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Extending the Barnes-Rivers operators to D = 3 topological gravity
Physics Letters B 301 (1993) 339-344 North-Holland
PHYSICS LETTERS B
Extending the Barnes-Rivers operators to D = 3 topological gravity F.C.P. N u n e s Umverstdade Federal do Espirtto Santo - IFQ, Vlldrla, ES, Brazil and Untverstdade Federal do Rto de Janetro - IF, 21910 Rto de Janetro, R J, Brazil
and G.O. Pires Centro Brasdetro de Pesqutsas Ftswas - DCP, 22290 Rto de Janetro, R J, Braztl
Received 3 November 1992
The spin-projector operators for symmetric rank-2 tensors are reassessed in connection with the issue of topologicallymassive gravity The original proposal by Barnes and Rivers is generahsedto account for D-dimensionalEinstein gravity and three-dimensional Cbern-Slmons massive gravitation
The possibility of building up a q u a n t u m - m e c h a n i c a l l y consistent gauge theory for the gravitational field seems to be actually reahsed in three-dimensional space-time. The early work by Deser, Jackiw and Templeton [ 1 ] brings about the issue of a massive dynamical theory for gravitation in 3D. Ever since, topologically masswe gravity has been fairly well investigated in a series of very interesting papers, till very recently it has been shown that it is not only renormahsable [ 2,3 ] but even more: massive 3D gravity is a finite q u a n t u m field theory [4 ]. The purpose of this letter is to reassess the set of Barnes-Rivers spin operators [ 5,6 ] in the framework of topologically massive gravity. These have been shown to be very relevant in the description of 4D q u a n t u m gravity [7,8]. We shall in this letter propose a set of operators that extend the original Barnes-Rivers projectors to include D-dimensional massless and massive gravity as well as 3D gravity with topological mass. The gravlton propagators shall be written down and the tree-level unltarity shall be discussed in terms of the residues of the propagators at their poles. The Barnes-Rivers spin-projectors, as introduced in refs. [ 5,6], form a complete set of spin-projector operators in the space of rank-2 tensors. For the symmetric case, they read as below: ..~. /)(2)
_l(O.~O~a+O.~O~
e ( os) u . , , a _= ~OF,~O,a, i
o(o)
--w~v,x2
) - ~1O u ~ O ~ ,
_. .....
=t'~',uvt~x2,
-,,~*~ -=- ½(Ou~o~a + Osaco~ + O.~coaa + O~aogu~) D(l)
(o)
=
1
eswltv,~c2 -- "--~
,/3
O,uvO)~c2,
o(o)
_
~t wslav,x2 :
~
1
toa~O,~,
( 1a, 1b) ( lc, ld, le, lf)
where Ou. and cou. are the usual transverse and longitudinal projectors on the space of vectors. The operators in ( l a ) and ( l b ) are respectively the spin-2 and -1 projectors. The remaining ones project out spin-0 components of rank-2 symmetric tensors. Let us now consider the E i n s t e i n - H I l b e r t action for gravitation and derive its propagator by means of the algebra of the B a r n e s - R w e r s operators, taken now in a D-dimensional space-time: 1
°~/gHE =
2K2 X / - - g ~ "
0370-2693/93/$ 06 00 © 1993 ElsevierSciencePubhshers B V All rights reserved
(2) 339
Volume 301, number 4
PHYSICS LETTERSB
11 March 1993
Adopting the viewpoint of expanding the metric field around the flat-space geometry,
g""(x) =rlu"(x) + xhU"(x) ,
(3)
where h u" is the field variable defining the expansion, and taking into account only the free sector of the expansion, one gets the following free lagranglan for the h u~ field: ~Pfree HE = 1 0 ; t h # v
ORh,UV_10~huu Oah". + ~ Oah~u Ouh". - ½0ah~u O.h"u .
(4)
To give meaning to the integration measure in the generating functional of Green's functions, it is necessary to fix the gauge invariance, ahu.(x) = au(.(x) + a . ( . ( x ) ,
(5)
by introducing the De Donder gauge-fixing term: 1
~gf = ~-~ FuFU ,
(6a)
where
Fu[hp,~]=Oa(h.Zu -~vu,, l~al.,. ,,,.~
(6b)
The Hilbert-Einsteln lagranglan with gauge-fixing term can be rewritten in terms of the operators ( i a ) - ( l f ) according to
50(z) = ½hu" C:U.,~h 'a ,
(7)
where
6~.,~=0(_½p,2)_
1 ptm,)+4ot-3 p~o)_ l__~_p~wO)+ x f ~ ,o, ~ p ( o ) ] 2a 4or 4or ~ P'~ + -.,s lu,,,,~a •
(8)
The associated propagator is obtained from the generating functional ~t:[z.~] = -½ f dDxdDyrU"eZ2:aZ ~x ,
(9)
SO that
(T[hu.(x)h~a(y) l ) =ie'22,~a~°(x-y) .
