Gauge-Fixing and Residual Symmetries in Gauge/Gravity Theories with Extra Dimensions

0.2 MB·English

by  C. S. Lim

KOBE-TH-07-09
Gauge-FixingandResidual Symmetries inGauge/GravityTheories withExtra Dimensions
C. S.Lim,1, TomoakiNagasawa,2,† SatoshiOhya,3,‡ KazukiSakamoto,3,§ andMakotoSakamoto1,¶
∗
1Department of Physics, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan
2Anan National College of Technology, 265 Aoki, Minobayashi, Anan 774-0017, Japan
3Graduate School of Science, KobeUniversity, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan
(Dated:February2,2008)
Westudycompactifiedpuregauge/gravitationaltheorieswithgauge-fixingtermsandshowthatthesetheories
possessquantum mechanical SUSY-likesymmetriesbetween unphysical degreesof freedom. Theseresidual
symmetriesareglobalsymmetriesandgeneratedbyquantummechanicalN =2supercharges. Also,wees-
tablishnewone-parameterfamilyofgaugechoicesforhigher-dimensionalgravity,andcalculateasacheckof
itsvalidityonegravitonexchangeamplitudeinthelowesttree-levelapproximation. Weconfirmthattheresult
isindeedx -independent andthecancellationofthex -dependenceisensuredbytheresidualsymmetries. We
alsogiveasimpleinterpretationofthevDVZ-discontinuity,whicharisesinthelowesttree-levelapproximation,
8
fromthesupersymmetricpointofview.
0
0
PACSnumbers:11.30.Pb,11.10.Kk
2

n
a I. INTRODUCTION KK-mode has its own superpartner. These functions are de-
J
fined as eigenfunctions of 2 2 matrix super-Hamiltonian.
6 Therefore it can be explained×that nth KK-mode of the 4d
One of the salient features in compactified higher-
component of the gauge field can absorb as its longitudinal
] dimensional field theories is the geometric “Higgs” mecha-
h degreesoffreedomitspartnernthKK-modeoftheextracom-
nismresultingfromthedimensionalreduction. Ifgaugepar-
t ponentof the field, andthen nth gaugebosonbecomesmas-
- ticles can propagate to extra dimensions, the corresponding
p sive.
higher-dimensionalgauge fields should be decomposed into
e In ordinary spontaneously broken gauge theories such as
h normal modes of extra dimensions and then, from the four-
[ dimensionalpointofview, theycanberecastintoaninfinite the Abelian Higgs model, if we work in the Rx gauge it is
inevitabletointroducethefictitiousparticles,i.e. thewould-
towerofmassivegaugebosonswhoselongitudinaldegreeof
1 beNambu-Goldstone(NG)bosons,soastomaintaintheuni-
freedomisprovidedbyabsorbingoneofextraspatialcompo-
v tarity of the S-matrix. The scalar componentof the massive
5 nentsofthefield. Thisgaugebosonmassgenerationisreal-
gaugefield,thewould-beNGboson,theFaddeev-Popov(FP)
4 izedwithoutinvokinganyscalarHiggsfields;itisessentially
ghostandtheantiFPghostareallindegeneratewiththesame
8 geometricalin nature. It also occursin compactifiedhigher-
0 dimensionalgravitytheoriesthesamemassgenerationmech- mass-squaredx m2. These spuriousdegreesof freedomcon-
. sistofamultiplet,knownastheBRS quartet[9], anddonot
1 anismwithoutinvokinganyfundamentalHiggsfields. There,
contributetothephysicalamplitudebycancelingeachother,
0 itseemsthatthereisnoexplicitsymmetrybreaking.However,
andthentheunitarityoftheS-matrixismaintained.
8 itwasshownintheKaluza-Klein(KK)theorythatthespon-
0 However,thereisnosuchscalarHiggsfieldsinpuregauge
taneous symmetry breaking certainly occurs [1, 2, 3, 4, 5]:
: theory with extra compact dimensions, nor in compactified
v thereexistsaninfinite-dimensionalKac-Moody-likesymme-
i tryatthe4dLagrangianlevel,butitisbrokendowntothe4d higher-dimensionalpure gravity. Thus the unphysicalpolar-
X
izationstatesofhigher-dimensionalgauge/gravitationalfields
translation and internal U(1) symmetry by the vacuum con-
r figurationM4 S1,andthenthenon-zerogravitonmodesbe- must be canceled among themselves. Furthermore, in order
a
× to realize this cancellation these spurious degrees of free-
comemassive.Thoughitwouldbethebesttodescribethege-
dom must be in degenerate with the same gauge-dependent
ometric“Higgs”mechanismbytheargumentalongthisline,
mass-squared. In view of this, we have to guaranteeat least
itseemstobehardtoextendtheanalysistoothermorecom-
two things: the first is the same number of degrees of free-
plicatedcompactifiedgravitytheories. Itislessobvioussuch
domfortheunphysicalcomponentsof4dgauge/gravitational
analysisisapplicabletocompactifiedgaugetheories.
field and the would-be NG bosons. These degrees of free-
Recently,analternativeviewofthismassgenerationmech-
dom have to appear in pairs. The second is the degeneracy
anism is given by [6, 8]: it is best described by using the of mass, that is, the same x -dependentpole of the propaga-
hiddenquantummechanicalsupersymmetryin4dmassspec-
tors between these pairs. In view of this, it is sufficient to
trum. Thepointisthatmasseigenfunctionforeachnon-zero
considerpuregauge/gravitationaltheoriesuptoquadraticor-
der and computethe propagatorsin Rx gauges. It should be
notedatthisstagethat,fromtheknowledgeofthesupersym-
metricstructureinthe4dmassspectrum,themassmatricesof
∗Electronicaddress:[email protected] thesepairsshouldbegivenbytheN =2super-Hamiltonians.
