Fierz-Pauli Higher Derivative Gravity
A.Aioly (1)
, S.Ragusa (2)
, H. Mukai (3)
,and E. de Rey Neto (1)
(1)
Instituto deFsiaTeoria,
UniversidadeEstadual Paulista, Rua Pamplona145,01405-900 S~aoPaulo,SP,Brazil
(2)
DepartamentodeFsiaeInformatia,
Instituto deFsiadeS~aoCarlos,
Universidadede S~aoPaulo,C.P.369,
13560-250 S~aoCarlos, SP,Brazil
(3)
Departamentode Fsia,
Funda~aoUniversidadeEstadual deMaringa,
Av. Colombo5790,87020-900 Maringa,PR,Brazil
e-mail: aiolyift.unesp.br
Reeived20Deember,1999. Revisedversionreeivedon31May,2000
We onsider a model for gravity in whih the linear part of the four-derivative terms
R
RR
p
gd 4
xand R
R 2
p
gd 4
x are inludedinto the Fierz-Pauli gravitationalation.
Uni-tarity isdisussed at thetree-level. Theissue ofthe gravitational deetionofalight rayis also
onsidered.
I Introdution
Einstein'seld theoryaountsverywellforallknown
marosopi gravitational phenomena. The theory is
dened byation
S= Z
d 4
x p
g 2R
2
; (1)
where 2
=32G, G beingNewton's onstant, is the
Einstein's onstant. Thepossibility of a osmologial
onstant,whihexperimentallymustbeverysmall,was
ignored.
However,asaquantumtheoryitislesssatisfatory,
sine puregravityhasan S matrix whih, despite
be-ing niteat one-looplevel[1℄,divergesat thetwo-loop
order[2℄.
Thus,itisrathernaturalforthequantumeld
the-orist to view (1) as a long-distane approximation to
someationwhihexhibitsbetterultravioletbehavior,
and denes ameaningful quantum theory. Now, as is
wellknown, (1)isnotthemostgeneralationallowed
for the symmetries (generalovariane)of thetheory.
Thegeneralationisgivenby[3℄
S= Z
d 4
x p
g
2
2
R+ 0
R 2
+ 0
R
R
+ 0
R
R
Æ
R
Æ +:::
;
d
where 0
; 0
; 0
; et., are suitable parameters. The so
alled higherderivative gravity, in turn, is dened by
theation
S = Z
d 4
x p
g
2R
2
+
2 R
2
+
2 R
R
; (2)
DeserandvanNieuwenhuizen[4℄arguedthatsuh
the-ory would be renormalizable but ontain ghosts due
to the k 4
propagators. On the other hand,
Anto-niadis and Tomboulis [5℄ pointed out that the
pres-eneofamassivespin-2ghostinthebarepropagatorof
quadrati gravity is inonlusive, sine this exitation
isunstable. Stelle[6℄provedrigorouslythe
sarily has ghosts or tahyons [7℄. This result led us
to onsider a model forgravity whih ontains, in
ad-dition to the linear partof the four-derivatives terms
R
R
R
p
gd 4
x and R
R 2
p
gd 4
x, the Fierz-Pauli
gravitationalation. Thismodel,however,asfarasthe
tree-levelunitarityisonerned,fellfarshortofour
ex-petations. Butontheotherhand,ithasitsvirtuesas
wellwhihwethinkmakeitsstudyworthwhile. Indeed,
itagreesasymptotiallywithNewton's law-besidesit
predits an aeptable value for the gravitational
de-etion. It alsoallowsusto ndtheeetiveoupling
onstantforFierz-Pauligravity.
Having heked that the linear part of the
four-derivative term R
R 2
p
gd 4
x is the Ahille's heel of
Fierz-Pauligravitywith higherderivatives,we
onsid-eredafterwardamodelwhereinonlythelinearpartof
the four-derivative term R
R 2
p
gd 4
x is inludedinto
theFierz-Pauligravitationalation. Theresulting
the-oryisunitaryatthetreelevelandinadditiongives,in
the limitof tinymasses,a valueof1.31 arsefor the
deetion ofalightbeampassingnear totheSun.