(10)
So, using the rank-2 identity in the space of symmetric rank-2 tensors, one gets
i (_2p(z)_2otP~t) - 2 D - 5 p}o) - 2 2 D a - 4 o t - D + (T[hu~(x)h~(y) ] ) = ~ D-2 D-2
1 p(wO)
+ 2 D~_32 P}°) + 2 ~_32 P(°) ws "~ .],u#.r.~c~D(x--y)
(11)
or, in momentum space,
(T[hu.(-k)h,~a(k)])=
~7 qu,~q,,a+quarh,,¢- D_----~rlu,,q~-(l-ct)(rl/,,~og,'.~+q,..°gua+rl~ °9,.~) ,
(12)
where the projectors have been replaced by eqs. ( l a ) - ( l f ) , and the gauge-fixing parameter has been kept arbitrary Adding to the Hllbert-Elnstein action a Proca-hke mass term yields the following expression for the gravlton propagator: 340
Volume 301, number 4
(T[hu.(-k)h~a(k)])+21
+ m2(D - 1 ) + m 2 ( k-2i k_ 2m 2
PHYSICS LETTERS B
1(
k Z _ - m 2 rlu~rl~a+rluaq~ +
_
11 March 1993
rlu~q~
((k2-m2)[(2m2-1)(D-1)+l]-m4D)~u.~O~a ?HZ(kZ--m 2) ) ( rluxoJva+quat~vx+rlv~uK+rlvrogua+
D+~21 _ (quvOg.~+rlr~Oguv) ) .
(13)
An e x t e n s i o n o f the B a r n e s - R i v e r s o p e r a t o r s can be p r o p o s e d in o r d e r to account for D = 3 topologically m a s s i v e gravity [ 1 ]. It can be shown that one needs to a d d two o p e r a t o r s to table 1, $1 u~,,a - ~ ( - [] ) {~u,~a 0~o'~ ~ + Eu~ 0~co~ ~ + ~,~a 0~tn~u + ~
0a~o'~u }
(14a)
and
S z u . , ~ - ~ i-]{~u~rl~. + E ~ q a ~ + ~.~rl~ u + E~,~r/au } 0 ~' ,
(14b)
whtch can be f o u n d by a n a l y s i n g the bilanear part s t e m m i n g f r o m the g r a v n a t l o n a l C h e r n - S ~ m o n s term: =
3"l, ~ ' ~ p , "
(15)
F i x i n g the gauge as in ( 6 ) , the b f l l n e a r t e r m c o m i n g f r o m the H f l b e r t - E i n s t e i n a n d C h e r n - S i m o n s actions looks as follows:
~q~(2)=½h~V{[]( I e ( 2 ) + ~l~P(l)zOl m
4 aP- ~3 ( ° ) + 4 a
~ 1 P(°) - ~4o~P~(°) _ X3fp(o~)'q_4k2(s+ , s2)4t~s o~ ] -
~-
~u.,,~ h~a " (16)
Again, the a s s o c m t e d p r o p a g a t o r can be r e a d o f f w l t h the help o f the o p e r a t o r algebra d~splayed m table 1: Table 1 Mult~phcaUve table for the Barnes-Rxvers spin-proJector operators m D d~mensxons Operator product
p(z)pt2)=p(2)+~(D_4)p(o) P~')P") =P") p(Z)ptO) = { (4_D)pjO)
Extensmn to the case of 3D massive gravity
Tensorml identity
o2o2-t-az" ~. _n3/ip(o)~.~_as- 4-.*~t!/~(l)_p(2))
(p(2)a_po).a_la(o)a_l~(o)~ _ l t . ~ ~y r I_M.3 .t t l v t C ] . t ~--$ i-W ]ltv,lO--~tUlgrqgtfl¢
S2Sl =~D3P 0) S, $2 = ~[--13p~)