†Electronicaddress:[email protected]
Therefore it is reasonable to expect that these free field the-
‡Electronicaddress:[email protected]
§Electronicaddress:[email protected] ories would be invariant under some SUSY-like transforma-
¶Electronicaddress:[email protected] tionsgeneratedbysuperchargestorotatetheseunphysicalde-
2
grees of freedomwith differentspins. This is the main sub- Theactionweconsideris
ject of this paper. We study the compactified pure Abelian
1
gauge theory and pure gravity to quadratic order of gravita- S= d4x dDy√ G GMKGNLF F , (2)
MN KL
tionalfluctuations. We showthatthesefreefieldtheoriesex- ZM4 ZK − (cid:26)−4 (cid:27)
hibit residual global SUSY-like symmetries between the un- whereF =¶ A ¶ A andG=det(G )(= g). The
physicalcomponentsof4dgauge/gravitationalfieldsandthe MN M N− N M MN −
actionisinvariantundertheU(1)gaugetransformation
would-beNambu-Goldstonebosons,whichareoneoftheex-
twraesepsatacbelicsohmnpeowneRnxtsgoafutgheeshinigfihvere--ddiimmeennssiioonnaallfigerladvsi.tyA. lWsoe, AM(x,y)7→A′M(x,y)=AM(x,y)+¶ Me (x,y), (3)
checkitsvaliditybycomputingthelowesttree-levelgraviton wheree isanarbitraryfunction.Forthefollowingdiscussion
exchangeamplitude. Ofcoursetheresultdoesnotdependon itishighlyconvenienttousetheelegantmathematicaltoolof
thegaugeparameterx . Notethatwewillnotarguewithuni- thedifferentialforms.NoticethatontheD-dimensionalmani-
tarityboundsofsomespecificmodelsasdiscussedin[10,11]. foldK,Am andAibehaveasascalarandvectorfieldssuchthat
Inthispaperwewillrestrictourselvestofreefieldtheories. they can be regardedas 0-formsand a 1-form on K, respec-
The rest of this paper is organized as follows. In Sec- tively.Forfurtherdiscussionitisalsoconvenienttointroduce
tion II we study the pure Abelian gauge theory with extra theinnerproductofdifferentialformsonthemanifoldK. The
D-dimensions compactified on a Riemannian manifold and innerproductoftwok-formsonK isdefinedby
show that in the Rx gaugethe scalar componentof the four-
dimensionalgaugefieldandthewould-bescalarNGbosonare w (k),h (k) := w (k) h (k)= dDy√g 1w (k)gIJh (k),
in degeneratewith the same gauge-dependentmass-squared, K ∧∗ K k! I J
and they form a multiplet under the SUSY-like transforma- (cid:0) (cid:1) Z Z (4)
tion generated by the supercharge found in the analysis of where istheHodgestaroperatorand
∗
the 4d mass spectrum. In Section III we extendthe analysis
1 1
to five-dimensionalgravity with the Randall-Sundrumback- w (k):= w (k) dyi1 dyik =: w (k)dyI, (5a)
ground and obtain the similar results. In this section we es- k! i1···ik ∧···∧ k! I
tablishtheone-parameterfamilyofgaugechoicesanalogous gIJ :=gi1j1 gikjk. (5b)
···
totheRx gaugeinspontaneouslybrokengaugetheories.Asa I and J denote k-tuples of indices. With this definition, (2)
checkofvalidityofthisRx gauge,wecomputetheonegravi- and(3)canbewrittenasthefollowingform:
tonexchangeamplitudeinthelowesttree-levelandshowthat
the result is indeedx -independent. From the resultobtained
S= d4xL , (6a)
by this computation, we give a simple interpretation of the K
M4
Z
appearance of the van Dam-Veltman-Zakharov(vDVZ) dis- 1 1
continuityfromsupersymmetricviewpoint. Sinceitseemsto LK = Fmn ,Fmn ¶ m A(1) dAm ,¶ m A(1) dAm
−4 −2 − −
beunfamiliartoparticletheorists,wedevoteAppendixAtoa
briefreviewofthevDVZ-discontinuity. WeconcludeinSec- 1(cid:0)dA(1),dA(cid:1)(1) , (cid:0) ((cid:1)6b)
−2
tionV.
(cid:0) (cid:1)
and
II. PUREABELIANGAUGETHEORYWITHEXTRA Am A′m =Am +¶ m e , (7a)
7→
DIMENSIONS A(1) A(1) =A(1)+de , (7b)
′
7→
Inthissectionwe showthatanycompactifiedpureAbelin whereA(1)=Adyiandd=dyi¶ .
i i
gaugetheoriesintrinsicallypossesshiddenquantummechan- Toclarifythesupersymmetricstructure,wewillfollowthe
ical SUSY-like symmetry between a scalar component of methodofref.[6]. TheHodgedecompositiontheoremtellsus
4d gauge field and one of scalar components of higher- that any k-form on K can be uniquely decomposed into the
dimensional gauge field, which should be regarded as a sumoftheharmonick-form,theexactk-formandthecoexact
would-bescalarNGboson. k-form on K. Noting that there is no exact 0-form, we can
LetusconsiderthepureAbeliangaugetheorywithextraD- write
dimensionscompactifiedon a RiemannianmanifoldK with-
outboundary.Wedenotethe(4+D)-dimensionalgaugefield A(m0)=Am ,0(x)h (0)+(cid:229) Am ,n(x)w n(0), (8a)
byAMm (x,y)= Am (x,y),Ai(x,y) (m =0,1,2,3;i=5,···,D+ n6=0
4), x are the coordinates of the ordinary four-dimensional A(1)=j +h+F
Minkowskispa(cid:0)cetimeM4andyi(cid:1)arethecoordinatesofK.The
bulkmetricisgivenby =(cid:229)b1 j (x)h (1)+(cid:229) h (x)f (1)+ (cid:229) F (x)w (1), (8b)
ds2=GMNdxMdxN =h mn dxm dxn +gij(y)dyidyj, (1) i=1 i i n6=0 n n n′6=0 n′ n′
where h mn is the flat spacetime metric whose signature is where h (0) = KdDy√g −1/2 and b1 is the first Betti num-
( ,+,+,+). ber. Aswe shallseeimmediately,b givesthenumberof4d
− (cid:2)R (cid:3) 1
3
massless scalar bosons, which are identified as Higgs fields d h =m(0)e and the others are equal to zero. Hence, from
n n n
inthegauge-Higgsunificationscenario. Forexample,b1=1 thehigher-dimensionalgaugeinvariantpointofview,hnhasto
for K =S1, b1 =0 for K =S2, b1 = n1 =n for K =Tn = appeartoLK inthecombinationAm ,n (1/m(n0))¶ m hn,which
S1 S1(Ku¨nnethformula),etc. is realized in (13) as it should be. B−y (partially) fixing the
×···× (cid:0) (cid:1)
w n(k) andf n(k+1) (k=0,1)arecoexactk-formandexactk+ gaugeen= (1/m(n0))hn(withe0leftarbitrary)allthewould-
1-formanddefinedastheorthonormalsetsoftheeigenmodes beNGboso−nsh aregaugedaway. Thisistheunitarygauge
n
ofthepositivesemi-definiteisospectralHamiltoniansd†dand of the theory. The four-dimensional particle content of the
dd†: theoryisnowobvious: themasslessphotonAm ,0,theinfinite
(0)
d0†d d0d† fw(kn(+k)1) = m(nk) 2 fw(kn(+k)1) , (9) tmowasesrleossf mscaaslsairvebogsaoungsefbioasnodnsthAem ,innfiwniitthemtoawssermonf m, tahsesivbe1
(cid:20) (cid:21)" n # " n # scalarbosonsF withmassm(1).
