)(3~ ) 1 (~ co ) (15) ~ o = ~ ~ - - 2 1~-co (~(fi)~--

1 - -~ (D~A, -- D~A~)(D~A~ - - D~A, ,) - -

-~- R ~ [2Yt~:. ~ Yt~ -[ ,th~:.(I)~ D~)+h~ --h~#D/,D~h~,#] -4-

144

where

and

J . JULV:E and ) I . TONIN

3 + (o h " ' z ; , , , + (1--2w)h!~(D,D,)+h~.--

oo) ] 2 1 -- c f sc f -t- (1_ -~ ~o)(DoAq)9 +

+ 4_R~,,R~.h~.oho~ ' + 2R~,,h ..u~.o,o~ -F (R~,h~") ~" - - ~ (/~,,/~'~)(h~ah~#) -1-

.2[ ( I ) ] -t- ~ (3 - - 2co)hu, hu" + 2 (1 -]-(.~ coy- 1~0c~ f~ T'

+ a 2[ , ,[]h,, ~v[]~o-- 40) l + e o

] (I)oAo)cf - i - - ~ ( D , D , . ) + ~., + -~ -g R ~ , , ; , , -l-

all , ,1], } + a -~ h , , h , = -g cf ~ -~- ... ~/~ ,

h ~,

~ D -~ ~hu

[] ~ g~D~D~ ,

(D~,D~)+ = �89 -F D,,D~,) ,

1 [ /~i ,~5~ -F -~' - ~ - ~ R ~'

is the Weyl tensor. The dots stand for terms which do not give contributions polar in ~ to one-

loop order. They are terms proport ional to D~,R, DoRl,,, DI, D , R , D e D ~ R ~ and terms proport ional to C~,,.# o,, C~,,# and .R~,,R. Contributions from the last three terms are absent since they only can arise f rom contractions with g,, a.nd/or 6~, and C ~ and/~,~ are traceless. The first (second) te rm should contract with a rank-one (three) tensor of dimension 1 which cannot be con- structed. The thi rd and four th terms would give rise to contributions propor- t ional to ~/gr~ R and "v~D~D~R"", which are total derivatives.

Notice tha t for b = 0, ~2 _--0 and ). = 0 the internM field ~o disappears. This is a consequence of the fact tha t the combination I~,~R~ ' ~ - �89 ~ is con-

formal invariant.

Q U A N T U ~ I G R A V I T Y )VITII I I I G I [ E R DJ ' ]RIVATIV~ TERMS 1 4 ~

We shall add to the Lagrangi~m density eq. (15) the gauge-fixing term

(t6)

where

b -- a Be.| 3

(~7) ~'u, = D u A l - - D ~ A u ,

I § B = D o A o []q~l

4(~ + o~)

The corresponding Faddeev-Popov term is

(t8)

+ ~ 1 - -5 (1 + o,)(:~ - 4~) R ~ . ( D . D . ) j +

~- 3-a- $~(1 -{- co) 2~U(DuD,)+u~ R

1 R ~ } ~ 1 7 6 ,

The dots again represent irrelevant terms. Since we are interested only in eounterterms of the form of the original Lagruugian {i.e. with no Feddeev- Popov external fields), we have not introduced a background field part in the )~I ) fields.

The polar one-loop contributions are expected to be independent of the gauge parameters ~ , G, G. We have checked this gauge independence by explicit c~lcul~tion. ]n the following we shall report only the calculation choosing for the gauge parameters the values

#~= 1, G-- O, & = ~-.

Let us ~ssume furthermore a > 0, co > 0 and perform the field redefinition

ha, -* 2 1 + w #v ~ ~ --~ (0

Then the total Lagrangian density c~n be wa'itten in the form

(19) s V/if{�89 (s h~)(,.-q h~p)-]-[V(hh)~]"~h~(DuD~)+ho~-',- [U(hh)~]h~,~ho.~ -~

- - �89 (~ q~)( .-q q)) + [ V(hq))~'~],~hc,~(D#D~) . q~ + [V(q)q~)]u~q~(DuD~)+ q~ + [U(q~q~)]q~ ~ +

+ (B ~')(~ u~,) + [V(~u)~y'qi~,I)uD, ue + [U(~u)~]~uq + ...},

10 - I I Nuovo Cimento B.

146 J. JULVE and ~. TONIN

where

= - ~ ~ ~,~ , 2 ( / 7 , ~ , , 5 ~ , + ~ ' , ~ [ V ( h h ) ~ y '~ - - 2C%,~g~.. -t- 2Rqr~og T ~- -q-.~, - -

R ~ (l_ + e)) ~ 1 ). I -t- to [ U ( ~ ) ] 18 to + 6 a w '

[V(~/u)~] ~ = Rqg + R~g ~" + -~ ( g ~ ' ~ § g~5~) ,

[~(~u)~] = o.

