Spontnous vs Explicit Lorntz Violtion nd Grvity Robrt Bluhm Colby Collg Third Summr School on th Lorntz- nd CPT-Violting Stndrd-Modl Extnsion, Indin Univrsity, Jun 208
Min gol of my tlk... è xmin issus tht ris whn th SME includs grvity.g., QED sctor of SME in Minkowski spctim L SME L 0 b 5 2 H ic D id 5 D SME cofficints ct s fixd bckground filds, b, H, c, d,... In curvd spctim, th SME cofficints brk both diffomorphisms nd locl Lorntz invrinc è symmtry brking cn b spontnous or xplicit Wnt to undrstnd th diffrncs btwn ths two typs of symmtry brking in th prsnc of grvity
Outlin I. Diffomorphism & Lorntz symmtry in GR II. Spctim symmtry brking è bckground filds nd grvity è spontnous & xplicit symmtry brking è consistncy issus with xplicit brking III. Excittions tht occur with symmtry brking IV. Exmpls of grvity modls with Lorntz brking è Bumblb modl (spontnous brking) è Mssiv grvity (xplicit brking) V. Conclusions
I. Diffomorphism & Lorntz symmtry in GR In GR, Einstin s qs cn b obtind from n ction 2 3 whr th mtric Z is th dynmicl grvittionl fild.g., considr thory with vctor mttr fild S Z d 4 x p p 2 4 g Einstin-Hilbrt trm 2 4 4 L 6 G R L M(g,A ) mtric dynmicl vctor fild Vrying with rspct to th mtric fild givs 5 3 5 G 8 GT è Einstin qs G R 2 Rg è Einstin tnsor
Th nrgy-momntum tnsor is dfind gnriclly s T 2 p g ( p gl M ) It dpnds on th mttr filds, which lso hv qs of motion,.g., vry with rspct to th vctor fild g L M @L M D A @D A @L M @A 0 è Eulr-Lgrng qs for mttr D A @ A A è covrint drivtivs ffin connction
In GR, th Einstin qs nd th mttr qs of motion r not ll dynmiclly indpndnt On th on hnd, sinc th Einstin tnsor obys gomtric idntity D G 0 è contrctd Binchi idntity Consistncy with G 8 GT D T 0 rquirs tht è covrint nrgy-mom. consrvtion must hold Howvr, thr r lso idntitis tht dirctly link th Einstin qs nd mttr qutions of motion. è du to locl spctim symmtry in GR Diffomorphism invrinc
Diffomorphisms è mp th spctim mnifold to itslf x x è lvs th ction invrint 4 infinitsiml trnsltions Undr diffomorphisms th coordint systm stys fixd but th dynmicl filds trnsform è chngs r givn by Li drivtivs,.g., g g L g L g D D A A L A L A (D )A D A Th ction in GR is invrint undr ths trnsfs ( S) di s 0 è lik gug symmtry
Cn driv n idntity from diffomorphism invrinc,.g., for vctor mttr fild ( S) di s Z d 4 x p " g " D ( 8 G G T ) L M D A A With ( S) di s 0 for ll w gt tht D ( 8 G G T ) L M D A A D ( L M A A )0 D ( L # M A ) A è 4 off-shll (Nothr) idntitis showing tht whn L M 0 & using th Binchi idntitis D G 0 A it utomticlly follows tht D T 0
This is n importnt rsult sying whn diffomorphism invrinc holds th 4 qutions D T 0 utomticlly hold whn th mttr filds r on shll L M 0 A è th mtric dos not nd to ply rol è it s th mttr filds qs. tht ssur nrgymomntum consrvtion in GR In fct 4 mtric mods cn b gugd wy s rsult of th diffomorphism invrinc
Do not confus diffomorphisms with infinitsiml gnrl coordint trnsformtions (GCTs) è though it s sy to do so s w cn s....g., considr chng of coordints x x 0 (x) A 0 (x 0 ) @x @x 0 A (x) è but th vctor stys fixd x 0 x using opposit sign è plug in nd xpnd Mthmticlly find tht th functionl chng of th vctor fild (t sm x) is givn by th Li drivtivs 0A (x) A 0 (x) A (x) L A è th ction dos not chng undr GCTs ( S) GCTs 0