(cid:0) (cid:1) n′ n′
wheretherelativephaseischosentobe
0 d† w n(k) =m(k) w n(k) . (10) A. Gauge-FixingandResidualSymmetry
d 0 f (k+1) n f (k+1)
(cid:20) (cid:21)" n # " n #
As already mentioned in Section I, in this paper we are
Nowweintroducethefollowing2 2matrixoperators interested in the Rx gauge. Since there is an undesirable
×
d†d 0 0 d† 1 0 quadratic term mixing Am and A(1) in (6b) (or Am ,n and hn
H= , Q = , ( 1)F = , in (13)), we have to get rid of this mixing term by correctly
0 dd† 1 d 0 − 0 1 choosinga gauge-fixingfunction. Sucha gauge-fixingfunc-
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) − (cid:21)
(11)
tion is already introduced, for example, in [7] in the context
whichsatisfytheN =2supersymmetryalgebra offive-dimensionalgaugetheory.TheD-dimensionalversion
ofsuchagauge-fixingfunctionshouldbechosenas
Q ,Q =2d H, [H,Q ]=[H,( 1)F]=0,
a b ab a
{ } −
( 1)F,Qa =0, a,b=1,2, (12) F[A]=¶ m Am x d†A(1)=¶ m Am x d†h. (14)
{ − } − −
whereQ2=i( 1)FQ1. ( 1)F isthe“fermionic”numberop- Byaddingagauge-fixingterm (1/2x )F[A]2 to(6b)andin-
erator which a−ssigns to t−he “bosonic” state w (k) a quantum tegratingbypartsweobtainthe−gauge-fixedLagrangian
n
number+1andtothe“fermionic”statef (k) aquantumnum-
n
1
berF−ro1m. theorthonormalityoftheeigenmodes1 ,itiseasyto L = 2 Am ,[h mn ((cid:3)−d†d)−(1−1/x )¶ m ¶ n ]An
writedownthe4dreducedLagrangian +(cid:0)1 j ,(cid:3)j +1 F ,[(cid:3) d†d]F (cid:1)
2 2 −
LK = 1 Fmn ,0 2 +1(cid:0)h,[(cid:3) (cid:1)x dd†(cid:0)]h , (cid:1) (15)
−4 2 −
(cid:0) (cid:1) 2 (cid:0) (cid:1)
+n(cid:229)6=0−41(cid:0)Fmn ,n(cid:1)2−21(cid:0)m(n0)(cid:1)2 Am ,n−m1(n0)¶ m hn!  twaihnertehe(cid:3)g=au¶gme¶bmo.soInnv(Ae0)rmt,in2ngprtohpeakgianteotriccotenrtmainoipnegraatoterrwmeporbo--
+i(cid:229)=b11(cid:26)−21 ¶ m j i 2(cid:27)  t−piookrn2tio=ofnxathletmow(n0ko)muk2lnd,/-wb(cid:0)hemicNnhG(cid:1)isbeowxsiaotchnt’ltsyh(ethhge’assu)a,gmaees-dsietipmsehpnoldeuelpndot.lpeIonpleoansaiy-t
+ (cid:229) 1(cid:0) ¶ m F (cid:1) 2 1 m(1) 2F 2 , (13) freefield(cid:0)theor(cid:1)ieswhentwodifferenntfieldsareindegenerate
−2 n′ −2 n′ n′ andhavethesamedegreesoffreedom,thereobviouslyexists
n′6=0(cid:26) (cid:0) (cid:1) (cid:0) (cid:1) (cid:27) asymmetrytorotatethem. Nowthisisindeedthecase. The
whereFmn ,n=¶ m An ,n ¶ n Am ,n. Itisworthmentioningatthis termproportionaltokm kn / m(n0) 2 withthegauge-dependent
point that each KK-m−ode transforms under the U(1) gauge pole at k2 =x m(0) 2 in the gauge boson propagatorsug-
transformation (7a) (7b) as d Am ,0 = ¶ m e0, d Am ,n = ¶ m en, geststha−tthescalarcnompon(cid:0)entof(cid:1)thegaugefieldhasthesame
mass-squaredx (cid:0)m(0) (cid:1)2 tothewould-beNGboson’s. Indeed,
n
ifwedecomposeAm intothetransverseandlongitudinal2(or
(cid:0) (cid:1)
1 Theeigenmodesarenormalizedas
w n(k),w m(k) =dnm, fn(k+1),fm(k+1) =dnm,
(cid:0)h i(1),h (j1)(cid:1)=dij. (cid:0) (cid:1) 2iWs,etrhaenrsevuersseethmeetaenrsmpsetrrpaennsvdeicruselaarntdol¶omng(iotrudkimn)alanindalocnogviaturidainntalwpaayr,atlhlealt
(cid:0) (cid:1) to¶m (orkm ).
4
puregauge)part H iswrittenbyproductofthesupercharges,itisquitenatural
thesespuriousdegreesoffreedomaremutuallyrelatedbythe
Am =A(mT)+¶ m A(L), (16) globaltransformationgeneratedbyQ1.
(3)Sincetheresidualsymmetry(20)isaspacetimesymme-
theLagrangianbecomes trybetweenaspin-1andaspin-0particle,onemightsuspect
that(20)wouldconflictwith the Coleman-Mandulatheorem
1 1
L = 2 A(mT),[(cid:3)−d†d)]A(T)m −2x A(L),(cid:3)[(cid:3)−x d†d]A(L) [12]. However, the residual symmetry rotates only the un-
physicaldegreesoffreedom.Hencethisisnotthecase.
+(cid:0)1 j ,(cid:3)j +1 F ,[(cid:3)(cid:1) d†d]F(cid:0) (cid:1) (4) This residual symmetry can be regarded as a relic
2 2 − of higher-dimensional gauge symmetry by using the gauge-
+1(cid:0)h,[(cid:3) (cid:1)x dd†(cid:0)]h . (cid:1) (17) fixing condition F[A]= 0. It is easy to see that the trans-
2 − formation(20)isindeedthehigher-dimensionalgaugetrans-
(cid:0) (cid:1) formationwithanidentificatione =q d†h=q (1/x )¶ m Am =
TocanonicalizetheLagrangianweredefinethefieldas3
q ((cid:3)/x )A(L). Inviewofthisfact,onemightexpectthatcom-
pactifiedgaugetheorieswithgaugeinvariantinteractionterms
(cid:3)
A(L) A˜(L):= A(L), (18) would also possess some residual symmetries such as (20).