p}O)e(2)= ~_(4_D)p(O) ptZ)p}O)= ~ ( 4 _ O ) p } O ) pLo)p(2)= ~(4_D)p(o) P~°)Pts°) = ~( D - 1)Pst°) pLo)eLo) =eLo)
St S, = ~ ( - [] 3)p~)
S2P (2) =82 "-~-S1 P{2)$2 =$2 +s~
e ~ ° ) v ~°, = ~ ( o -
l )v~ °)
s~p~' = -s~
p~O)ptO) = ~ ( O-- 1 )p~O) p}O)p(O) =p(O) p(O) p~O) = p(O) p(O)p(O) = ps(O) p(O) p(O) = ~( D - 1 )p(O)
P~ )3"2= - S ,
St P~ ) =S, p ~ )S 1 = S 1
341
Volume 301, number 4
[h,,.(x)h,a(y)]
(T
PHYSICSLETTERS B
1 (
2(#/K2) 2
p(2)+2ap~)
) = ~ \ (#/x-~5)~ 6~[Z
11 March 1993
- 4 [ ( # / K 2 ) 2+481-q ] p}o) [(#/K2) 2+64[ ~ ]
16(#/K 2 )
+4(~-l)P~)-2x/3PJ~)-2x f ~ P ~ ) - [] [ ( # / K 2 ) 2 + 6 4 D ]
(SI
+$2)) 63(x-Y).
(17)
By choosing c~= 1 (Feynman gauge), we can write (T
-i( 41(#/x2)k" ] ) = ~ - k64k2_ (#/~c2) 2 (eu~ O~. + e~,.~Ox. +
[hu.(-k)h,a(k)
(~/K2) 2 - 64k 2_ ( # / / ~ 2 ) 2
~.,~.~0,~u + ~.,~,¢0.~u)
(rlu,~q~a+quarl~K--2rlu.q,a)
64k2
-- 64k 2_ ( # / / ~ 2 ) 2 ( ?~,uxo,)v2 + 71/15.(~)vx "1- J/vx(d)/a2 "1- F/v2 (J)/o¢ "4- {~,uv Omt -- 2~/v.o&a - 2rM ¢ou. )
)
.
( 18 )
As it can be seen, the Hilbert-Einstem action in D = 4 leads to a massless dynamical pole in the hv~-propagators, whereas the Elnstein-Chern-S~mons D= 3 action ymlds two poles: a massless non-dynamxcal excitation along with a non-tachyonlc massive dynamical mode, 2
.0
/19,
as already known from ref. [ 1 ]. Couphng the propagator to external currents, r "~, compatible with the symmetries of the theory, and then taking the imaginary part of the residues of the amplitude at the poles, one can probe the necessary condition for umtarlty at tree level and count degrees of freedom described by the field The current-current transition amphtude in momentum space ~s written as
sd= r*~'~(k)(T[hu~( -k)h,a(k)]
~ r~a(k),
(20)
where only the spin-projectors P (2), p co) and $2 shall contribute due to the transversahty of ru"(k). Now, defining the following set of independent vectors in momentum space:
kU=(k°;k),
/~=(k°;-k),
e,~=(0;G),
t=l,.
,D-2,
(21)
we can write the symmetric current tensor ru~(k) as
ru~(k) =a(k)kuk.+b(k)k(u~) +c,(k)ko, G) +d(k)Fcuk~+ e,(k)/~(uG.)+fj(k)E'(uG),
(22)
and then extract some relatmns mvolwng the above coefficients when ~mposmg its conservation for on-shell momenta k". So, for the Einstein theory an D d~mensions, the amphtude e/reads
;d=~22r.u,,(k)(_2p(2)(k)+2(5-D) D-2 P°(k))"~"~'~r~z(k),
(23)
then, at the pole k2=0, 5-D ImRes~'=[21rl,.12--~(l+~_2) Manipulating with 342
lr~.121.