Diffomorphisms r prticl (ctiv) trnsformtions whil infinitsiml GCTs r obsrvr (pssiv) trnsfs è It mns somthing to sy diffomorphisms r symmtry in GR ( S) di s 0 è.g., it ruls out hving fixd bckground filds in GR sinc thy xplicitly brk diffomorphisms At th sm tim th ction hs to b invrint undr GCTs to b obsrvr indpndnt (physiclly vibl) ( S) GCTs 0 This is mthmticl sttmnt bout covrinc nd coordint indpndnc tht must lwys hold
Intrstingly, cn still driv Nothr idntitis from ( S) GCTs 0 è mthmticlly gt th sm idntitis btwn th mttr qs of motion nd nrgy-mom. consrvtion L M A 0 which must hold du to obsrvr indpndnc D T 0 Oftn viwd s rsult of th fct tht th choic of coordints hs no physicl consquncs In GR, hv both ( S) di s 0 nd ( S) GCTs 0 Gnrl covrinc nd diffomorphism invrinc go hnd in hnd in GR with no incomptibility
In contrst, th SME hs fixd bckground filds,.g.,, b, H, c, d,... SME coffs è putting thm dirctly into curvd spctim will brk diffomorphism invrinc ( S) di s 60 è yt w still must hv coordint indpndnc ( S) GCTs 0 Must b crful tking th SME to curvd spctim è diffomorphism brking & gnrl covrinc r potntilly in conflict with ch othr This potntil conflict is somthing Aln worrid bout whn dvloping th grvity sctor of th SME
Still hv two mor importnt qustions bout GR è whr is th Lorntz symmtry in GR? è how do w includ frmions in GR? In GR, Lorntz symmtry is locl symmtry tht holds in infinitsiml locl inrtil frms whr g b è locl Lorntz frms Vctors nd tnsors hv bsis sts & componnts with rspct to ithr th spctim or locl Lorntz frms ~ ~ ~. Th filds rlting ths componnts r th virbin virbin.
r lso indics virbin risd nd lowrd using(4) gs whil thth L An invrs cn givs,grk whn using risd virbin b introducd. rwrittn g on th virbin b b,it g.th mtric using cn.th nd b indics r lowrd..b virbin nd invrs virb b bb b on th virbin r risd nd lo Grk indics s r risd nd lowrdth using b. Th virbin nd invrs virbin lso o b using risd th th virbin r risd nd lowrd g,, g whil Ltin indics r nd lowrd using Th virb For th mtric g. bgg.. b ntz symmtry using.b b b nd ndics on th th virbin virbin risd,, whil dics on r nd lowrd using whil b lowrd,. lowrd using b. Th virbin nd invrs virbin lso oby, b b, isd nd lowrd. Th virbin nd invrs virbin isd nd lowrd using virbin nd invrs virbin b b rbin cn lso b introducd. It givs th invrs mtric, b whr th invrs virbin obys Th Grk indics on th virbin r risd nd lowrd u mnt, both th di omorphism nd locl Lorntz trnsform,. (5) 3. Locl Lorntz symmtry b b lowrd. th indics rthory. risd nd Th virbin nd i symmtry ymmtris of th Undr di omorphism, virbin using Locl Lorntz b,. 3.. g. b Locl symmtry b b b Lorntz Using vctors, virbin trtmnt, both th di omorphism nd locl Lorntz spctim gntz virbin trtmnt, both th di omorphism nd locl Lorntz trnsfor symmtry,. trtmnt, bundr di omorphis whil th b tions (LLTs) bcom symmtris ofnd th thory. th di omorphism, ndics on th r lowrd using g, Using virbin both componnts (with trnsform Spctim virbin risd Grk indics) (LLTs) symmtris th di omorphism, trnsforms L s stof(@spctim vctors, @ Undr (6) th virb of ) bcom thory.. l Lorntz symmtry mnt, both th nd locl Lorntz tions bcom symmtris of th thory. lso Und isd nd using vctors,. Th virbin ndtrnsforminvrs virbin lforms Lorntz undr diffomorphisms,.g., b(llts) s lowrd st ofdi omorphism spctim Undrs virbin LLT,3. th virbin trnsforms Ls locl Lorntz vctor