7→ sx However,thisisanopenquestion.
so that (A˜(L),h)-sector can then be recast into the following
2 2matrixform III. 5DGRAVITYWITHRANDALL-SUNDRUM
× BACKGROUND
t
1 A˜(L) 1 0 d†d 0 A˜(L)
, − (cid:3) x . (19)
2 (cid:20) h (cid:21) (cid:20) 0 1(cid:21)(cid:18) − (cid:20) 0 dd†(cid:21)(cid:19)(cid:20) h (cid:21)! We would like to further extend the previous analysis to
compactifiedhigher-dimensionalpuregravitytoquadraticor-
We have thus separated the Lagrangian into the gauge- der of gravitationalfluctuations. Because of authors’ inabil-
independentand-dependentparts. Thegauge-dependentpart itytoanalyzethegravityingeneraldimensionswith general
canbewrittenintermsof2 2matrixformandthemassma- backgroundmetric,weshallrestrictourselvestoconsiderthe
×
trixisexactlytheHamiltonianin(11),asexpected. Itisnow perturbationtothewell-knownRandall-Sundrummetric[13]
obvioustheLagrangianisinvariantunderthetransformation asalesstrivialexample.
Letusconsiderthefive-dimensionalgravitywithsingleex-
A˜(L) A˜(L)′ =q 0 d† A˜(L) . (20) tra dimension compactified on an interval (0,p R) with two
h 7→ h′ d 0 h 3-branes which are located at the end points of the interval.
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
Theactionrespectingthisconfigurationis
Thissymmetrytransformationisgeneratedbythesupercharge
Q in(11),andhenceSUSY-like. p R
1 S= d4x dy √ G(M3R L )+√ g ( s )d (y)
Severalcommentsarenowinorder: 0 − − − UV − UV
Z Z (cid:26)
(1) This is a symmetry between the scalar component of
thegaugefieldAm andthewould-beNGbosonh,bothareun- +√ gIR( s IR)d (y p R) , (21)
− − −
physicaldegreesoffreedomofthetheory. Thusifweaddto (cid:27)
theLagrangianthegaugeinvariantinteractiontermsorextend whereM is the five-dimensionalPlanck mass, R is the Ricci
tonon-Abeliangaugetheories,andthencalculatetheS-matrix scalar4 evaluatedbythe5dmetricGMN andgUmn V (gImnR)isthe
elements,thesespuriousdegreesoffreedommustbecanceled metricinducedontheUV(IR)brane. Thebulkcosmological
soastomaintaintheunitarityoftheS-matrix.Wemaythere- constantL andthebranecosmologicalconstantss ands
UV IR
foreconcludethatthesecancellationsarecompensatedforthe aretunedtogiveawarpedbackgroundsolutionwithasliceof
symmetry(20),thoughitismuchlessobviouswhetherresid- AdS :
5
ual symmetry such as (20) might exist in a fully interacting
theory. ds2= e−2kyh mn dxm dxn +dy2, (physicalcoordinate)
(2)AsalreadymentionedinSectionI,(20)isanexpected ( 1 2(h mn dxm dxn +dz2), (conformalcoordinate)
1+kz
symmetry. From the unitarity point of view, these unphys- (22)
(cid:0) (cid:1)
ical gauge-dependent part must have its partners to ensure
the cancellation of each contributions to the S-matrix ele-
ments. In view of this, these partnersmust be in degenerate
with the same x -dependent mass-squared. Since the super- 4Inthispaperweusetheconventions:
HamiltonianH isonlythecandidateforthemass-matrixand
1
G AMN:= 2GAB(¶NGBM+¶MGBN−¶BGMN),
RKLMN:=¶MG KLN−¶NG KLM+G ALNG KAM−G ALMG KNA,
3Thedefinition(18)iswell-definedonlyintheregioninside(outside)the RMN:=RAMAN=¶AG AMN−¶NG AMA+G AMNG BAB−G BMAG ANB.
light-coneandx >0(x <0.)
5
wcuirthvastuUrVes=ca1le2kdMefi3naenddass kIR==−(12LkM)/31,2wMh3e.rIetkisijsusthteamAadtS- Kmn ;r 5=−21(h mr ¶ n +h nr ¶ m −2h mn ¶ r )(¶ z+3A′), (27b)
ter of choice which coordinateps(−physical or conformal)one Kmn ;f = 3h mn (¶ z+3A′)(¶ z+2A′), (27c)
will use, however, the conformal coordinate will turn out to 2
beadvantageoustoevaluatevariousquantities,especiallythe 1
gravityactionbecauseitutilizestheveryusefulformulaofthe Km 5;rs =−2(h mr ¶ s +h ms ¶ r −2h rs ¶ m )¶ z, (27d)
conformaltransformation. Thusin whatfollowswe will use 1
theconformalcoordinatezforcomputationalconvenience. Km 5;r 5=−2(¶ m ¶ r −h mr (cid:3)), (27e)
We investigate the gravitational fluctuations around the 3
background Km 5;f =−2¶ m (¶ z+2A′), (27f)
3
GMN =e2A(z) h MN+h¯MN(x) , A(z)= ln(1+kz). (23) Kf ;rs = 2h rs (¶ z+A′)¶ z, (27g)
−
3
It should be no(cid:2)ted that the ex(cid:3)plicit expression for the warp Kf ;r 5=−2¶ r (¶ z+A′), (27h)
f(a1c/tko)r(eAp(kzR) i1s):valid on the open interval 0 < z < zc := Kf ;f = 23(cid:3)−3(¶ z+A′)(¶ z+2A′). (27i)
−
A (A)2=2keA[ d (z)+d (z p R)], (24) Theseexpressions5 arealsofoundin[15,16]andconsistent
′′ ′
− − − withtheEinsteinequations[17].
One can easily verifythatthe action is invariantunderin-
where prime () indicatesthe derivativewith respectto z. In
′
finitesimalgeneralcoordinatetransformations
the following analysis, however, the delta functions on the
right hand side of Eq.(24) can safely be ignored because of
xM xˆM=xM+x M, (28)
theboundaryconditionsforthemetricfluctuationsconsistent 7→
withthegeneralcoordinateinvariance.Thereforewewilluse
whichtransformthefluctuationsatthelinearizedlevelas
therelationA′′ (A′)2=0inthewholeofthispaperinstead
−
of(24). hmn hˆmn =hmn ¶ m x n ¶ n x m h mn (¶ z+3A′)x 5, (29a)
Thefluctuationsh¯ shouldbeparameterizedas 7→ − − −
MN hm 5 hˆm 5=hm 5 ¶ zx m ¶ m x 5, (29b)
7→ − −
h¯MN = hmn −(h1n/52)h mn f hfm 5 , (25) f 7→fˆ=f −2(¶ z+A′)x 5. (29c)
(cid:20) (cid:21) SupersymmetricStructure:Forthelaterdiscussionwehere
briefly summarize the quantum mechanical supersymmetric
which enables us to identify the spectrum of the linearized
theory. Noticethatthe limitL 0givesk=0 andz =p R structure of the theory. As shown in ref.[8] the fluctuations
c
andhencethebackgroundgeom→etry(22)justreducestoM4 areexpandedas
(0,p R),asitshould. × ¥
Thequadraticactiontakestheform hmn (x,z)= (cid:229) h(mnn)(x)f(n)(z), (30a)
n=0
¥
S(2)=M3Z d4xZ0zcdze3A41hMNKMN;KLhKL hm 5(x,z)=n(cid:229)=1h(mn5)(x)g(n)(z), (30b)
zc ¥
=M3 d4x dze3A f (x,z)= (cid:229) f (n)(x)k(n)(z), (30c)
0
Z Z
1 n=0
hmn Kmn ;rs hrs +2hmn Kmn ;r 5hr 5+hmn Kmn ;f f
×4 where we have excluded the vector zero-modeg(0) from the
(cid:26)
+2hm 5Km 5;rs hrs +4hm 5Km 5;r 5hr 5+2hm 5Km 5;f f expansion(30b),sinceasweshallseeimmediatelyitwillbe
incompatiblewiththesupersymmetricstructure.