ru,.(k) as expanded
above, one gets
(24)
Volume 301, number 4
PHYSICS LETTERSB
1
Im R e s d = 2 [ ' f s 1 2 - ~ ( 1
11 March 1993
+5-D
(25)
~--~_2) If, 12] -
For D = 4 dimensions, (26)
Im Res J = 2 ( ½ IA, -fz212+21A212) > 0 . Upon solwng the eigenvalue problem of the M-matrix of (26): ½I m R e s N = ( f t ,
f~2 f~3)
-½ 0
½ 0
(27)
[f22] ,
kfs3/
M one gets two non-vanishing elgenvalues that describe the two on-shell degrees of freedom of the massless graviton. For D = 3 dimensions, Im Res ~¢=2( [fj I2- [ft 12) = 0
( l = J = 1) ,
(28)
confirming, as it ~s known, that the Einstein theory is non-dynamical m three d~mensions. For the Einstein-Chern-Simons theory, ~¢=
i k2 [64k 2 - (/t/x2) 2 ]
{ (~z)2 [ ( /~z ) 2 ] 16 (/t/~c2) k2 )u~,~ ×z~.(k) 2 e(2)(k)- 4 -192k 2 e}°)(k)+ S2(k)~ r,~.~(k).
(29)
At the pole k2=0, Im Res ~4 { 2(/~/x212 = {m° k64k~_ ~ )
4[ (/I/K 2) 2-48k2] 16(it/k2)k '~ ) 2 (I r,~ 12- {I "r",, 12) - 3 [64k 2 - (,u/x 2) 2] I "c"%,12+ 64k 2_ (,u/x2) 2 ~,,~ r*u,dr~
,. { 64k21j]2 '~ = nk~=Ok64k m - T--.772-.,2 - - ( l t / x ) 2,] =0 '
(30)
which is therefore shown to be non-propagating. At the pole kZ= (/1/8K2) 2, Im Res d =
lim ( 2(~//K2)2 I2 4[(/t/Kz)2--48k2] IrUul2+ 16(#/x2)k" ) k2=(,u/SxZ)2\ k 2 ( I r ~ 1 2 - { [r~u ) 3k 2 kz eu~z*U':r~
=641J]2>0,
(31)
giving one degree of freedom. Here, attenuon must be prod to the stgn of the Hllbert-Elnstem lagranglan in D=3: a minus sign has to be chosen in (16) m order to guarantee a ghost-free massive propagator in three dimensions, although, w~th our choice of memc, the opposite sign ~s the one needed to ensure that the massless grav~ton ~s not a ghost. To conclude, we have set the spin-projector operators to deal with D-dimensional Einstein gravity and D = 3 topologically massive gravitaUon. Their mulUphcatwe table has been used in the derivation of the graviton propagators in a general gauge. 343
Volume 301, number 4
PHYSICS LETTERS B
11 March 1993
H a v m g an m i n d the c o u p l i n g o f a M a x w e l l - C h e r n - S t m o n s gauge field to E i n s t e i n - C h e r n - S l m o n s gravity, the p r o p a g a t o r ( 1 7 ) wall be e m p l o y e d to e x p h c l t l y calculate o n e - l o o p c o r r e c t i o n s to the c o u p l e d gauge-gravity system. T h e s e results shall be p r e s e n t e d a n d discussed in a further w o r k [ 9 ] We are i n d e b t e d to Dr. J.A. H e l a ~ e l - N e t o for p a t i e n t discussions a n d careful g m d a n c e . We are also grateful to O.M. Del C t m a for a helpful dtscusslon. We express o u r gratitude to the m e m b e r s o f the T h e o r e t i c a l Physics G r o u p o f the Unxversxdade C a t 6 h c a de Petr6polas for k i n d hospltahty. M.A. A n d r a d e is a c k n o w l e d g e d for the kind help an typesetting with L A T E X . T h e a u t h o r s are grateful to the C N P q and C A P E S for the i n v a l u a b l e financial support.