trnsforms st of spctim vctors, ymmtris of th thory. th Lorntz di omorphism, (@ ) @ Locl symmtry. both di omorphism.nd locl Lorntz tr bin trtmnt, th, L (@ ) @ bin trtmnt, both th di omorphism nd locl tr b spctim vctors,. Lorntz b b L (@ (x). (7) locl Lorntz v trnsforms di omorphism, Altrntivly, undr LLT, virbin s b th th bcom symmtris of thory. Undr th Using virbin trtmnt, both th di omorphism nd bcom symmtris of th thory. Undr di omorphism, th ntivly, undr LLT, th virbin trnsforms s locl Lorntz vctor L (@ ) @. (6) b whil locl Lorntz componnts (with Ltin indics) Altrntivly, undr trms LLT, th virbin trnsform s st of spctim vctors, Lorntz trnsformtion cn b writtn in of six locl (x). tions (LLTs) bcom symmtris of th thory. Undr di Lorntz st of spctim vctors, lllt, symmtry b b In cs,. infinitsiml locl trnsform trnsfs,.g. th trnsforms locl Lorntz vctor slorntz b (x) locl b b (x)trnsforms svirbin b undr. this n (x). of spctim trnsformtion cn in b locl b Lorntz s st vctors, b An infinitsiml b writtn trms L (@ ) @. Lorntz (@cn writtn bcoms L th b di omorphism virbin @. bin trtmnt, both nd locl trnsf nnfinitsiml of cofficints th Lorntz trnsformtion b in trms of six l ) locl (x). (7) (x) (x) s. In this cs, n infini b b b infinitsiml b locl b Lorntz b An infinitsiml trnsformtion cn L (@ ) cints (x) (x) s. In this cs, n infinitsiml bcom symmtris of th thory. Undr di omorphism, th vil b b bth s locl LLTs LLT, th virbin b b b bcoms Lorntz vct y, undr trnsforms Lorntz trnsformtion of virbin (x). (8) cofficints b (x)intrms b (x) locl b bvc. In trnsformtion cn b writtn of s six,s Lorntz undr virbin trnsforms s locl LLT, th b virbin b Lorntz ntz trnsformtion of th bcoms st of spctim vctors, Lorntz infinitsiml th trnsforms b locl bcoms trnsformtion of virbin this bvirbin Altrntivly, LLT, th s lo b (x) s b b undr cs, n (x). b. In b (x). b b b (x) dn out tcn spctim th cofficints b (x) formlism dpnd (x).virbin lso incorport frmions in ch bcoms point,. b of th virbin b L (@ ) @ (x). b. b (x) ToSinc mintin covrinc whn drivtivs scond r out point,b th ml loclllts trnsformtion cnr busd, writtn incofficints trms of Lorntz crrid b t ch spctim out t(x) (8) (x) dp LLTs r crrid ch. spctim point, th cofficints
(i @ m) 0 0 A (x) A 0 (x) Lby A (x)r A In spcil rltivity, frmions dscribd (i @ Lgrngin m) 0 Lfrmion i @ Lfrmion i @ @ (i Dirc q Lfrmion i @ m m m) 0 S( ) m i b 4 b p Undr 4 globl Lorntz trnsformtion B C S dx g R L(g i, f b, D LADL(,k.ABC id, k,..) b b 2 Z [, ] 4 b S( ) p A B 4 S d x g R L(g, f, D D D 2 C k ABC ( 2 b Z S S p b b b 3, C S S 4 A B d x Z S g R L(g, f, @ @ @ k ABC p spctim Z go B uscth virbin d4 xto g4 to L(g DGR, D A B k ABC ( )) prcurvd 2, f, D Ain C d x2 g R L(g, f, @ @ @ p k ABC ( ) Z S Lfrmion2 (i AD B m C) 4 p ( ))g dx g R L(g, f, D D 3 D k ABC Z 3 2 è Lorntz symmtry bcoms locl symmtry Ap A drivtiv A of covrint with nw typ Z @ < (k ) p F g 0 4 A B C dx g R L(g, f, @ @ @ k ABC ( )) 2 32
Covrint drivtivs now involv spin connction @ D @ i 4 b b, b b 24 componnts Acting on th virbin (ssuming mtric thory) givs D @ b b 0 This qution rlts th spin connction to th virbin, its drivtivs, nd th ffin connction Hv two possibilitis (two possibl gomtris) è spin connction is not indpndnt (Rimnn) è spin connction is indpndnt (Rimnn-Crtn)