InSe. II weexamineindetailthequestionof the
tree-level unitarity of quadrati gravity. Our aim in
so doing was in rst plae to provide a presription
unitarityofwhatevertheoryofgravitationwithhigher
derivativeswewished; and seondly, to present a
rig-oroustreatmentoftheissuerelatedtotheunitarityof
higherderivativegravityatthetreelevel,llingin this
wayagap inthe literatureonthe subjet. InSe. III
wedisuss Fierz-Pauli gravitywith higherderivatives.
\Fierz-Pauli+R 2
gravity"isanalyzedinSe. IV.
In our notation the signature is (+ ). The
urvature tensor is dened by R
Æ
=
Æ
+:::,
the Rii tensor by R
= R
, and the urvature
salarbyR =g
R
, whereg
is themetritensor.
Naturalunitsareusedthroughout.
II Tree-level unitarity and
bending of light in the
on-text of higher derivative
gravity
Intheweakeldapproximation,i. e.,g
=
+h
,
where
= diag(+1; 1; 1; 1), the higher
deriva-tivegravityLagrangianreduesto
L
g =
b
4 h
2h
2h
A
;
2
F 2
+(1+4) A
; 2
2 i
1
2 h
h
2h
+A 2
+(A
)
2 i
; (3)
where A
h
;
;h;F
A
; A
; ;b
2
2 ;
. Indies arelowered(raised)using
(
).
Inorderto verifywhether ghostsandtahyonsareabsentin(3), werequirethatthepropagatorhasonlyrst
order poles at k 2
M 2
=0with realmasses M (notahyons)and with positive residues(noghosts) [8℄. Let us
thendeterminethepropagatorforhigherderivativegravity. Todothat,weaddto(3)thegauge-xingLagrangian
L
gf =
1 ( A
)
2
+ b
4 h
2 A
; 2
2
+
3 F
2
i
:
TheresultingLagrangian,
L=L
g +L
gf ;
anbeastintothebilinearform
L= 1
2 h
O
; h
;
where theoperatorOisgivenby[9℄
O=x
1 P
1
+x
2 P
2
+x
0 P
0
+x
0
P 0
+
x
0
P 0
x 1 b=2 3 k 4 +2 1 k 2 =b ; x 2 b=2 k 4 +2k 2 =b ; x 0 b=2 4k 4 4k 2
=b+12k 4 +3 2 2 k 4 +12 1 2 k 2 =b ; x 0 b=2 2 k 4 2 2 k 4 8 1 k 2 =b+ 2 2 k 4 +4 1 2 k 2
=b+4
1 k 2 =b ; x 0 b=2 2 k 4 4 1 k 2 =b+ 2 2 k 4 +4 1 2 k 2 =b ; and P 1 ; = 1 2 ( ! + ! + ! + ! ) ; P 2 ; = 1 2 ( + ) 1 3 ; P 0 ; = 1 3 ; P 0 ; =! ! ; P 0 ; = ! +! ; with = k k =k 2 ; ! =k k =k 2 : P 1 ;P 2 ;P 0 ; P 0 and P 0
aretheBarnes-Riversoperators[10℄ andk
isthemomentum ofthegravitonexhanged.
Thenthepropagatorinmomentum spaeis
O 1 = 1 x 1 P 1 + 1 x 2 P 2 + 1 x 0 x 0 3 x 2 0 h x 0 P 0 +x 0 P 0 x 0 P 0 i : (4)
Thehoie =0,givesthepropagator
O 1 = m 2 1 k 2 (m 2 1 1 k 2 3 ) P 1 + m 2 1 k 2 (m 2 1 k 2 ) P 2 + m 2 0 2k 2 (k 2 m 2 0 ) P 0 + m 2 1 k 2 ( 2 1 m 2 1 2 k 2 ) P 0 : (5) with m 2 0 2 2
( 3+) ; m 2 1 4 2 : d
Note that all parts of the propagator (5) behave like
k 4
. ThishoieorrespondstotheJulve-Toningauge
[11℄.