References [ 1] S Deser, R. Jacklw and S Templeton, Ann Phys 140 (1982) 372 [2] S Deser and Z Yang, Class Quantum Grav 7 (1990) 1603 [3] B Keszthelyl and G. Kleppe, Phys Lett B 281 (1992) 33 [4] F Delduc, C Lucchesl, O. Plguet and S P Sorella, Nucl Phys B 346 (1990) 313, A Blasl, O. Piquet and S P Sorella, Nucl Phys B 356 ( 1991 ) 154 [5] R J Rivers, Nuovo Cimento 34 (1964) 387 [ 6 ] K.J Barnes, Ph D thesis ( 1963 ), unpubhshed [7] P van Nleuwenhmzen, Nucl Phys B 60 (1973) 478 [8] I Antomadls and E T Tombouhs, Phys Rev D 33 (1986) 2756 [9] F.C P Nunes, G O Plres and F A B Rabelo de Carvalho, work m preparation
344
PHYSICS LETTERS B
Extending the Barnes-Rivers operators to D = 3 topological gravity F.C.P. N u n e s Umverstdade Federal do Espirtto Santo - IFQ, Vlldrla, ES, Brazil and Untverstdade Federal do Rto de Janetro - IF, 21910 Rto de Janetro, R J, Brazil
and G.O. Pires Centro Brasdetro de Pesqutsas Ftswas - DCP, 22290 Rto de Janetro, R J, Braztl
Received 3 November 1992
The spin-projector operators for symmetric rank-2 tensors are reassessed in connection with the issue of topologicallymassive gravity The original proposal by Barnes and Rivers is generahsedto account for D-dimensionalEinstein gravity and three-dimensional Cbern-Slmons massive gravitation
The possibility of building up a q u a n t u m - m e c h a n i c a l l y consistent gauge theory for the gravitational field seems to be actually reahsed in three-dimensional space-time. The early work by Deser, Jackiw and Templeton [ 1 ] brings about the issue of a massive dynamical theory for gravitation in 3D. Ever since, topologically masswe gravity has been fairly well investigated in a series of very interesting papers, till very recently it has been shown that it is not only renormahsable [ 2,3 ] but even more: massive 3D gravity is a finite q u a n t u m field theory [4 ]. The purpose of this letter is to reassess the set of Barnes-Rivers spin operators [ 5,6 ] in the framework of topologically massive gravity. These have been shown to be very relevant in the description of 4D q u a n t u m gravity [7,8]. We shall in this letter propose a set of operators that extend the original Barnes-Rivers projectors to include D-dimensional massless and massive gravity as well as 3D gravity with topological mass. The gravlton propagators shall be written down and the tree-level unltarity shall be discussed in terms of the residues of the propagators at their poles. The Barnes-Rivers spin-projectors, as introduced in refs. [ 5,6], form a complete set of spin-projector operators in the space of rank-2 tensors. For the symmetric case, they read as below: ..~. /)(2)
_l(O.~O~a+O.~O~
e ( os) u . , , a _= ~OF,~O,a, i
o(o)
--w~v,x2
) - ~1O u ~ O ~ ,
_. .....
=t'~',uvt~x2,
-,,~*~ -=- ½(Ou~o~a + Osaco~ + O.~coaa + O~aogu~) D(l)
(o)
=
1
eswltv,~c2 -- "--~
,/3
O,uvO)~c2,
o(o)
_
~t wslav,x2 :
~
1
toa~O,~,
( 1a, 1b) ( lc, ld, le, lf)
where Ou. and cou. are the usual transverse and longitudinal projectors on the space of vectors. The operators in ( l a ) and ( l b ) are respectively the spin-2 and -1 projectors. The remaining ones project out spin-0 components of rank-2 symmetric tensors. Let us now consider the E i n s t e i n - H I l b e r t action for gravitation and derive its propagator by means of the algebra of the B a r n e s - R w e r s operators, taken now in a D-dimensional space-time: 1
°~/gHE =
2K2 X / - - g ~ "
0370-2693/93/$ 06 00 © 1993 ElsevierSciencePubhshers B V All rights reserved
(2) 339
Volume 301, number 4
PHYSICS LETTERSB
11 March 1993
Adopting the viewpoint of expanding the metric field around the flat-space geometry,
g""(x) =rlu"(x) + xhU"(x) ,
(3)
where h u" is the field variable defining the expansion, and taking into account only the free sector of the expansion, one gets the following free lagranglan for the h u~ field: ~Pfree HE = 1 0 ; t h # v
ORh,UV_10~huu Oah". + ~ Oah~u Ouh". - ½0ah~u O.h"u .
(4)
To give meaning to the integration measure in the generating functional of Green's functions, it is necessary to fix the gauge invariance, ahu.(x) = au(.(x) + a . ( . ( x ) ,
(5)
by introducing the De Donder gauge-fixing term: 1
~gf = ~-~ FuFU ,
(6a)
where
Fu[hp,~]=Oa(h.Zu -~vu,, l~al.,. ,,,.~
(6b)
The Hilbert-Einsteln lagranglan with gauge-fixing term can be rewritten in terms of the operators ( i a ) - ( l f ) according to
50(z) = ½hu" C:U.,~h 'a ,
(7)
where
6~.,~=0(_½p,2)_
1 ptm,)+4ot-3 p~o)_ l__~_p~wO)+ x f ~ ,o, ~ p ( o ) ] 2a 4or 4or ~ P'~ + -.,s lu,,,,~a •
(8)
The associated propagator is obtained from the generating functional ~t:[z.~] = -½ f dDxdDyrU"eZ2:aZ ~x ,
(9)
SO that
(T[hu.(x)h~a(y) l ) =ie'22,~a~°(x-y) .