In Rimnn-Crtn gomtry, thr s n dditionl gomtricl quntity clld th torsion T Einstin s GR hs zro torsion (Rimnn spctim) è But whn spin nd frmions r includd in grvity thory it is nturl to hv both curvtur nd torsion è Cn thn hv both propgting mtric xcittions nd propgting spin connction Thoris of grvity in Rimnn-Crtn spctim hv prllls with non-blin gug thoris è virbin & spin connction bcom gug filds ssocitd with diffos nd LLTs
T p k 8 G g g k k k formlism... To summriz so fr, GR in virbin S L f Hs 6 0 k totl of 0 symmtris (4 diffos nd 6 LLTs) S S 8 G S 6 0 thr 6 r 0 0 Nothr idntitis nd thrfor 6 0 k k S th mttr filds rk on D G tht 0.g., whn shll gt ( S) 0 k S 0 GCTs S For diffomorphisms 0 0 k k di inv ( S)di s 0 D G 0 S 0 ( k S)GCTs 0 D T 0 4 conditions ( S) 0 ( S) 0 di inv T T di s ( S) 0 di s di s L S)0di slorntz For invrinc 6 0 f (locl D T 0 T T G G ( S)LLTs 0 ( S)LLTs 0 ( S)di s 6 0 L ( S)GCTs 0 6 conditions y x (G T )( b (X ) f b ) [b] x y 2 f ( S) 0 G G GR(isS)lso mthmticlly obsrvr ( 0S)di spssiv 6 0 LLTs invrint LLTs 6 ( S)GCTs undr 0 0 GCTs nd chngs of Lorntz bss pssiv G( S) G LLTs Z L L ( S)pssiv 4 p 0 y x LLTs S) d x g L k 6 0 (T(S) 6 0 (G S )( ) (X ) f 0. ( S) 0 di s di s ( S) 0 b b [b] GCTs LLTs x 0 y k 2 fpssiv Z L 4 p 3 2
II. Spctim symmtry brking To dvlop th grvity sctor of th SME, w nd to includ th SME cofficints s bckground filds, b, H, c, d,... è this brks diffomorphisms nd LLI To considr ths in gnrl wy, lt s dnot k è gnric bckground fild è hs fixd spctim dirctions First, must look t how bckground filds trnsform undr prticl & obsrvr trnsformtions
Undr prticl diffomorphisms & LLTs: è bckground filds sty fixd k prticl k. è likwis in locl Lorntz frms k bc prticl k bc Must thrfor hv bckground virbins: ē ē prticl ē Th bckground virbins oby: k ē ē b ē c k bc, è whr ll sty fixd undr prticl trnsfs
Howvr, undr obsrvr GCTs & pssiv LLTs: è spctim componnts trnsform undr GCTs k obsrvr k L k è locl Lorntz frm coordints trnsform k bc obsrvr k bc p k pbc p b k pc Bckground virbin componnts lso trnsform: ē ē obsrvr ē (D )ē obsrvr ē bē b. (GCTs) (LLTs) Undr ths trnsformtions th ction is unchngd
In grvity thoris with fixd bckground filds, it s th prticl symmtris tht r brokn è prticl diffs & LLI r both brokn ( S) di s 60 ( S) LLTs 60 è but obsrvr GCTs nd LLTs rmin mthmticl invrincs of th ction ( S) GCTs 0 ( S) pssiv LLTs 0 è rquirmnt of obsrvr indpndnc mthmticl trnsfs è diffomorphism brking & gnrl coordint invrinc r potntilly in conflict è likwis with brking of prticl LLTs
Spontnous vs xplicit brking Th ffcts of bckground filds in grvity dpnd on whthr th symmtry brking is spontnous or xplicit è With xplicit ppl brking th bckground tnsor is nondynmicl nd origints in th Lgrngin k, ē è nondynmicl è No xcittions of th bckground fild L L(, k, ē,...) è No qutions of motion for th bckgrounds L L 6 0 k ē 6 0 GR dos not prmit nondynmicl bckgrounds thy r objcts with prior gomtry
è With spontnous brking th bckground tnsor origints s dynmicl vcuum xpcttion vlu k hk i ē h i è Obys vcuum qutions of motion L k ppl vcuum 0. è Excittions bout th vcuum occur k k k è vcuum solutions of dynmicl filds L ē vcuum 0 dynmicl bckground xcittions L L(,k k k )