Moreover, if we define the fields [~ related to the background fields g~, by g~ =: ~ -4- f~ , we can write

i ~ r) 'a f l~qa 1~ r /af lT, qa

Vg{(~ ~)-" = {(D~) ~ + ~z~ + ~2~ + . . . ,

v~E-] ~" [] u~ = ~,"(~ ~,)(~u,) ]- ~ z ~ u ~ + ~=ZD~ ~ + . . . ,

m

where [] = ~ ~ ~,, and g~z ~ _Q., Z~, Z (Z~ , Z~, Z) are linear (quadratic) in ]~ , and the dots represent terms of higher order i n / ~ .

T h e Lagrangian equation (19) contains kinetic terms quartic in derivatives and vertices between two internal lines and a background-f ield-dependent ex- pression. We shall speak of Z-vertices which come from the covaz'iant d'Alem- ber t ian operator, V-vertices which are proport ional to b~ckground curvature tensors or u~ and U-vertices which are quadrat ic in these quantities or pro- portiona.1 to 2,.

Since the polar terms are general invariant , it is sufficient to ex t rac t the polar contributions up to second order in ]~, to get the complete s t ructure of the counterterms. Accordingly we need the expressions of the vert ices up to quadrat ic terms in ]~,~, and we should compute those one-loop F e y n m a n graphs up to second order in ],~ as well. Moreover, we can choose ]~ to be trace-

QVA~,'TV~ GRAWTV W*TR me.hER DERIVATiVe. T~R~S 147

less. If we keep this in mind, the relevant Feynman graphs a.re given by

7) ~ h

8) ~}~ 9) ~u

lo)

11) C ~ )

12)

13)

where �9 O stands for the background-field-dependent vertices.

148 z. JULVE a ~ 5[. TONIN

--•163 --

~ . =

A~f,o -

O,

By a straightforwar4 calculation the polar contributions coming from these graphs can be easily worked out. I t is clear that the graphs in each figure give invariant contributions. In fact, we get

8

- s ~ + R"

--As176 = ~ - k \ 3 6 0 1 s - R~ ,

e 8 '

' / g [ . ] -}-a) R2 I 1 ~"~(-~ ] a 36 24 w a

~ S - F R~ ,

[ 1 . 3 ~ R + 9 ~] - - l O S - - ~ ( o - - 2 c o ) R ~ - - ~ a 2 a '

[~8 (1 + ~)" R-" ~ +~o~] ~o 6 ~ '

( 7) + 1-f -~m a ~ '

A~,r n "v/g[ 1 1 ( l+m) '~ 2 1 (1-- 0)2 (.~) 2] = - R + , e ~ ( 1 + eo)2R 2 + ] 2 eo a 16 ~)~

A,~2 V~[ l_~(1-i-(~)'Rz i(i-I-o)) 2~2 11-}~o(~) 2] e o~ 6 ~o a 8 oJ

8

where S----R~,R,,--i�89 2, s = 8 = ' - v , an4 --A.~f~ represents the contribution of the graph(s) in figure i).

Summing up all the partial contributions A~Lfi, we finally obtain the com- plete expression for the one-loop countcrterms corresponding to the action eq. (2)

QUANTUM GRAVITY WITI~ I I IGIIER DERIVATIVE TERMS 149

4. - R e n o r m a l i z a t i o n group equat ion.

In the previous sections a renormalizable quan tum field theory of gravi ty has been discussed and the divergent one-loop counterterms have been ob- tained. However , as has been pointed out, this theory does not guarantee nni ta r i ty due to the possible appearance of massive ghosts (and/or tachions). In this section we shall discuss their renormalization group equations and we shall speculate on a possible mechanism to avoid the ghosts.

The theory under discussion depends on four finite renormalized parameters -- (a, b, ~2, ),) which need to be defined by means of a suitable normalization

prescription. In the D E scheme in which we ha~e choosen to work, a conve- nient normalizat ion prescription is the ' t H o o f t ' s one (10): ~ are defined b y

requiring the counterterms to contain only poles in 3. In the ] )R scheme, a factor #-3, where tt has dimension of mass, appears in f ront of the action for di- mensional reasons and the ' t Hooft ' s prescription forces ~ to become functions of #. In other words, the theory is described by a t ra jectory in the coupling constant space parametr ized by the normalizat ion mass #. This t ra jectory, i.e. the functions ~(#), can be obtained in principle from the renormulization group equations. The behaviour of the coupling constants ~t /z -> 0 (# --> co) determines the asymptot ic bchaviour of the ver tex functions when the mo- menta tend together to 0 (or to oo). For instance, the renormalization group

equat ion for the inverse propagator gives

(20) ]'(~)(#qo; a, b, z ~, 2) = # ' ~ T TM qo; a(#), b(#), #2 , #~ ] ,

where the functions a(/t), b(/~), ~2(/~), t(tt ) are the solutions of the differential equations

d a (21a) dt f l '

db (21b) dt -- 6 '

d~2 ~2 (21c) dt --- ?~ --a-'