(10)
So, using the rank-2 identity in the space of symmetric rank-2 tensors, one gets
i (_2p(z)_2otP~t) - 2 D - 5 p}o) - 2 2 D a - 4 o t - D + (T[hu~(x)h~(y) ] ) = ~ D-2 D-2
1 p(wO)
+ 2 D~_32 P}°) + 2 ~_32 P(°) ws "~ .],u#.r.~c~D(x--y)
(11)
or, in momentum space,
(T[hu.(-k)h,~a(k)])=
~7 qu,~q,,a+quarh,,¢- D_----~rlu,,q~-(l-ct)(rl/,,~og,'.~+q,..°gua+rl~ °9,.~) ,
(12)
where the projectors have been replaced by eqs. ( l a ) - ( l f ) , and the gauge-fixing parameter has been kept arbitrary Adding to the Hllbert-Elnstein action a Proca-hke mass term yields the following expression for the gravlton propagator: 340
Volume 301, number 4
(T[hu.(-k)h~a(k)])+21
+ m2(D - 1 ) + m 2 ( k-2i k_ 2m 2
PHYSICS LETTERS B
1(
k Z _ - m 2 rlu~rl~a+rluaq~ +
_
11 March 1993
rlu~q~
((k2-m2)[(2m2-1)(D-1)+l]-m4D)~u.~O~a ?HZ(kZ--m 2) ) ( rluxoJva+quat~vx+rlv~uK+rlvrogua+
D+~21 _ (quvOg.~+rlr~Oguv) ) .
(13)
An e x t e n s i o n o f the B a r n e s - R i v e r s o p e r a t o r s can be p r o p o s e d in o r d e r to account for D = 3 topologically m a s s i v e gravity [ 1 ]. It can be shown that one needs to a d d two o p e r a t o r s to table 1, $1 u~,,a - ~ ( - [] ) {~u,~a 0~o'~ ~ + Eu~ 0~co~ ~ + ~,~a 0~tn~u + ~
0a~o'~u }
(14a)
and
S z u . , ~ - ~ i-]{~u~rl~. + E ~ q a ~ + ~.~rl~ u + E~,~r/au } 0 ~' ,
(14b)
whtch can be f o u n d by a n a l y s i n g the bilanear part s t e m m i n g f r o m the g r a v n a t l o n a l C h e r n - S ~ m o n s term: =
3"l, ~ ' ~ p , "
(15)
F i x i n g the gauge as in ( 6 ) , the b f l l n e a r t e r m c o m i n g f r o m the H f l b e r t - E i n s t e i n a n d C h e r n - S i m o n s actions looks as follows:
~q~(2)=½h~V{[]( I e ( 2 ) + ~l~P(l)zOl m
4 aP- ~3 ( ° ) + 4 a
~ 1 P(°) - ~4o~P~(°) _ X3fp(o~)'q_4k2(s+ , s2)4t~s o~ ] -
~-
~u.,,~ h~a " (16)
Again, the a s s o c m t e d p r o p a g a t o r can be r e a d o f f w l t h the help o f the o p e r a t o r algebra d~splayed m table 1: Table 1 Mult~phcaUve table for the Barnes-Rxvers spin-proJector operators m D d~mensxons Operator product
p(z)pt2)=p(2)+~(D_4)p(o) P~')P") =P") p(Z)ptO) = { (4_D)pjO)
Extensmn to the case of 3D massive gravity
Tensorml identity
o2o2-t-az" ~. _n3/ip(o)~.~_as- 4-.*~t!/~(l)_p(2))
(p(2)a_po).a_la(o)a_l~(o)~ _ l t . ~ ~y r I_M.3 .t t l v t C ] . t ~--$ i-W ]ltv,lO--~tUlgrqgtfl¢
S2Sl =~D3P 0) S, $2 = ~[--13p~)