With spontnous symmtry brking must ccount for th xcittions bout th vcuum solutions Th xcittions k occur s è msslss NG mods è mssiv Higgs-lik xcittions Whn th xcittions r includd th ction rmins invrint undr both diffs & LLTs ( S) di s 0 ( S) LLTs 0 è Bcus of this th potntil inconsistncy with th Binchi idntity nd dynmics is vdd D T 0 nd T T occur nturlly s in GR è SSB mintins dsirbl fturs of GR
It is for this rson tht th grvity sctor of th SME usully ssums th symmtry brking is spontnous S SME,frmion Z d 4 x 2 i $ D M M m b 5 2 H b b c b b d b 5 b è th SME cofficints r typiclly ssumd to origint s vcuum xpcttion vlus, b, H, c, d,... è vvs Must thn ccount for th NG nd mssiv mods è voids inconsistncy issus with xplicit brking
D A Z @ A A Z p L 4 p ( S)di s d G x g T L k 4 6 0 S dx g R L(g, f, k 2 Howvr, with xplicit 2diffomorphism brking 3 @ f pf Z Z D f p p 4... 6 2 ( gl)zl things r diffrnt 7 4 R 4 5 0 (S S) d gx g (D T ) L k GCTs x T L(g, f, k ) Gd 4 p T p k 2 gg R L(g g gs d x 8 GT, k Considr n ffctiv Lgrngin with bckground 2 p G T ( gl) Z 2 gx x b for simplicity p T p 4 S d gx g R L(g, k ) 8 G 2 3 2 8 G Z p f 0 L 6 7 L 4 Z ( S)GCTs d x g 4(DT ) p L k 5 f 0 4 gk R L(grquirs, k ). ind. è undr GCTs x Sx d xobsrvr, f D T 0 2 8 G L 2 @L @L 3 L Z p D, 6 L f @L 07 4 L @L f @D f @f 4 ( S)GCTs d x g (DG LDT ) @LD L k @L 5 0 k 0 0 D @x 0 Df @f @D f, A (x ) G 0 A (x) @D f @f @x f (G idntity L must vnish nondynmicl Don T0 shll 0 brokn sym L D G 0 f L G G 0 D T0 0 6 0. L k 6 0 f xg x (x) k 8 GT S0 k hk i. L D T 0 6 0 DT 0
ppl Howvr, with xplicit diffomorphism brking Z ppl things r diffrnt... Considr n ffctiv Lgrngin with bckground S Z è undr GCTs ( S) GCTs Z d 4 x p g d 4 x p g ppl 2 R L(g, k ) Z 2 6 4 x x 2 6 4(D G D T ) ppl for simplicity 8 G f 0 obsrvr ind. rquirs 3 7 5 L 7 k 5 0 k L è potntil inconsistncy btwn th Binchi idntity, dynmics, & covrint nrgy-momntum consrvtion 3
@ A But thr is loophol tht vds th 0 inconsistncy @ A è th intgrnd is totl drivtiv (D T ) (L k ) Z L ppl D k This llows th intgrl to vnish ( S) GCTs Z d 4 x p g 2 2 6 4 0 @ 6 4(D G D T ) L g g 3 7 5 A L 7 k 5 0 k L 3 vn with nonvnishing (nondynmicl) vrition nd brokn symmtry L k 6 0. L k 6 0 è s long s th totl drivtiv dos not vnish
6 In gnrl, thoris with xplicit diffomorphism brking cn b consistnt or inconsistnt dpnding on how th mtric tnsor coupls to th bckground fild In most css with sufficint intrctions cn hv vn whn ( S) GCTs 0 L 6 0 k nd D T 0 conflict is vdd Why, wht is hppning? g hs 4 xtr dgrs of frdom du to diffomorphism brking Extr mtric mods cn coupl to th bckground fild nd tk vlus tht stisfy th 4 conditions D T 0
Simplst xmpl tht is inconsistnt: tim-dpndnt cosmologicl constnt S Z d 4 x p g (x) thory is inconsistnt @ (x) 6 0 D T 60 è no coupling to mtric componnts But mor gnrlly, whn thr r sufficint couplings btwn th mtric nd th nondynmicl bckground, thn th inconsistncy is vdd. 4 xtr mtric dgrs of frdom (du to loss of diffomorphism brking) cn stisfy D T 0
To summriz so fr... W v sn tht potntil inconsistncy cn ris in grvity thoris with fixd bckground filds Dpnds on whthr diffomorphisms nd LLI r brokn spontnously or xplicitly è spontnous brking vds th inconsistncy (thory bhvs similrly to GR) è with xplicit brking thr is loophol tht usully llows th inconsistncy to b vdd Wnt to look t ths typs of brkings mor closly How r thy diffrnt? How r thy similr? è nd to look t th diffrnt typs of xcittions tht occur with spontnous vs xplicit brking