Wearenowreadytoprobethetree-levelunitarityof
higherderivativegravity. Toaomplishthisweouple
thepropagator(5)toexternalonservedurrents,T
,
ompatiblewiththesymmetriesofthetheoryandthen
we examine the residue of the urrent-urrent
ampli-in momentumspaeanbewrittenas
A=g 2
T
(k)O
; ( k)T
(k) ;
wheregistheeetiveouplingonstantofthetheory.
Now,takinginto aountthat k
T
=0,weometo
theonlusionthatonlythespin-projetorsP 2
andP 0
A=g 2
T
m 2
1
k 2
(m 2
1 k
2
) P
2
+
m 2
0
2k 2
(k 2
m 2
0 )
P 0
; T
: (6)
Thus,wehavetwopolesforboththespin-2setor,i. e.,
k 2
=0 ; k 2
=m 2
1 ;
andthespin-0setor,namely,
k 2
=0 ; k 2
=m 2
0 :
In summary, the theory ofquadrati gravity yieldsa masslessand twomassiveexitations. Abseneof tahyons
requires that <0and3+ >0.
From(6)wepromptlyobtain
A=g 2
T
T
T 2
=2
k 2
T
T
T 2
=3
k 2
m 2
1 +
T 2
6(k 2
m 2
0 )
; (7)
where T
T
=T
.
Nowweexpandthesouresin asuitablebasis. Thesetofindependentvetorsinmomentum spae,
k
= k 0
;k
; ~
k
k 0
; k
; "
i
(0;~
i
) ; i=1;2 ;
where~
1 and~
2
aremutuallyorthogonalunitvetorswhiharealsoorthogonaltok,servesourpurpose. Aordingly,
thesymmetriurrenttensorT
( k)anbewrittenas
T
=ak
k
+b ~
k
~
k
+ ij
" (
i "
)
j +dk
(
~
k )
+e i
k (
" )
i +f
i
~
k (
" )
i :
Theurrentonservation,k
T
=0,givesthefollowingonstraintsfortheoeÆientsa,b,d,e i
andf i
ak 2
+
k 2
0 +k
2
d=2=0 ; (8)
b
k 2
0 +k
2
+dk 2
=2=0 ; (9)
e i
k 2
+f i
k 2
0 +k
2
=0 : (10)
IfwesaturatetheindiesofT
withmomentak
,weobtaintheequationk
k
T
=0,whihyieldsaonsisteny
relationfortheoeientsa,b andd
ak 4
+b
k 2
0 +k
2
2
+dk 2
k 2
0 +k
2
=0 : (11)
From(7),(8), (9)and(10)weonludethattheresidueofAatthepole k 2
=0is
R es A
k 2
=0 =g
2
1
2
11
2
+2 12
2
+ 1
2
22
2
k 2
=0 >0 :
So,theneessaryonditionfortree-levelunitarity,i. e.,R esA>0atthepole,isensuredasfarasthepolek 2
=0
isonerned.
Similarly,wendtheresidueatthepole k 2
=m 2
R es A k 2 =m 2 1 = g 2 ab k 2 0 +k 2 2 +b 2 k 4 +bdk 2 k 2 0 +k 2 + ij 2 1 2 k 2 0 +k 2 e i f i k 2 2 f i 2 1 3 h ak 2 +bk 2 ii +d k 2 0 +k 2 i 2 k 2 =m 2 1 = g 2
(a b)k 2 2 + ij 2 + k 2 2 h e i 2 f i 2 i 1 3
(b a)k 2 ii 2 k 2 =m 2 1 ;
whereusehasbeenmadeof(8),(9),(10)and(11). Thisexpressionanalsobewritten as
R es A k 2 =m 2 1 = g 2 ( 2 3
(a b)k 2 2 + " ij 2 ii 2 3 # + k 2 2 h e i 2 f i 2 i 2 3
(a b)k 2 ii k 2 =m 2 1 : d
Now,assumingasusualthatT 0,wegetthat ii
0,
whih implies that R es Aj
k 2
=m 2
1
<0. So, we have a
massivespin-2 ghost in the bare propagatorof higher
derivative gravity. This ghost is nontahyoni sine
we have assumed that < 0 m 2
1 >0
. In short,
quadratigravityisnonunitary.
