Regular gravitational lagrangians

Physics Letters B 276 (1992) 31-35 North-Holland

PHYSICS LETTERS B

Regular gravitational lagrangians Norbert Dragon Theoretical Physics Division, CERN, CH- 1211 Geneva 23, Switzerland Received 27 June 1991

The Einstein action with vanishing cosmological constant is for appropriate field content the unique local action which is regular at the fixed point of affine coordinate transformations. Imposing this regularity requirement one excludes also WessZumino counterterms which trade gravitational anomalies for Lorentz anomalies. One has to expect dilatational and SL(D) anomalies. If these anomalies are absent and if the regularity of the quantum vertex functional can be controlled then Einstein gravity is renormalizable.

Symmetries restrict very effectively the arbitrariness of lagrangians. It is not always appreciated that also the domain of regularity of the lagrangian is essential for the resulting restrictions. For example, if scale invariance leads to the condition

(~1 ~ 1

"~ I~2

~(~1, ~2)=0 ,

(1)

then the general solution ~ ( ~/),, ci02) = f ( ~JS,/ ~ 2 )

(2)

contains infinitely many parameters, while the solution which is regular at the fixed point (q~l, q~2) = (0, 0) of the differential operator in ( 1 ) 5~( q~t, qb2) = const.

(3)

contains only one parameter. For a compact internal symmetry group G it is well known [ 1 ] how the requirements that the lagrangian 5 ° be regular at a fixed point of a subgroup H = G affect 5e: the fields have to consist of Goldstone fields which parametrize the coset G / H and of H-multiplets, H is realized linearly and the H-multiplets have to couple to H-invariants. The tightest conditions on 50 follow i f H is chosen maximal H = G [ 1 ]. Imposing the requirements that the lagrangian be regular at the fixed point gmn = q,~n of Poincar6-transOn leave of absence from Institut f'tir Theoretische Physik, Universitat Hannover, W-3000 Hannover 1, FRG.

formations all local actions which are invariant under general coordinate transformations and their anomalies have been determined [2]. The result is unaltered if one requires regularity at the fixed point o f the De Sitter group [ 3 ] which is also a maximal symmetry group for an invertible metric. Here I investigate the restrictions which follow for gravitational lagrangians if one requires them to be regular at the fixed point of the affine coordinate transformations

x'm(x) = M m n x n + a

n,

MeGL(D), a~ D .

(4)

This is the highest-dimensional subgroup of general coordinate transformations (unique up to the choice of coordinates) which can act linearly on an affine connection FkF. I assume such a connection to be one o f the gravitational fields. Fk~m transforms under infinitesimal coordinate shifts ~ as sFklm = OkOl~mq_ ~r OrFklrn_jr Ok~r l"~rlm -1- Ot~ r Fkrm -- Or~ m F k l r ,

(5)

with the characteristic inhomogeneous piece OkO/~m. The conditions for a fixed point SOn, ...OmFkl m = On, ...Ores Fklm = 0 ,

i = 0 , 1 .....

(6)

show that the second and higher Taylor coefficients of ~m are determined in terms of the D (D + 1 ) coefficients ~m, Os~m. There are further restrictions on the

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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parameters of the fixed point group if the Riemann curvature Rkhn n,

6 February 1992

does not vanish. Rktm" transforms as a consequence of (5) as a tensor, i.e., by its Lie-derivative

iant solution and which, moreover, restrict near-by solutions to obey the four-dimensional gravitationally self-coupled wave equation. Rather, I want to investigate theories with an additional tensor density Gmn= G nm, transforming according to

SRklm n ~ ~ R k h n n

sGm"=L/2cGm"+d(Or~ r) G m~ .

Rkl,nn=cok rtmn--OlFk,nn--FkmrF~rn + Fi,n~rkrn ,

(7)

~_~r OrRklmn t_ Ok~r Rrhnn_t_ Ol~r Rkr,n n + O m ~ r Rl,.lr n - ~r~ n Rklm r .

(8)

The fixed point equation SRktm~= 0 imposes a linear relation among ~m and Os~m, leaving a lower-dimensional subgroup H, unless

(14)

Let us investigate the dimensional charge assignment d ( G " n ) = 1. If D > 1 then the only fixed point of (0k~...0k, Gm,, i = 0, 1, ... ) under the affine transformations (4) is a vanishing G ran. It is a matter of taste whether one still calls G mn a metric even if one allows G mn to vanish. I f G .... is invertible and D > 2 one can construct a metric g,,n from Gmn:

OrRkhn n = 0 , (~kSRrhn n "[- ~ / R k r m n "]- (~mSRklr n -- (~rnRklm s ~- 0 .