III. Excittions tht occur with symmtry brking Considr n ffctiv grvittionl fild thory with bckground fild nd convntionl mttr filds S Z d 4 x p ppl g 2 R L(g,f ppl, k ) Z ppl 8 G Eulr-Lgrng: dynmicl mttr filds L f D @L @D f @L @f è on shll th qs of motion hold L f 0 Wnt to considr possibl xcittions
With spontnous symmtry brking... è xcittions r msslss NG nd mssiv Higgs mods è th NG mods ppr s gug xcittions diffomorphism NG mods (up to 4) k k (D ) k (D ) k dynmicl D k ( k ) mssiv è NG xcittions mintin diffomorphism invrinc è with SSB th symmtry is hiddn, but still holds b Lorntz NG mods (up to 6) k bc ' k bc j k jbc j b k jc ( k bc ) mssiv
Whil with xplicit symmtry brking... k è nondynmicl, hs no xcittions Sinc diffs & LLI r xplicitly brokn, thr r xtr mtric nd virbin mods comprd to GR (would b gugd wy with unbrokn symmtry) ( S) di s 60 ( S) LLTs 60 è 4 xtr mtric mods è 6 xtr virbin mods Ths xtr mods hv qutions of motion tht cn stisfy th rquirmnts for consistncy D T 0 T T è 4 qutions è 6 qutions k è th nondynmicl bckground dos not hv qs of motion, but th xtr mods do
Stucklbrg Filds It is common with xplicit brking to introduc Stucklbrg filds, which rstor th brokn symmtry è givs n ltrntiv wy to s th xcittions To rstor diffomorphism brking, introduc 4 sclrs A è lt thm b dynmicl A 0,, 2, 3 Thy r dfind by substituting into th ction: S Z Z ppl k (x) D A D B D C k ABC ( ) d 4 x p g ppl 2 R L(g,f,D A D B D C k ABC ( )) Z ppl
Th Stucklbrg sclrs trnsform undr prticl diffs A A D A è so now thr s diff invrinc ( S) di s 0 Stucklbrg trick dds 4 filds, but rstors 4 locl syms, so th totl numbr of dgrs of frdom is unchngd Cn lwys pick (unitry) gug whr A A x which givs bck th bckground fild D A D B D C k ABC ( ) k (x) so th xplicit-brking thory nd th gug-fixd Stucklbrg thory Z r quivlnt ppl D A A
Look t infinitsiml xcittions in th Stucklbrg filds A A (x ) 4 infinitsiml xcittions Ths r NG xcittions bout A A x Substituting this into th Stucklbrg xprssion givs D A D B D C k ABC ( ) ' k (D ) k (D ) k D k Ths xcittions r lso NG mods
Compring this to th NG xcittions with SSB... k k (D ) k (D ) k dynmicl D k ( k ) mssiv è w s thy r th sm NG xcittions è but th mssiv xcittions r missing in th Stucklbrg pproch with xplicit brking ( k ) mssiv 0 using Stucklbrg Th Stucklbrg trick puts in th NG mods, which is th minimum ndd to rstor diff brking
With spontnous diff brking, th bckground k is th vv of dynmicl tnsor with n rbitrry numbr of componnts è Only if w pplfrz out ll th mssiv mods do w gt thory with only 4 NG mods But vn thn th thoris r diffrnt bcus whil L k vcuum L k 6 0 0. with SSB with xplicit brking nd Stucklbrg trick Stucklbrg trick involvs SSB for th sclrs but not for th originl bckground tnsor A k
Dos using th Stucklbrg trick gurnt tht th potntil inconsistncis ssocitd with xplicit symmtry brking r vdd? No A 60 must hv xcittions tht giv solutions for D T 0 è Thr is no gurnt tht ths solutions xist Th Stucklbrg pproch is diffrnt from thory with spontnous brking whr th bckground tnsor is vcuum solution. è Still th Stucklbrg nd NG xcittions r th sm è This llows th SME to b pplid with xplicit brking nd grvity (s long s th thory is consistnt)
IV. Exmpls of grvity modls with Lorntz brking Will look t two xmpls of grvity thoris with bckground filds tht brk spctim symmtris Bumblb modls (spontnous brking) B hb i b q è dynmicl vctor fild è vcuum vlu Mssiv grvity (xplicit brking) k k q è nondynmicl bckground (g k ) 2 è grviton mss trms g g