(21~) d-)- = 7~ a

(10) G. 'T HOOFT: .Yucl. Phys., 61B, 455 (1973).

150 J. JULV~ and M. TO~N

Here t------in #, and fl, ?, ~ arc the Cal lan-Symanzik functions which depend on the pa ramete r s a, b, u s, t . In the DI~ scheme with ~t Hoof t ' s prescript ion they

do not depend on the dimensional pa ramete r s u s and i , and can be obta ined

as the residues of the simple poles in v which appear in the counter tcrms. I f we define / s = ]l/a[, M s = ~ z s / s and recall w = - - b / a , eqs. (23) yield

d/2 (22a) dt -- • fl( :j: p ' w ) / ' ,

d(D (22b) dt

(22c)

(22d)

_ • [ ~ ( • is, ,~) + ~o/~(• ~o)]p,

d M 2

dt - • I t : ( • I s, .,) + / ~ ( • 12, ,~)]Msls,

dt dt

- -• b 's(+ 12, o~)).l 2 :J: 7'~(• I s, ~o)M,] .

The sign ~- (- -) is for the case a > 0 (a < 0). The values of the CS functions a t p = 0 can be obta ined b y looking a t the

one-loop counter terms. Therefore, f rom the explicit calculations in the previous

section we get

S~ ~ ~ , riO, c,) =

(~(0, co) -

~,,(0, o)) =

~'s(O, ~ ) -

8~2

8u 2 ~ - ~ 8co 3 ~ '

11+] 8xe s + 6 ~, '

32z~ s 4 ~ �9 ~,~(0, ~o) = - - -

Genera]ly the t ra jec to ry $(#) interpolates between two fixed points of

eqs. (22) : an inf ra- red-s table (IS) fixed point a t /x ~ 0 and an ultra 'violet-stable

(US) fixed point a t # - > c~.

The po in t /2 = 0 is a fixed point for eq. (22a). Since fl(0) > 0, it is an IS (US)

one for a > 0 (a < 0)~ and in fact 1~ goes to zero as 1/ln/~ when/~ -> 0 (/~ -> ~ ) . A t the fixed point one can see f rom eq. (22c) t ha t the mass pa rame te r M s be- haves as some power of In/x. The finite p a r a m e t e r M s represents the squared

mass of the ghost (tachion) only a t the tree level. The correct mass can be

calculated in principle in te rms of this and the other pa ramete rs of the theory.

:Now the negat ive values of a which correspond to an asympto t ica l ly free

theory should be excluded since in this case we expect the tachion which ap-

QUANTUi~f GRAVITY W ITI I HIGI tER ]):ERIVATIV%~ TERMS 151

pears a t the tree level to remain in the complete propagator 2~,~;Q~(q). Let us

define the quant i ty

/ ~ , ; o . ( q ) = ~ [A, , , ,o . (q)] -~ .

At a fixed small value of q2 the mass term should dominate this quant i ty , which, therefore, at this point is positive for a ~ 0. In an asymptot ical ly free theory the mass M 2 behaves a.s some power of In q2 as q~ --~ c~ and then F,,;q~(q) di-

verges as q~. Therefore, it will vanish at some negative value of q~ so tha t the taehion cannot be avoided.

Vice versa, a ~ 0 corresponds to an infra-red-free theory and this is welcome since the infra-red divergences duo to massless gravitons may cancel as iu QED. For a > 0 a ghost is expected in the spin-two par t of the propagator.

However, it may happen tha t the t rue mass of the ghost turns out to be in- finite and the ghost disappears. Indeed the te rm -- n~R in the original action dominates the inverse propagatol at small q~, whereas at large q2 it is the te~m a ( R~ , , R ~ ' - -~2 -R ~) the leading one. I t is this crossover which is responsible for the ghost. However~ the crossover arises only if the total dynamical di- mension of M is > 0. I f it were negative, the te rm - - n 2 R should dominate the inverse propagator a t small q2 as well as at infinite q~ ~nd then the ghost

should be presumably absent. In conclusion we conjectttre t ha t the ghost difficulty could bc avoided and

the uni tar i ty could be restored if the parameters of the theory approach an US (necessarily non-Gaussian) fixed point, say ]2 = ].2 and co = co*, and at this fixed point the anomalous dimension of M is < - 1.