p}O)e(2)= ~_(4_D)p(O) ptZ)p}O)= ~ ( 4 _ O ) p } O ) pLo)p(2)= ~(4_D)p(o) P~°)Pts°) = ~( D - 1)Pst°) pLo)eLo) =eLo)
St S, = ~ ( - [] 3)p~)
S2P (2) =82 "-~-S1 P{2)$2 =$2 +s~
e ~ ° ) v ~°, = ~ ( o -
l )v~ °)
s~p~' = -s~
p~O)ptO) = ~ ( O-- 1 )p~O) p}O)p(O) =p(O) p(O) p~O) = p(O) p(O)p(O) = ps(O) p(O) p(O) = ~( D - 1 )p(O)
P~ )3"2= - S ,
St P~ ) =S, p ~ )S 1 = S 1
341
Volume 301, number 4
[h,,.(x)h,a(y)]
(T
PHYSICSLETTERS B
1 (
2(#/K2) 2
p(2)+2ap~)
) = ~ \ (#/x-~5)~ 6~[Z
11 March 1993
- 4 [ ( # / K 2 ) 2+481-q ] p}o) [(#/K2) 2+64[ ~ ]
16(#/K 2 )
+4(~-l)P~)-2x/3PJ~)-2x f ~ P ~ ) - [] [ ( # / K 2 ) 2 + 6 4 D ]
(SI
+$2)) 63(x-Y).
(17)
By choosing c~= 1 (Feynman gauge), we can write (T
-i( 41(#/x2)k" ] ) = ~ - k64k2_ (#/~c2) 2 (eu~ O~. + e~,.~Ox. +
[hu.(-k)h,a(k)
(~/K2) 2 - 64k 2_ ( # / / ~ 2 ) 2
~.,~.~0,~u + ~.,~,¢0.~u)
(rlu,~q~a+quarl~K--2rlu.q,a)
64k2
-- 64k 2_ ( # / / ~ 2 ) 2 ( ?~,uxo,)v2 + 71/15.(~)vx "1- J/vx(d)/a2 "1- F/v2 (J)/o¢ "4- {~,uv Omt -- 2~/v.o&a - 2rM ¢ou. )
)
.
( 18 )
As it can be seen, the Hilbert-Einstem action in D = 4 leads to a massless dynamical pole in the hv~-propagators, whereas the Elnstein-Chern-S~mons D= 3 action ymlds two poles: a massless non-dynamxcal excitation along with a non-tachyonlc massive dynamical mode, 2
.0
/19,
as already known from ref. [ 1 ]. Couphng the propagator to external currents, r "~, compatible with the symmetries of the theory, and then taking the imaginary part of the residues of the amplitude at the poles, one can probe the necessary condition for umtarlty at tree level and count degrees of freedom described by the field The current-current transition amphtude in momentum space ~s written as
sd= r*~'~(k)(T[hu~( -k)h,a(k)]
~ r~a(k),
(20)
where only the spin-projectors P (2), p co) and $2 shall contribute due to the transversahty of ru"(k). Now, defining the following set of independent vectors in momentum space:
kU=(k°;k),
/~=(k°;-k),
e,~=(0;G),
t=l,.
,D-2,
(21)
we can write the symmetric current tensor ru~(k) as
ru~(k) =a(k)kuk.+b(k)k(u~) +c,(k)ko, G) +d(k)Fcuk~+ e,(k)/~(uG.)+fj(k)E'(uG),
(22)
and then extract some relatmns mvolwng the above coefficients when ~mposmg its conservation for on-shell momenta k". So, for the Einstein theory an D d~mensions, the amphtude e/reads
;d=~22r.u,,(k)(_2p(2)(k)+2(5-D) D-2 P°(k))"~"~'~r~z(k),
(23)
then, at the pole k2=0, 5-D ImRes~'=[21rl,.12--~(l+~_2) Manipulating with 342
lr~.121.