+f Kf ;rs hrs +2f Kf ;r 5hr 5+f Kf ;f f , (26)
(cid:27)
where
1
Kmn ;rs = (h mr ¶ n ¶ s +h ms ¶ n ¶ r +h nr ¶ m ¶ s +h ns ¶ m ¶ r )
−2
+h mn ¶ r ¶ s +h rs ¶ m ¶ n
1
+ (h mr h ns +h ms h nr 2h mn h rs )
2 −
((cid:3)+(¶ +3A)¶ ), (27a) 5Thenormalizationisdifferentfrom[8]by1/2tocanonicalizetheoperator
× z ′ z Kmn ;rs .
6
The orthonormal sets f(n) ¥ g(n) ¥ and k(n) ¥ tions
{ }n=0 { }n=1 { }n=0
aredefinedastheeigenfunctionsoftheSchro¨dinger-likeequa-
(¶ +3A)¶ 0 f(n) f(n)
− z 0 ′ z ¶ z(¶ z+3A′) g(n) =m2n g(n) , (31a)
(cid:20) − (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
(¶ +2A)(¶ +A) 0 g(n) g(n)
− z 0′ z ′ (¶ z+A′)(¶ z+2A′) k(n) =m2n k(n) , (31b)
(cid:20) − (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
withtheboundaryconditions whereQ =i( 1)FQ andQ¯ =i( 1)F¯Q¯ .
2 1 2 1
− −
¶ f(n)(z)=0, g(n)(z)=0, (¶ +2A)k(n)(z)=0, (32) We emphasizethat(32)istheuniqueboundaryconditions
z i i z ′ i
toensuretheHamiltoniansH andH¯ tobeisospectral(except
wherez =0orz . Notethatthebackgroundsolutionenables
i c forthezeromodes),orthesupersymmetriesinthemassspec-
ustofactorizetheHamiltonianforthevector-modeg(n)intwo
trum. Indeedthe hermiticity of the Hamiltoniansand super-
different ways (except on the boundaries); ¶ (¶ +3A)=
(¶ +2A)(¶ +A). − z z ′ chargesisensuredifandonlyiftheboundaryconditions(32)
− z ′ z ′ areimposed(seeref.[8]).
Theseeigenfunctionsaremutuallyrelatedthrough
0 (¶ +3A) f(n) f(n)
¶ z − z0 ′ g(n) =mn g(n) , (33a) UthneitthaeroyrGy.aTuogeth:iNseexntdwweewfiilrlstidneontteiftyhathteuspianrgticthleecmonotdeentdoef-
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
0 (¶ +2A) g(n) g(n) compositions(30a)– (30c) the linearized generalcoordinate
¶ z+A′ − z0 ′ k(n) =mn k(n) . (33b) transformations(29a)–(29c)read
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)
Nowitisclearthatthereisnonontrivialsolutionofthevector
dzeifrfoe-rmenotdiaelgeq(0u)awtiohnicsh has to satisfy the two different linear d h(mnn)=−¶ m x n(n)−¶ n x m(n)+mnh mn x 5(n), n∈Z≥0, (38a)
(¶ z+3A′)g(0)=0=(¶ z+A′)g(0). (34) d h(mn5)=−mnx m(n)−¶ m x 5(n), n∈Z>0, (38b)
− df (n)= 2m x (n), n Z , (38c)
Wehavealreadyincludedthisfactintotheexpansion(30b). − n 5 ∈ ≥0
Itshouldbenotedthattheeigenfunctionssatisfy theiden-
tity
wherethegaugefreedomx areassumedtobeexpandedas
M
2A(z)g(n)(z)=m f(n)(z)+k(n)(z) , n Z , (35)
′ n >0
− ∈
which plays an essen(cid:0)tial role in the(cid:1) proof of gauge- ¥ ¥
independenceof one graviton exchangeamplitude in the Rx x m (x,z)= (cid:229) x m(n)(x)f(n)(z), x 5(x,z)= (cid:229) x 5(n)(x)g(n)(z).
gauge;seeSectionIV.Nowweintroducethefollowing2 2 n=0 n=1
matrixoperators × (39)
Ifwemoveontothecoordinateframebychoosing
(¶ +3A)¶ 0
H= − z 0 ′ z ¶ (¶ +3A) ,
z z ′
(cid:20) − (cid:21)
Q1= ¶0z −(¶ z+0 3A′) , (−1)F = 10 01 , (36a) x 5(n)= 2m1nf (n), x m(n)= m1nh(mn5)−2m12n¶ m f (n), n∈Z>0,
(cid:20) (cid:21) (cid:20) − (cid:21) (40)
(¶ +2A)(¶ +A) 0
H¯ = − z 0′ z ′ (¶ +A)(¶ +2A) ,
z ′ z ′
(cid:20) − (cid:21)
Q¯1= ¶ z+0A′ −(¶ z+0 2A′) , (−1)F¯ = 10 01 , (36b) f (n) andh(mn5)withn>0areallgaugedaway,leavingonly
(cid:20) (cid:21) (cid:20) − (cid:21)
whichsatisfythetwoN =2superalgebras
1
Qa,Qb =2d abH, [H,Qa]=[H,( 1)F]=0, hˆ(mnn)=h(mnn) ¶ m hn(n)+¶ n h(mn)
{ } − −mn
{Q(¯−,1Q)¯F,Q=a}2=d 0H,¯, [H¯,Q¯ ]=[H¯,( 1)F¯]=0, (37a) +21 h mn (cid:0)+2¶mm ¶2n f (n),(cid:1) n∈Z>0, (41a)
{ a b} ab a − (cid:18) n (cid:19)
( 1)F¯,Q¯a =0, a,b=1,2, (37b) hˆ(mn0)=h(mn0), fˆ(0)=f (0), (41b)
{ − }
7
sothatthequadraticactionbecomes6 notstraightforwardtofindsuchagaugeduetotheappearance
ofthreeundesirablequadraticmixingterms.