Exmpl: Bumblb Modls Grvity thoris with vctor fild nd potntil trm tht inducs spontnous Lorntz brking dynmicl vctor fild SSB potntil B V (B B ± b 2 ) vv hb i b ppl Not: BB modls do not hv locl U() gug invrinc (dstroyd by prsnc of th potntil V) Bumblbs: thorticlly cnnot fly (nd yt thy do) hv bn studid in vrious forms L 6 G R L B V (B B ± b 2 ) Z ppl kintic trms potntil
Kintic trms nd intrprttions: () vctor-tnsor thoris of grvity Will-Nordvdt kintic trms L B B B R 2 B B R 4 B B 2 2D B D B 2 3D B D B no mttr couplings (grvittionl intrctions only) xpct propgting ghost mods (2) modifid Einstin-Mxwll thoris Kostlcky-Smul (KS) modls L B 4 B B cn includ intrctions with mttr ± ± B D B D B no propgting ghost mods
Potntil Trms: Th potntil trm V (B B ± b 2 ) inducs spontnous Lorntz violtion ppl Th minimum with V 0 occurs whn B hb i b vcuum vlu with prfrrd spctim dirctions Mttr intrctions gnrt SME cofficints s vvs B b SME cofficint Z ppl è Illustrts how SME coffs cn ris from SSB Also hv xcittions of th vctor fild bout th vcuum
Excittions with V 0 V 0 60 è msslss mods è mssiv mods Exmpls of potntils: () Lgrng-multiplir potntil V (B B ± b 2 ) frzs out mssiv mod λ pprs s n xtr fild ppl (2) Smooth qudrtic potntil V 2 ppl(b B ± b 2 ) 2 llows mssiv (Higgs) fild Both ld ppl to spontnous Lorntz brking
In gug thory SSB hs wll known consquncs: () Goldston Thorm: whn globl continuous sym is spontnously brokn msslss Nmbu-Goldston (NG) mods ppr (2) Higgs mchnism: if th symmtry is locl th NG mods cn giv ris to mssiv gug-boson mods..g. W,Z bosons cquir mss (3) Higgs mods: dpnding on th shp of th potntil, dditionl mssiv mods cn ppr s wll.g. Higgs filds Cn ths occur with spontnous Lorntz violtion?
NG mods: Appr s virtul xcittions bout th vcuum solution gnrtd by th brokn symmtry trnsformtions msslss xcittions tht sty in th potntil minimum V 0 If NG mods xist, thy might possibly b: known msslss prticls (photons, grvitons) nonintrcting or uxiliry mods gugd into grvittionl sctor (modifid grvity) tn (Higgs mchnism) Dpnding on th choic of potntil nd kintic trms, Bumblb modls cn xhibit ths diffrnt fturs
KS Modls in Rimnn spc: (Mxwll kintic trm, no torsion) Ths dscrib modifid Einstin-Mxwll modls NG mods propgt lik msslss photons cn coupl to chrgd mttr filds sttic mssiv mod modifis Nwtonin potntils Einstin-Mxwll solution is lrg-mss limit Einstin-Athr modls: (Will-Nordvdt typ kintic, Lgrng-multiplir potntil) Ths giv modifid grvity thoris NG mods mix in s vctor-tnsor grvittionl filds no couplings to mttr filds modify grvittionl wvs
Higgs Mchnism: 2 typs of NG mods (Lorntz & diffs) potntilly 2 typs of Higgs mchnisms diffomorphism mods: cn Higgs mchnism occur for th diffos? dos th mtric (or virbin) cquir mss? (D B ) 2 ( b ) 2 connction dpnds on drivtivs of mtric no mss Z trm pplfor th mtric (or virbin) itslf No convntionl Higgs mchnism for th mtric (no mss trm gnrtd by covrint drivtivs)
Lorntz mods: go to locl frm (using virbin) b gug filds of Lorntz symmtry (D B ) 2 ( b b b ) 2 Gt qudrtic mss trms for th spin connction Z ppl suggsts Higgs mchnism is possibl for ω αβ only works with dynmicl torsion llowing propgtion of dynmiclly indpndnt ω αβ Lorntz Higgs mchnism for th spin connction cn occur but only in Rimnn-Crtn gomtris offrs nw possibilitis for modl building but finding vibl modls is chllnging