(9)

( 10 )

if one uses the first Bianchi identity Rklm n "Jr-Rlmk n nu R,,kl ~= O .

( 11 )

Summing the cyclic permutations of r, l, m in (10) one finds that the Ricci tensor R .... = ½( R , , k , k+ Rnkm k)

(12)

and the dilatational curvature rmn = R m n k k = R m k n k - R n k m k

( 13 )

have to vanish at a fixed point o f a D ( D + 1 )-dimensional subgroup of the general coordinate transformations acting on FkF. But then (lO) implies D'Rk~m~=Rkm~" SO the Riemann curvature vanishes at the fixed point. Therefore Fk/" is a pure gauge which we can choose such that Fk/" vanishes. Inserted into SFk/"= 0 this means that the fixed point transformations are generated by linear inhomogeneous transformations, i.e., they are the affine transformations (4). So we have shown that the affine group is the highest-dimensional fixed point group of the affine connection and the fixed point is F k F = O. From FkF' alone one can only construct C h e r n Simons forms as lagrangians in odd dimensions [2 ]. I do not know [4] whether some of these lagrangians are acceptable, i.e., yield D > 4-dimensional theories which allow for a four-dimensional Poincar6 invar32

Gm"=x/-g g "n ,

gm~ = ( G - l ) m n (det G kl) l/(D-2)

Contracting with 6~k one obtains D" R~Im" -- Rrml n -I- (~rnRlsm s = 0

ifdet G " " ¢ O,

(15)

However, ( 15 ) should only be understood as an appeal to the reader's background in differential geometry: G'n" is defined by its transformation law (14). It is an argument of the local action and may have any value including values with det ( G mn) = O. A field G .... is suited to discuss questions such as whether the dynamics can enforce a change in the signature of the metric gmn if one starts from an invertible G " ' with signature ( - 1, 1, 1, 1, ...). Just as the ordinary invertible metric g,,,n, the tensor density G "n measures in a lagrangian the kinetic energy such as £~kin( q ) ) = G m " ~ , , c I ) 2 , ~

(16)

of a scalar field. If G .... becomes noninvertible this means that components of the gradient of q) can be nonzero without a kinetic energy being required for that fluctuation. This makes perturbation theory hard, but the world has not necessarily come to an end where det (Gin") = 0. For the dimension assignment d ( G '~") = 1 and for even D with D >~4 the s-invariant local action which is regular for vanishing G m~ and l~kl m is unique: it is given by the lagrangian 5')=c~ Gm~R,,,,( F ) "4"-c2 + d Y

(17)

(where the value of c~ is relevant only together with a nonvanishing vacuum expectation value of G mn ). This follows along the lines of ref. [2 ] because, up

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to the subtlety o f C h e r n - S i m o n s forms which occur in o d d dimensions, the lagrangian can be constructed from tensor densities. In the case at h a n d the tensors are the covariant derivatives

The condition that the resulting S L ( D ) invariant has weight d = 1 requires

~k,...~k~G ran, ~k,...~k, Rjlm ~ ,

These equations have to be satisfied with nonnegative integers. If N e e 0 then N ~ = I , N ~ = 0 by (23) and N ~ + 2NR = D . N ~ = 0 , however, for the following argument: no more than two indices o f each factor Rk;m" can be contracted to gm,...mD because o f the first Bianchi identity ( 11 ). Therefore all covariant derivatives and exactly two indices o f each Rktm n have to be contracted to g,~,...mD because N~ +2NR = D . But then the second Bianchi identity

i=0, 1,...,

(18)

where the covariant derivative 2,~ o f a tensor density T o f weight d contains a piece ( - d ' F k / " T), e.g., ~k G " " = O;:G""+F*/"Gt"+Fk/'G "l - dF~/G m , .

( 19 )

NO inverse o f G mn can a p p e a r in the lagrangian 5 ° because G2,1 is not regular at the fixed point o f the affine coordinate transformation. The variables (18) have to be coupled to S L ( D ) singlets which must have weight d = 1 - a p a r t from a possible constant contribution to 50. This means that - possibly by help o f the numerical tensors _~,,...~ and (,,,,...m~ which are completely antisymmetric and have values (0, 1, - 1 ) - all indices have to be contracted. e,,,,...m,, is a tensor density o f weight d = 1 and _e,,...,o has d = - 1 , s~ .......... = 0 = ~

........" + 0.~" ~m,...,.D,

S~-ml ...... = O = ~ Q ~ ~-. . . . . . . D -

0 . ~ n ~-. . . . . . . . . .