¥
S(2)=MP2lZ d4x(n(cid:229)=014hˆ(n)mn Kmn(n);rs hˆ(n)rs +38fˆ(0)(cid:3)fˆ(0)), tioOnusrimstirlaatregtoytthoefiRnxdgoauutgtheeinapspproonptrainaeteougsaluygbe-rfioxkeinnggfauungce-
theoriesisbasedontheobservations:
(43)
(1) In 4d general relativity the so-called harmonic gauge,
whereM isthe(5daveraged)4dPlanckmassscaledefined which is the counterpart of the Lorentz gauge in ordinary
Pl
as unbroken gauge theories, is just the linearized version of
¶ m (√ ggmn )=0. Therefore we could expect that the har-
M2 =M3 zcdze3A= M3(1 e 2p kR), (44) monic−gauge conditions for the warped Randall-Sundrum
Pl 0 2k − − metricwould also be givenbysomethinglike the conditions
Z
¶ (√ GGMN)=0tolinearorderofh¯ .
whichreducestoMP2l−L−→−→0 p RM3asitshould,and M(2)−In order to fix the gauge arbitraMriNness x M we have to
impose five independentconditions. The gauge-fixingfunc-
1
Kmn(n);rs =+−h2m(nh ¶mrr ¶¶sn ¶+s h+rsh ¶msm ¶¶nn ¶ r +h nr ¶ m ¶ s +h ns ¶ m ¶ r ) stthiaoamnteffoobrroxxumndmmauruyssttcbboeenedexixtpipoaannnddteoeddxbmbyy,thi.eeg.o(nr){t(hzfo)(nn)o¥(rzm)}.a¥nl=As0es,twwwehitehsrhetahalesl
1 5 { }n=1
+2(h mr h ns +h ms h nr −2h mn h rs )((cid:3)−m2n). saedeapimtabmleedtioatetlhye,tghaisugise-afikxeinygobfusnercvtiaotnionfrtoomsetlheectltihneeatreirzmeds
(45) ¶ (√ GGMN).
M
−
Notice thatthe gravitonmass appearsas the so-calledFierz- Inwhatfollowswefirstinvestigatetheharmonicgauge(or
Pauli[14]form,whichistheonlyformthatdoesnotintroduce ’t Hooft-Feynman gauge from the four-dimensional sponta-
ghosts[18]. Thefour-dimensionalfieldcontentsarenowob- neouslybrokenpointofview)forthewarpedgravityandthen
vious: the massless graviton h(mn0), the infinite tower of mas- establish the one-parameter family of gauge choices analo-
goustotheRx gaugeinspontaneouslybrokengaugetheories.
(n) ¥
sive gravitons hmn with increasing masses {mn}n=1 and the The analysis presented below is based on the knowledge of
massless scalar boson f (0) known as the radion. It is worth gauge-fixingin4dgeneralrelativity,whichisbrieflysumma-
mentioningatthispointthatM isnotdirectlyrelatedtothe rizedinAppendixAtogetherwithabriefreviewofthevDVZ-
Pl
NewtonconstantG ;M isjustintroducedasthecoefficient discontinuity.
N pl
of the massless graviton Lagrangian, whereas G has to be
N
determinedasthecoefficientoftheinversesquarelawpartof ’t Hooft-Feynman Gauge: As in 4d general relativity, we
thestaticgravitationalforce,whichnowincludesthemassless willfirstcalculate¶ M √ GGMN tolinearorder:
−
radion’scontributionaswellasthemasslessgraviton’sone.
(cid:0) (cid:1)
1
¶ M √ GGMn =e3A ¶ m hmn + ¶ n h ¶ z+3A′ hn 5
− − 2 −
A. Gauge-FixedActiontoQuadraticOrder (cid:16) (cid:17) +O(cid:20)(h¯2), (cid:0) (cid:1)(46(cid:21)a)
3 1
Oncefixedtheactiontotheunitarygauge,allthefictitious ¶ M √ GGM5 =e3A 3A′+ A′(h f )+ ¶ zh ¶ m hm 5
massive vector- and scalar-modes would disappear from the − 2 − 2 −
(cid:16) (cid:17) (cid:20)
dimensionalreducedactionsothatwecouldnotseethecan- 3
celation mechanism of the would-be NG bosons which ap- −2 ¶ z+2A′ f +O(h¯2), (46b)
(cid:21)
pear in the other gauges, nor the expected SUSY-like sym- (cid:0) (cid:1)
metries between the unphysicaldegreesof freedom with the where h=h mn hmn . First look at the right hand side (RHS)
samegauge-dependentmass-squared.Inviewofthiswehave of Eq.(46a). The three terms in the brace can be expanded
toestablishtheone-parameterfamilyofgaugechoicesanalo- inonly f(n)-modessothatitseemstobeadoptedasagauge-
gousto the Rx gaugein the previoussection. However, it is fixingterm. NextlookatRHSofEq.(46b). Sincewearenow
interestedingauge-fixingforquadraticpartoftheaction,the
firstterm3A shouldbeignored,becauseitgives0thand1st
′
order of the fields. The last three terms in the brace can be
6Theeigenfunctionsarenormalizedas expanded in g(n)-modes, however, the second term can not.
M3 zcdze3Af(n)(z)f(m)(z)=MP2ldnm, Tfohrugsa,ufgoer-tfihxeinpgreasnendte,xwaemwiniellwadhoetphteornitlywtohrekslawstetlhlroerentoetr.ms
Z0
M3 zcdze3Ag(n)(z)g(m)(z)=MP2ldnm, Ourmodifiedharmonicgaugeconditionsaretherefore
Z0
M3Z0zcdze3Ak(n)(z)k(m)(z)=MP2ldnm. Fm [h]=−¶ l hlm +21¶ m h− ¶ z+3A′ hm 5=0, (47a)
(42) F5[h]= 1¶ zh ¶ m hm 5 3 (cid:0)¶ z+2A′(cid:1)f =0. (47b)
2 − −2
(cid:0) (cid:1)
8
Letusfirstexaminehowtoimplementthesegauge-fixingcon- ThesearetheanalogueofEq.(A8). Noticethatwhenoneim-
ditions. To see this, it is sufficient to calculate d Fm [h] and pose the gauge-fixing conditions (47a) and (47b), however,
d F [h]. Using the transformation law of the fluctuations at there still remains gauge freedom generated by on-shell pa-
5
KK-modelevelweget rameters x m(n) and x 5(n) which satisfy ((cid:3)−m2n)x m(n) =0 and
¥ ((cid:3) m2)x (n) =0. This is also the analogue of residual on-
d Fm [h]= (cid:229) (cid:3)−m2n x m(n) f(n), (48a) shel−lganuge5freedomin4dgeneralrelativity.
n=0
¥ h(cid:0) (cid:1) i
d F [h]= (cid:229) (cid:3) m2 x (n) g(n). (48b)
5 − n 5 Addingtotheaction(26)agauge-fixingterm
n=1
h(cid:0) (cid:1) i
Itisobviousthatonecanalwaysimposetheconditions(47a)
and(47b)bychoosingx Masoneofsolutionstothefollowing
differentialequationsforeachn:
((cid:3)−m2n)x n(n)=¶ m h(mnn)−12¶ n h(n)−mnhn(n5), n∈Z≥0, LGxF=1=M3e3A(cid:26)−12 Fm [h] 2−12 F5[h] 2(cid:27), (50)
(49a) (cid:0) (cid:1) (cid:0) (cid:1)
1 3
((cid:3) m2)x (n)= m h(n)+¶ m h(n) m f (n), n Z .