Ofourse,R es Aj
k 2
=m 2
0
>0. Thus,thesalar
mas-sivepartileisaphysialone. Notethat wehave
sup-posedthat3+>0 m 2
0 >0
.
Letus takefor granted,for awhile, that the
mas-sivespin-2partileofnegativeresidue,i. e.,theghost,
is unstable [6℄. Now,aording to Feynman, The test
of all knowledge is experiment [12℄. In this vein, we
should expetthathigherderivativegravity,viewedas
a lassial theory, passed the tests suggested by
Ein-stein for general relativity. Sine we are dealingwith
the linearized version of higher derivative gravity, we
will onentrateourattention onlyonthe issueof the
gravitationaldeetion ofalightray. Letusthen
on-sider the interation between a xed soure, like the
Sun,andalightray. Theassoiatedenergy-momentum
tensorswillbedesignatedrespetivelyasT
andF
.
Theurrent-urrentamplitudeforthisproessisgiven
by
~
A = g 2 T m 2 1 k 2 (m 2 1 k 2 ) P 2 + m 2 0 2k 2 (k 2 m 2 0 ) P 0 ; F = g 2 T + 2k 2 1 2 ( + ) 1 3 k 2 m 2 1 + + 6(k 2 m 2 0 ) F : (12) d
Taking into aount that the energy-momentum
tensorforlight(eletromagnetiradiation)istraeless,
whileT =Æ 0 Æ 0 T 00
forastatisoure,weget
~
A=g 2 T 00 F 00 1 k 2 1 k 2 m 2 : (13)
Note that the harmless massive salar mode gives no
ontribution at allto the gravitational deetion.
In-identally, it does not ontribute either to the light
deetion in the framework of R+R 2
gravity[13,14℄.
is alwayssmaller than that predited by general
rela-tivity (
E
= 1:75 00
). Fig. 1 shows a qualitative plot
of deetion versus m
1
for rays passing by the Sun.
In the measurement of the solar gravitational
exp
theor
=1:00010:0001forthe deetion at thesolar
limb,where
theor
isthedeetionpreditedbygeneral
relativity. So,
Q:G: <
measured .
6
-m
1
Q:G:
E =1:75
00
Figure1 Deetion vs. m
1 .
III Fierz-Pauli higher
deriva-tive gravity
To arrive at the Lagrangian for Fierz-Pauli higher
derivative gravity we add the linear part of the
La-grangianontainingthefour-derivativeterms
2 R
2 p
g
and
2 R
2
p
g,namely,
L
hd =
b
4 h
2h
2h
A
;
2
F 2
+(1+4) A
; 2
2 i
;
where b ~ 2
2 ;
, ~
2
being the \Einstein's onstant" for Fierz-Pauli gravity [16℄, to the Fierz-Pauli
La-grangian,i. e.,
L
FP =
1
2 h
h
2h
+(A
)
2
+(A
)
2 i
1
2 m
2
h 2
2
:
Inmomentum spaetheresultingLagrangiananbewritten as
L= 1
2 h
b
2 k
4
+k 2
m 2
P 2
m 2
P 1
+ 2bk 4
2k 2
+6bk 4
+2m 2
P 0
+m 2
P 0
i
h
Sine this theoryis not gaugeinvariant owingto theProa-likemassterm, we donot need to introdue any
gaugexingterminto (14)in orderto ndthepropagator. As aresult, allwehavetodoin this aseisto invert
theoperator
O=
b
2 k
4
+k 2
m 2
P 2
m 2
P 1
+ 2bk 4
2k 2
+6bk 4
+2m 2
P 0
+m 2
P 0
:
Usingthealgorithm presented in Ref. [9℄ foromputing thepropagatorfor higherderivativegravitytheories,
wepromplyobtain
O 1
=
1
b
2 k
4
+k 2
m 2
P 2
1
m 2
P 1
2bk 4
+6bk 4
2k 2
+2m 2
3m 4
P 0
+ 1
3m 2
P 0
: (15)
Ifwetakeb==0in(15)wereoverthepropagatoronerningFierz-Pauligravity[16℄,namely,
O 1
; =
1
2 (
+
)
1
3
k 2
m 2
;
wherewehaveomittedthetermsproportionaltothegravitonmomentum.