(20)

because a n t i s y m m e t r i z a t i o n o f D + 1 indices vanishes in D dimensions (r~; denotes omission o f m J :

ifN_,>~0,

N,-N~=I

or

ifN~>~0,

N c ; + N ~ = 1 respectively.

~kRm, fl + ~tR,,,kr ~+ ~ , , R k t f = 0

(23)

(24)

makes each contribution with covariant derivatives vanish. So if N, = 1 then N . = 0 and NR = ½D, i.e., only products of Chern forms for S L ( D ) × dilatations remain. They are topological densities o f the form d Y (17). Therefore no term with e-contributes to the equations of motion. If N, = N_~= 0 then N~; = 1 and N,~ + 2NR = 2 has the two solutions (N~, NR) = (2, 0) or (0, 1 ). The first solution leads only to a derivative term ~,,~,Gmn=Om(.~,G""

)

(25)

because cJ,,Gm" is a vector density o f weight d = 1 and because Fkzm = Ftk'". The second solution yields the Einstein term

D

E (-)'

G""Rmk~ k .

0,.o~ " ' e " ° ' ~ ...... = 0 ,

(26)

1=0 D

( - ) ; Om,~m° -e..... ,~......o = 0 .

(21)

Consider now N , > 0 : combining (22), (23) one derives

i=0

N . +2NR + ( 0 - - 2 ) (N, - 1 ) = 4 - D , We can always arrange the contributions to 5 ° to contain either only gm,...m,, or only e,l..., o because each product o f these tensors can be written in terms o f Kronecker O's. The condition that all indices be contracted yields

which has no solution for D>~ 4 and N_, >~ 1 - the case D = 4 , N~_= l, N ~ = N R = O yields a vanishing contraction. Consider the low dimensions D = 2 and D = 3: D = 2 is special because the weight-0 field

i f N , >~0, N ~ + 2 N R + D N , _ = 2 N ( ; , ~ 2 = ~-kl~-mn

if N,->~0,

N~+2NR=2NG+DN,,

(27)

GkmGl~

(28)

(22)

where N~, NR, No, N,, N~; denote the numbers o f derivatives, curvatures, g,~,...,,,o, _e~,...,,,,and metrics G mn.

can a p p e a r arbitrarily in the lagrangian which therefore contains infinitely m a n y arbitrary parameters. F o r D = 3 , (27) reads 33

Volume 276, number 1,2 N,~ +2NR + (N, - 1 ) = 1,

PHYSICS LETTERS B (29)

with the solutions D=3:

( N ~ , N R , N,,N~;)

= (1,0, 1, 2) or (0, 0, 2, 3 ) .

(30)

The first solution gives only vanishing contractions, the second one gives the three-dimensional cosmological constant q~3 = fktm-~k'l'm' G kk'Gtl' G ...... '

( 31 )

(it coincides with ~ i f G m" is invertible). This concludes the proof of the uniqueness of c~ as given by (17). It is easy to see that (17) describes matter-free Einstein gravity wherever G"" is invertible. Just use ( 15 ) as a definition for the metric; then (17) is the Palatini form of the Einstein action. In the form (17) the action can, however, also be used at space-time points where gmn ( 15 ) is not defined. If one wants to include matter in a regular lagrangian one has either to derive it from higher-dimensional G mn and spontaneous compactification, or one has to choose different gravitational fields with different charge assignments. No such charge assignment, however, will make a D = 4 dimensional model unique up to finitely many parameters if Yang-Mills fields are to be included with a usual kinetic energy

G k t G " ' F ~ , ~ F i , c~j .

(32)

For this term to be regular G ~"" can have at most d = ½. It could have d < ½if scalar fields with positive weight provide the necessary dimensional charge. But then the scalar field qbD = _e....... _e.......... G ...... ...G . . . . .