− n 5 −2 n m 5−2 n ∈ >0
(49b)
weget
Sx(2=)1=M3 d4x 0zcdze3A 14hmn 21(h mr h ns +h ms h nr −h mn h rs ) (cid:3)+(¶ z+3A′)¶ z hrs .
Z Z (cid:26) (cid:20) (cid:21)
1 3(cid:0) (cid:1)
+ hm 5 h mn (cid:3)+¶ z(¶ z+3A′) hn 5+ f (cid:3)+(¶ z+A′)(¶ z+2A′) f . (51)
2 8
(cid:27)
(cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:3)
Thedimensionalreducedactionis
¥
Sx(2=)1=MP2l d4x (cid:229) 41hmn (n) 12(h mr h ns +h ms h nr −h mn h rs )((cid:3)−m2n) hrs (n)
Z (cid:26)n=0 (cid:20) (cid:21)
¥ ¥
+(cid:229) 21hm 5(n)[h mn ((cid:3)−m2n)]hn 5(n)+(cid:229) 83f (n)[(cid:3)−m2n]f (n) . (52)
n=1 n=0 (cid:27)
It should be emphasized the first line suggests that the residue of the graviton propagatoris common to any n Z so that
0
onemayexpectinthe’tHooft-FeynmangaugeitmaybepossibletoevadethevDVZdiscontinuity,whicharises∈inth≥eunitary
gauge.Forfurtherdiscussion,seeSectionIV.
RRRx Gauge:SofarwehaveusedthenotationLGxF=1andSx(2=)1, ofx : a/x =b/x =(ab)/x =1. Butobviouslyitisnotpos-
whichimpliesthat,asingaugetheories,our’tHooft-Feynman sible. At first glance, it seems to be impossible to eliminate
gaugejustcorrespondstothecasex =1insomewhatunder- thethreeundesirablemixingtermsbya singlegaugeparam-
lyingRx gauge,however,nomentionhasbeenmadeofhow eterx . However,we canconstructthe one-parameterfamily
we haveinsertedthe gaugeparameterx into the action. Ac- ofgaugechoicesanalogoustotheRx gaugeinspontaneously
tually,itishighlynon-trivialtoinsertx intothegauge-fixing broken gauge theories with the help of the boundary condi-
Lagrangian(50). Toseethedifficulty,considerthearithmetic tions(32).
1 (x+ay+bz)2 Oneofthecrucialkeysisthatwehavetwosquaredbrack-
2x ets in (50), Fm [h] 2 and F5[h] 2, the former fixes the four-
= 21x (x2+b2y2+c2z2+2axy+2bxz+2abyz). (53) ddiimmeennssiioonnaall(cid:0)ognaeugxe(cid:1)5.arSbiintrca(cid:0)eritnheesss(cid:1)cxamlaranfideldthef loanttleyratphpeeeaxrstrian-
thelatterbracket,wehavetoinsertx asacoefficientoff to
Since we already know the ’t Hooft-Feynman gauge, which canceloutbytheoverallfactor1/(2x ). Thenthecrossterm
correspondstox =a=b=1,wenowwanttochoosetheco- betweenhandhm 5 inthelatterbracketcometodependonx .
efficients a and b to make the three cross terms independent However,wecanmakeallthecrosstermsindependentofx by
9
insertingitintotheformerbracketasfollowingcombination: Uponintegratingbyparts,onecaneasilycheckthatthecross
termbetweenhandhm 5 isjustthesameas(50)withthehelp
LGF=M3e3A −12 FmRx [h] 2−12 F5Rx [h] 2 , (54) onfattehsethbeouunnddeasryiracbolnedmitiioxninsg(t3e2r)m.sT. hus (54) correctly elimi-
(cid:26) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
with
FmRx [h]=−¶ l hlm +21 2−x1 ¶ m h−x (¶ z+3A′)hm 5,
(cid:18) (cid:19)
(55a)
F5Rx [h]= 12¶ zh−¶ m hm 5−32x (¶ z+2A′)f . (55b)
Gauge-FixedActiontoQuadraticOrder:Addingtotheaction(26)agauge-fixingterm(54)weget
SR(2x)=M3 d4x zcdze3A 41hmn KmnRx ;rs hrs +21hm 5KmR;xr hr 5+38f KfR;xf f , (56)
0
Z Z (cid:26) (cid:27)
where
KmnRx ;rs =−21 1−x1 (h mr ¶ n ¶ s +h ms ¶ n ¶ r +h nr ¶ m ¶ s +h ns ¶ m ¶ r )
(cid:18) (cid:19)
1 2 1 2
+ 1−x h mn ¶ r ¶ s + 1−x h rs ¶ m ¶ n
(cid:18) (cid:19) (cid:18) (cid:19)
1 1 1 2
+2 h mr h ns +h ms h nr −2 1−2x (cid:18)2−x (cid:19) !h mn h rs !(cid:3)
1 1
+2 h mr h ns +h ms h nr − 2−x h mn h rs (¶ z+3A′)¶ z, (57a)
(cid:18) (cid:18) (cid:19) (cid:19)
KmR;xr =h mr [(cid:3)+x¶ z(¶ z+3A′)]− 1−x1 ¶ m ¶ r , (57b)
(cid:18) (cid:19)
KRx =(cid:3)+(3x 2)(¶ +A)(¶ +2A). (57c)
f ;f − z ′ z ′
Now it is straightforward to compute the propagatorsin the intothefollowingpolarizationstates
Rx gauge. The results are summarized in Appendix B. As
in a massive gauge boson propagator of spontaneously bro- hmn =h(mnTT)+¶ m h(nT)+¶ n h(mT)+2¶ m ¶ n h(L)+h mn h(tr),
ken gauge theories, the massive graviton propagator can be
(58a)
writtenintothesumofthegauge-independentand-dependent
part separately. The gauge-independentpart corresponds to hm 5=h(mT5)+¶ m h(5L), (58b)
thetransverse-tracelesspartofhmn , asitshould. Thegauge-
dependentpartcanfurtherbedecomposedintothesumofthe where h(mnTT) is the transverse-traceless mode (5 DOF), h(mT)
three independent parts; the transverse gauge freedom part
the transverse gauge freedom(3 DOF), h(L) the longitudinal
D (T), the longitudinalgauge freedom part D (L) and the trace
partD (tr). D (T) hasthesamesimplepolepositiontothetrans- gaugefreedom(1DOF),h(tr) the“trace”piece(1DOF),h(mT)
verse part of the vector propagator. D (L) and D (tr) have the thetransversemode(3DOF)andh(L) thelongitudinalmode
5
samesimplepolepositionstothelongitudinalpartofthevec- (1DOF). Notice thatthese decompositionsare justified only
torpropagatorandthescalarpropagator,respectively.Inview for the case that hmn and hm 5 are both massive. Indeed the
of thisfact we expectthat asin the case of the pureAbelian masslesstensordecompositionaremorecomplicated;seefor
gaugetheory,thesespuriousdegreesoffreedomhaveitspart- example Appendix of [19]. However, since we are now in-
nerswiththesamemass-squared,andconsistofmultipletsun- terested in the gauge-dependentmass terms of the would-be
derSUSY-liketransformationsgeneratedbythesupercharges. NG bosons, we do not need the massless tensor and vector
To this end we decompose the graviton and vector fields decompositions.