OurtasknowistondoutwhetherornotFierz-PaulihigherderivativegravitygivesanaeptableNewtonian
limit. Tosueedindoingthisweomputetheeetivenonrelativistipotentialfortheinterationoftwoidential
massivebosonsofzerospinviaagravitonexhange. Theexpressionforthepotentialis
U(r)= 1
4M 2
1
( 2) 3
Z
d 3
kM
N:R: e
ikr
; (16)
whereupon M
N:R
is thenonrelativisti limitof the Feynman amplitudefor theproesss+s ! s+s, where s
standsforaspinlessbosonofmassM. TheorrespondingFeynmandiagramisshowninFig. 2.
TheLagrangianfortheinterationofgravitywithafree,massivesalareld ~
,is
L
int =
~
k
2 h
~
~
1
2
~
~
M
2
~
2
:
FromthepreviousexpressiontheFeynman rulefortheelementaryvertexmayreadilybededued. Itisshown
in Fig. 3. Aordingly,theinvariantamplitudefortheproessshowninFig.2is
M= 2m 2
1
1
( k 2
M 2
1 )(k
2
M 2
2 )
~
k 2
2 h
(pq)(p 0
q 0
)+(pq 0
)(p 0
q)
+(pp 0
) M 2
qq 0
+(qq 0
) M 2
pp 0
+2 M 2
pp 0
M 2
qq 0
2
3 2M
2
pp 0
2M 2
qq 0
;
whereupon
M 2
1 =m
2
1 +
q
m 4
1 2m
2
1 m
2
; M 2
2 =m
2
1 q
m 4
1 2m
2
1 m
2
;
wherem 2
1
1
b =
2
~ 2
. Abseneoftahyonsrequires that<0andm 2
1 >2m
2
.
Inthenonrelativistilimitthisexpressionredues to
M
N:R: =
4
3 ~ 2
M 4
m 2
1
(k 2
+M 2
)( k 2
+M 2
)
U(r)= 4
3 M
2
~
G
1
p
1 2m 2
=m 2
1
e M
1 r
r e
M
2 r
r
;
from whih itfollowsimmediately theexpressionfor\Fierz-Pauligeneralizedpotential"
V(r)= 4
3 M
~
G
1
p
1 2m 2
=m 2
1
e M1r
r e
M2r
r
: (18)
d
It is easy to see that (18)tends to the Fierz-Pauli
potential,i. e.,
V
F:P: ( r)=
4
3 M
~
G e
mr
r ;
asm 2
1
!1. Comparisonofthis result,in thelimitof
anextremelysmallmass,withtheNewtonianpotential
V
N (r)=
MG
r
,showsthat ~
G= 3
4
G. Asaonsequene,
~ g 2
=g 2
= ~
G =G=3=4, where ~g is theeetive oupling
onstantforFierz-Paulitheory.
A ursory look at (18) allows us to onlude that
at the originthat expression tends to the nite value
4
3 M
~
G(M
2 M
1 )=
s
1 2m
2
m 2
1
. Itisinterestingtonote
that only in the absene of tahyons (both positive
andnegativeenergy)inthedynamialelddoes
Fierz-Pauli higher derivative gravity agree asymptotially
withNewton's Law.
Let us now examine the question of unitarity (at
the tree level) of Fierz-Pauli higher derivative theory.