ru,.(k) as expanded
above, one gets
(24)
Volume 301, number 4
PHYSICS LETTERSB
1
Im R e s d = 2 [ ' f s 1 2 - ~ ( 1
11 March 1993
+5-D
(25)
~--~_2) If, 12] -
For D = 4 dimensions, (26)
Im Res J = 2 ( ½ IA, -fz212+21A212) > 0 . Upon solwng the eigenvalue problem of the M-matrix of (26): ½I m R e s N = ( f t ,
f~2 f~3)
-½ 0
½ 0
(27)
[f22] ,
kfs3/
M one gets two non-vanishing elgenvalues that describe the two on-shell degrees of freedom of the massless graviton. For D = 3 dimensions, Im Res ~¢=2( [fj I2- [ft 12) = 0
( l = J = 1) ,
(28)
confirming, as it ~s known, that the Einstein theory is non-dynamical m three d~mensions. For the Einstein-Chern-Simons theory, ~¢=
i k2 [64k 2 - (/t/x2) 2 ]
{ (~z)2 [ ( /~z ) 2 ] 16 (/t/~c2) k2 )u~,~ ×z~.(k) 2 e(2)(k)- 4 -192k 2 e}°)(k)+ S2(k)~ r,~.~(k).
(29)
At the pole k2=0, Im Res ~4 { 2(/~/x212 = {m° k64k~_ ~ )
4[ (/I/K 2) 2-48k2] 16(it/k2)k '~ ) 2 (I r,~ 12- {I "r",, 12) - 3 [64k 2 - (,u/x 2) 2] I "c"%,12+ 64k 2_ (,u/x2) 2 ~,,~ r*u,dr~
,. { 64k21j]2 '~ = nk~=Ok64k m - T--.772-.,2 - - ( l t / x ) 2,] =0 '
(30)
which is therefore shown to be non-propagating. At the pole kZ= (/1/8K2) 2, Im Res d =
lim ( 2(~//K2)2 I2 4[(/t/Kz)2--48k2] IrUul2+ 16(#/x2)k" ) k2=(,u/SxZ)2\ k 2 ( I r ~ 1 2 - { [r~u ) 3k 2 kz eu~z*U':r~
=641J]2>0,
(31)
giving one degree of freedom. Here, attenuon must be prod to the stgn of the Hllbert-Elnstem lagranglan in D=3: a minus sign has to be chosen in (16) m order to guarantee a ghost-free massive propagator in three dimensions, although, w~th our choice of memc, the opposite sign ~s the one needed to ensure that the massless grav~ton ~s not a ghost. To conclude, we have set the spin-projector operators to deal with D-dimensional Einstein gravity and D = 3 topologically massive gravitaUon. Their mulUphcatwe table has been used in the derivation of the graviton propagators in a general gauge. 343
Volume 301, number 4
PHYSICS LETTERS B
11 March 1993
H a v m g an m i n d the c o u p l i n g o f a M a x w e l l - C h e r n - S t m o n s gauge field to E i n s t e i n - C h e r n - S l m o n s gravity, the p r o p a g a t o r ( 1 7 ) wall be e m p l o y e d to e x p h c l t l y calculate o n e - l o o p c o r r e c t i o n s to the c o u p l e d gauge-gravity system. T h e s e results shall be p r e s e n t e d a n d discussed in a further w o r k [ 9 ] We are i n d e b t e d to Dr. J.A. H e l a ~ e l - N e t o for p a t i e n t discussions a n d careful g m d a n c e . We are also grateful to O.M. Del C t m a for a helpful dtscusslon. We express o u r gratitude to the m e m b e r s o f the T h e o r e t i c a l Physics G r o u p o f the Unxversxdade C a t 6 h c a de Petr6polas for k i n d hospltahty. M.A. A n d r a d e is a c k n o w l e d g e d for the kind help an typesetting with L A T E X . T h e a u t h o r s are grateful to the C N P q and C A P E S for the i n v a l u a b l e financial support.
References [ 1] S Deser, R. Jacklw and S Templeton, Ann Phys 140 (1982) 372 [2] S Deser and Z Yang, Class Quantum Grav 7 (1990) 1603 [3] B Keszthelyl and G. Kleppe, Phys Lett B 281 (1992) 33 [4] F Delduc, C Lucchesl, O. Plguet and S P Sorella, Nucl Phys B 346 (1990) 313, A Blasl, O. Piquet and S P Sorella, Nucl Phys B 356 ( 1991 ) 154 [5] R J Rivers, Nuovo Cimento 34 (1964) 387 [ 6 ] K.J Barnes, Ph D thesis ( 1963 ), unpubhshed [7] P van Nleuwenhmzen, Nucl Phys B 60 (1973) 478 [8] I Antomadls and E T Tombouhs, Phys Rev D 33 (1986) 2756 [9] F.C P Nunes, G O Plres and F A B Rabelo de Carvalho, work m preparation
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