I t is interesting to compare this conjecture with the Weinberg's one tha t gravi ty is a (nonrenormalizable) (( asymptot ical ly safe ~ (AS) theory (~). A theory is AS if the t ra jec tory in the coupling constant space which characterizes the theory hits a, fixed point when # --~ c~, and at this fixed point only a finite number of operators has total dynamical dimensions <4 . l~enormalizable theories are a subset of AS theories. In our case we can assign dimension 4 to the quadrat ic t e rm in h of u~-R. If we formally define hi~ ~ ~ uh~,~, a factor ] / M ~ appears in f ront of the operators R~,~R~ ,~ and R -~. When the anomalous

dimension of M ~ is ~ - - 2 , 1 / M 2 has positive dynamical dimension and the operators quadrutic in the curvature tensor will have dimensions <4 . This

corresponds to an AS theory. Of course it is not easy to calculate non-Gaussian fixed points, so tha t not only we are not able to just ify our conjecture, bu t ei ther we cannot ver i fy if, in this theory, the total dynamical dimension of M is indeed negative.

A different possibility to avoid the ghost difficulty is to invoke the Lee-

(11) S. ~VEINBERG: lCOtUrCS given at the International School (~E. Majorana ~, ~Erice, July 1976.

152 z. zur~v~ and ~. T o . i x

W i c k m e c h a n i s m (12) b y showing t h a t t h e u n p h y s i c a l poles a re sh i f t ed off t h e

rea l axis . This p o s s i b i l i t y has b e e n d i scussed r e c e n t l y in ref. (~).

A d i f f icu l ty ar ises wi th eq. (22d). I n f ac t t h e i n h o m o g e n e o u s t e r m p ro -

p o r t i o n a l to M ~ on t h e r .h.s , of t h i s e q u a t i o n c o n t r a d i c t s t he va.uishing of t he

cosmolog ica l c o n s t a n t . As is k n o w n , a n o n v a n i s h i n g 2 is a r a t h e r d i s a p p o i n t i n g

f e a t u r e . H o w e v e r , th is d i f f tc td ty is n o t p e c u l i a r ~o the t h e o r y of g r a v i t y u n d e r

d i scuss ion . E v e n if in p u r e q u a n t u m g r a v i t y b a s e d on t h e c o n v e n t i o n a l E i n s t e i n

~c t ion t h e cosmolog ica l t e r m does n o t a p p e a r as a d i v e r g e n t c o u n t e r t e r m , i t

wil l do as soon as g r a v i t y is coup l ed to m a t t e r (~). Also in our case m a s s i v e f ields

w o u l d m o d i f y t h e l a s t t e r m in eq. (22d): f e rmion (boson) f ields w o u l d a d d

n e g a t i v e (posi t ive) c o n t r i b u t i o n s to t he CS func t ions for 2. One can s p e c u l a t e

t h a t all t h e i n h o m o g e n e o u s c o n t r i b u t i o n s a d d to g ive zero. Since th is cancel -

l a t i o n m u s t h o l d as an i d e n t i t y , i t shou ld come f rom deepe r g r o u p - t h e o r e t i c a l

reasons .

(~) T. D. L],m and G. C. WICK: Nucl. Phys., 9 B , 209 (1969); 10B, 1 (1969); Phys. Rev. D, 2, 1033 (1970). (~3) E. To~tnouHs: Phys. Lett., 70B, 361 0977).

_Vote added in proo]s.

After this paper was submit ted for publicat ion we have received a preprint by A. S~Lh~ and J. STRATHD:EE ([.C. 78/12) in which the same mechanism suggested here to avoid the ghost is proposed.

We thank Profs. A. SALAr~ and J. S'rR~THDE~ for having sent to us their paper before publication.

@ R I A S S U N T O

Si studia una tcoria quantist ica della gravitazione, rinornmlizzabile ma per turbat iva- mente non unitaria, la eui azione include termini quadratici nei tensori di curvatura. Usando il mctodo dei campi background, si calcolano i controtermini al l 'ordinc l ln 'ans~ nello schema della regolarizzazione dilnension,~le, e si ottengono le equazioni dcl gruppo di rinormalizzazione. Infine si discute la possibilit'~ che i fantasmi di Weyl, carat ter is t ici di questa teoria e che in principio violcrebbero l 'unitariet'~, non siano presenti. Si fa anche notare una difficolt.h collegata al tcrmine cosmologico.

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