10
B. ResidualSymmetries
Insertingthedecompositions(58a),(58b)into(56)weget
SR(2x)=M3 d4x 0zcdze3A 21hˆ(TT)mn (cid:3)+(¶ z+3A′)¶ z hˆ(mnTT)
Z Z (cid:26)
+12t(cid:20)hˆhˆ((TT))mm5(cid:21)(cid:20)−01 01(cid:21)(cid:18)(cid:3)−x(cid:2)(cid:20)−(¶ z+03A′)¶ z (cid:3)−¶ z(¶ z0+3A′)(cid:21)(cid:19)"hhˆˆ((mmTT5))#
1t hˆ(L) 1 0 (¶ +3A)¶ 0 hˆ(L)
+2 "hˆ(5L)#(cid:20)0 −1(cid:21)(cid:18)(cid:3)−x 2(cid:20)− z 0 ′ z −¶ z(¶ z+3A′)(cid:21)(cid:19)"hˆ(5L)#
1t hˆ(tr) 1 0 (¶ +3A)¶ 0 hˆ(tr)
+2 fˆ −0 1 (cid:3)−(3x −2) − z 0 ′ z (¶ z+A′)(¶ z+2A′) fˆ , (59)
(cid:20) (cid:21)(cid:20) (cid:21)(cid:18) (cid:20) − (cid:21)(cid:19)(cid:20) (cid:21)(cid:27)
where7 troducethefollowing2 2matrixoperators
×
hˆ(mnTT):= √12h(mnTT), hˆ(mT):=s(cid:3)x h(mT), hˆ(mT5):=h(mT5), Hˆ =(cid:20)−(¶ z+03A′)¶ z −(¶ z+A′0)(¶ z+2A′)(cid:21),
0 (¶ +3A)(¶ +2A)
hˆ(L):= 1 (cid:3)h(L) (3x 2)h(tr) , hˆ(L):= (cid:3)h(L), Qˆ1= (¶ z+A′)¶ z − z 0′ z ′ ,
x 3 − − 5 sx 5 (cid:20)− (cid:21)
(cid:0) (cid:1) ( 1)Fˆ = 1 0 , (62)
hˆ(tr):=√p3h(tr), fˆ := √3f . (60) − (cid:20)0 −1(cid:21)
2
it is easy to check that these operators satisfy the so-called
Thisisourfinalexpressionforthequadraticaction. Wehave higher-derivativesupersymmetryalgebra(HSUSYalgebra):
separatedtheactiontothegauge-independentand-dependent
part,justasdoneinthepureAbeliangaugetheory. Itisnow Qˆ ,Qˆ =2d Hˆ2, [Hˆ,Qˆ ]=[Hˆ,( 1)Fˆ]=0,
a b ab a
obviousthat(59)is invariantunderthefollowingthreeinde- { } −
pendentglobaltransformations ( 1)Fˆ,Qˆa =0, a,b=1,2, (63)
{ − }
d "hhˆˆ((mmTT5))#=q 1(cid:20)¶0z −(¶ z+0 3A′)(cid:21)"hhˆˆ((mmTT5))#, (61a) pwdeihfrfeaerlrgeeenQbˆtr2far=odmisi(ct−uhse1s)seFtdˆaQnˆi1dn.a[r2dT0sh,uis2p1ei,rsa2lag2ne,be2rx3at,e(n32d74ea,d)2,5(n3,o7n2bl6i)]n,.ethaAersssuuis--
hˆ(L) 0 (¶ +3A) hˆ(L) perchargesarenowthesecondorderdifferentialoperators,as
d "hˆ(5L)#=q 2(cid:20)¶ z − z0 ′ (cid:21)"hˆ(5L)#, (61b) tthheeyreshfaocutlodrifzraotmionthoefmthenetiHonamabilotovnei.aTnhfeorpothinetvisecotnocre-magoadien,
d (cid:20)hˆf(ˆtr)(cid:21)=q 3(cid:20)−(¶ z+0A′)¶ z −(¶ z+3A0′)(¶ z+2A′)(cid:21)(cid:20)hˆ(f(ˆ6tr)1(cid:21)c.) b−a¶c(z2k()g¶rzTo+huen3rdeAs′io)sl=untoi−orne(¶sAizd′′+u=a2l(AsA′y)′m)(2¶mz(e+extrcAye′p)g,teownneirtthahteethdbeobuhyneQdl¯pa1r.oiefOs)tnh.ee
may wonder about the absence of the symmetry generated
As expected, the transformations(61a) and (61b) are gener- by Q¯1. However, it is reasonable from the unitarity point of
atedbysuperchargeQ in(36a). Thetransformation(61c)is view. In ordinary spontaneously broken gauge theories, the
1
not, however,expressedby a first-orderdifferentialoperator. x -dependentpartofthegaugebosonpropagatormusthaveas
Thisisbecause(61c)isthesymmetrytransformationbetween itspartnersthewould-beNGbosonsandtheFaddeev-Popov
(the unphysical component of) hmn and f , which should be ghostswiththesamex -dependentpolesoastocanceloutthe
generatedbyproductoftwosupercharges. x -dependenceofanyphysicalamplitudes,andthenmaintain
Severalcommentsareinorderatthisstage: theunitarityofthetheory. Inthiscase,sincewedonotneed
(1) It can be regarded that (61c) is the so-called higher- theFaddeev-Popovghosts, theunphysicalpolarizationstates
derivative supersymmetric transformation. Indeed, if we in- ofthemassivegravitonfield(5DOF)musthaveasitspartners
thewould-beNGvector(4DOF)andscalar(1DOF)bosons.
Thus,wedonothaveanydegreesoffreedomtorelatebythe
symmetrytransformationgeneratedbyQ¯ .
1
7Onceagain,thesedefinitions(60)arewell-definedonlyintheregioninside (3) In special cases x =1,2, the four polarization states
(outside)thelight-coneandx >0(x <0.) in(61b)and(61c)arehappenedtobeindegeneratewiththe