( 33 )

has at most weight d = ½( D - 4), in particular for D = 4 it can appear arbitrarily in the lagrangian. If D>~9 then the standard lagrangian below is unique up to finitely many parameters and has no cosmological constant. One introduces scalars • with weight d = ¼, chiral spinors with weight d = 3, the affine connection Fk/~ and Yang-Mills-fields for G g a u g e X S O ( 1 , D - 1 ) L . . . . tz with weight d = 0 (they cannot have a weight d ~ 0 and a nilpotent s-transformation) and a vielbein E~ m with weight d = ~ (which need not be invertible as opposed to the usually em34

6 February 1992

ployed vielbein e~m with weight d = 0). Up to a finite number of terms which contain covariant derivatives of the vielbein, the resulting regular lagrangian is in D~>9

~ = rlabE,'"Eb"R ......s( F) V2 ( crP) + rlt'CEamEt,nRnmca( ff2 ) V'2( ~ ) -- ~rlabrI CdEakEblEcmEdnF~,,, F~, c,).

+ ½iEj"g'r";7,. 7'+ g'V~ (q,) 7" + ½qahE,,"Eh"~m~,,~-

V4(~)

(34)

( Vi (q~) are homogeneous polynomials of order i). S has a vanishing cosmological constant and remarkably coincides with a dilatationally invariant lagrangian in D = 4. For D = 8 a cosmological constant A.det E J can occur. The analogue of (27) now reads

N~+ 3N~+N~+2NR+(D-4)(N,-1) =S-D,

(35)

which excludes N~ >/1 in D >/9 dimensions. For N~ = 0 it exhibits the reason for the dilatational invariance: the equation

N,~ + 3 N ~ + N,~ + 2Nn = 4

(36)

coincides with counting the dimensions of fields in D=4. The quantization of (17), (34) has some uncommon features: the lagrangians are polynomial and (17) has only a qb3-interaction. The propagators, however, can only be defined for the spontaneously broken situation det ( ( G'~" ) ) ¢ 0. This makes it hard to control perturbatively the regularity requirement for the quantum vertex functional at vanishing fields. If this regularity can be maintained then the vertex functional is unique up to local contributions of the form ( 17 ), (34) and therefore renormalizable in the sense that only a finite number of parameters need be specified. One has to expect, however, that anomalies conflict with the invariance under general coordinate transformations. These anomalies can no longer be exchanged for Lorentz anomalies by regular WessZumino counterterms [ 5 ]. This is different from the situation [2,6] when G ' ~ is taken to be invertible. One has to expect the nonabelian S L ( D ) anomaly, a

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m i x e d d i l a t a t i o n a l a n d S L ( D ) a n o m a l y , a n d a purely dilatational anomaly. T h e restrictions which the absence of these a n o m alies i m p o s e s o n acceptable m o d e l s n e e d a separate investigation. Perhaps o n e can live with d i l a t a t i o n a l a n o m a l i e s because u n i t a r i t y does not require this part o f the gauge group [ 7 ]. A n o t h e r o b v i o u s d i r e c t i o n o f i n v e s t i g a t i o n will det e r m i n e regular supergravity lagrangians a n d their properties. In p a r t i c u l a r R - i n v a r i a n c e could possibly m a k e regular D = 4 - d i m e n s i o n a l theories with Y a n g Mills fields u n i q u e . I t h a n k W. Bernreuther, F. B r a n d t , U. Ellwanger, R. G r i m m , O. Piguet a n d R. Stora for s t i m u l a t i n g discussions a n d T. S c h w a n d e r for help in w r i t i n g this paper.

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References

[ 1 ] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239. [2] F. Brandt, N. Dragon and M. Kreuzer, Nucl. Phys. B 340 (1990) 187. [ 3 ] M. Maischak, Diplomarbeit (Hannover), to be published. [4] N. Dragon, in: Proc. XXIV Intern. Symp. Ahrenshoop, ed. G. Weigt (Institut ftir Hochenergiephysik, Zeuthen, FRG, 1991 ) p. 204. [ 5 ] J. Wess and B. Zumino, Phys. Lett. B 37 ( 1971 ) 95. [6] L, Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 (1984) 269; W. Bardeen and B. Zumino, Nucl. Phys. B 244 (1984) 421. [7] J.J. van der Bij, H. van Dam and Y.J. Ng, Physica A 116 (1982) 307; N. Dragon and M. Kreuzer, Z. Phys. C 41 (1988) 485; W. Buchmiiller and N. Dragon, Phys. Lett. B 223 ( 1989 ) 313; M. Kreuzer, Class. Quantum Gray. 7 (1990) 1303.

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