Tobeginwith weanalyzethepoles to getinformation
onthe physiality of the masses. As we havealready
mentioned, tahyons are exluded from the spetrum
whenever <0andm 2
1 >2m
2
. It istrivial tosee in
thisase that wehavetwo massivespin-twopartiles,
onewithsquaremassM 2
1
andanotherwithsquaremass
M 2
2
. One anverify promptly, that the residueof the
transition amplitudeat thepole k 2
=M 2
1
is negative,
whihimpliesthatFierz-Paulihigherderivativegravity
isnonunitary. Theresidueofthetransitionsamplitude
atthepole k 2
=M 2
2
,inturn, ispositive. Thereby,the
partileofmassM
2
isaphysialone,whilethatofmass
M
1
isaghost.
The transition amplitude for the interation
be-tweenastatisoureandalightbeamisgiven,inturn,
by(seeSe. II)
~
A = g~ 2
T 00
F 00
m 2
1
p
m 4
1 2m
2
m 2
1
1
k 2
M 2
1 +
1
k 2
M 2
2
= 3
4 g
2
T 00
F 00
m 2
1
p
m 4
1 2m
2
m 2
1
1
k 2
M 2
1 +
1
k 2
M 2
2
:
For m 2
extremely small, this expression tells us that 0<
F:P:H:G:
<1:31 00
, where
F:P:H:G:
is the deetion
s
s s
s
p 0
p
q 0
q k
Figure 2 Lowestorderontributiontothereations+s !s+s,
where sstandsforaspinless massiveboson.
p 0
p
V
( p;p
0
)= 1
2 ~
p
p
0
+p
p
0
p:p
0
+M 2
Figure 3 TherelevantFeynmanruleforboson-bosoninteration.
IV Final remarks
Despite its interesting properties, Fierz-Pauli higher
derivativegravitydidnotome uptoourexpetations
asfar asthe issueof unitarity isonerned. However,
it is easy to see that the massive spin-2 ghost that
appears in the bare propagator of Fierz-Pauli higher
derivative gravity is entirelydue to the linear partof
thefour-derivativeterm R
R 2
p
gd 4
x. So,itis
worth-whiletoonsideramodelforgravityin whihonlythe
linearpartof the four-derivativeterm R
R 2
p
gd 4
x is
inludedintothe Fierz-Pauligravitationalation. For
thesakeofsimpliity,letusallthistheory\Fierz-Pauli
+R 2
gravity". TheorrespondingLagrangianisgiven
by
L= ~
2
A
; 2
2 1
h
2h
+(A
)
2
+(A
)
2
1
m 2
(h 2
2
L= 1
2 h
h
k 2
m 2
P 2
m 2
P 1
+ 2k 2
+3~ 2
k 4
+2m 2
P 0
+m 2
P 0
i
; h
:
Thus,thepropagatorin momentumspaeis
O 1
= 1
k 2
m 2
P 2
1
m 2
P 1
3~ 2
k 4
2k 2
+2m 2
3m 4
P 0
+ 1
3m 2
P 0
:
d
Of ourse, the inlusion ofthe linearpartof the
four-derivativeterm R
R 2
p
gd 4
xintotheFierz-Pauli
grav-itationalationdoesnotimprovetheultraviolet
behav-ioroftheusualFierz-Paulitheory.
ItiseasytoseethatFierz-Pauli+R 2
gravityis
uni-taryatthetree-level. Ontheotherhand,thetransition
amplitude for the interation between a stati soure
andalightbeamisgivenby(seeSe. II)
~
A=~g 2
T 00
F 00
1
k 2
m 2
= 3
4 g
2
T 00
F 00
1
k 2
m 2
:
Inthelimitofanextremelysmallmass,Fierz-Pauli
+R 2
gravitygivesavalueforthebendingofalightray
passinglosetotheSunwhihis 3
4
ofthatpreditedby
generalrelativity. So,
1:31 00
=
F:P+R 2 <
measured :
Note thatthepreditionofbothFierz-Pauli+R 2
grav-ity and Fierz-Pauli gravity for the solar gravitational
deetion areoneandthesame(asexpeted).
Aknowledgment
E.deReyNetoisverygratefultoFunda~aode
Am-paroaPesquisadoEstadodeS~aoPaulo(FAPESP)for
nanialsupport.
Referenes
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20,69(1974).
[2℄ Mar H. Goro and Augusto Sagnotti, Phys. Lett. B
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