Summrizing, Bumblb modls offr numbr of scnrios dpnding on th typ of modl & th gomtry è KS modls giv Einstin-Mxwll solutions s rsult of SLV (not U() gug symmtry) è no Higgs mchnism for th mtric è cn hv Higgs mchnism in Rimnn-Crtn è mssiv (Higgs) mods cn ppr s wll Mor gnrlly find tht... Bumblb modls giv modifid grvity thoris with ffcts stmming from spontnous Lorntz brking SME couplings mrg nturlly s vv s no problm with consistncy issus
Exmpl: Mssiv Grvity Th qustion of whthr thr is consistnt thory of mssiv spin-2 grviton tht hs GR s its msslss limit hs bn round for ovr 75 yrs. 939 970 972 972 Firz & Puli find n pproximt (linrizd) thory for mssiv spin-2 prticls Vltmn, Vn Dm, & Zkhrov show tht th m 0 limit of th linr thory dos not gr with GR Vinshtin shows tht th xct nonlinr thory will gr with GR in th m 0 limit Boulwr & Dsr show th nonlinr Firz-Puli modls r unphysicl (contin ghost stts) Not until 200 did d Rhm, Gbdz, & Tolly (drgt) find mssiv grvity thory tht is ghost-fr
Cnnot form qudrtic mss trm using just th mtric g g Instd introduc bckground symmtric two-tnsor k k è nondynmicl bckground Cn thn mk mss trms è (g k ) 2 g k g k Mssiv spin-2 grviton hs 5 dgrs of frdom But most mss trms introduc ghost mod è ghost ppr s 6 th mod (Boulwr-Dsr ghost) è drgt found potntil for th mtric nd bckground tht limints th Boulwr-Dsr ghost
drgt Mssiv Grvity è Hs both mtric nd virbin formultions In th mtric formultion Th ction is dfind in trms of r squr-root mtrix q L MG 6 G q g k 0 @R r 4 m2 U( must obyè grviton mss g k ) q A g k It is not obvious tht th squr roots q xist è but sufficint conditions hv bn found
Th originl drgt ction usd Minkowski bckground è othr bckgrounds wr shown to b ghost fr too Th bckgrounds xplicitly brk diffomorphisms è thr r thrfor up to four xtr mtric mods, which must stisfy crtin consistncy contrints Th consistncy conditions hv th form (on shll) 2D k L MG k k L MG k D k 0 whr L MG k 60 nondynmicl è It s th xtr mtric mods tht must mk ths hold
Usully thr is nough coupling btwn th mtric nd th bckground so tht th 4 conditions cn hold è but thr is on notbl xcption If n nstz is md tht k Minkowski nd th mtric is sptilly flt FRW d 2 dt 2 2 (t) dx 2 dy 2 dz 2 thn th consistncy condition rducs to d (t) 0 dt è No flt FRW solution with Minkowski bckground ppl è no-go rsult But flt FRW solutions hv bn found using othr bckgrounds k 6
In th virbin formultion of drgt mssiv grvity è Th mtric hs nturl squr root g b b. è dynmicl virbin ppl Cn lso pick bckground virbin ppl obying k v v b b ppl nd dfin products s v è nondynmicl virbin v Cn writ th ction using -forms nd wdg products: dx v v dx forms
4 6 f S d x R dx 2 2 22 2 22 3 33 4 44 m m 2 3 4 m 0 0 0 is thn 2 322 433simply: 44 drgt ction using 4virbins 2 4 dx 44m 44 4 0 2 3 4Z 4 Z 2 3 4 S d4 x R m2 0 2 3 4 2 8 4 2Z22 3 p 4 44 33 44 2 4 3 4 2322 433 dx d x g R L 33 3 2 3 4 S 2 3 2 3 4 2 3 4 Z 333 4 44 22 2 0 2 3 4 2 2 4 2 3 4 p 3 Z 4 2 2 22 2 322 433 4p 2 Z 44 3 (2) 2(2) L(g (2) 2 34d S 4 x g R p, 2 S dx g R 24 (2) S2 d x g R 22 22 33 33 44 44 2 2 3 2 3 4 3 2 3 4 2 32 43 4 2 3 4 23 (2)33 3 2 3 4 3 2 3 4 3 2 3 4 Z 3 p 33 4 4 44 2 22 22 33 4 3 4R )4 ) L(g 4 4 2 3 4 m2 4 2 3 4 S d x g 2 3 3 0 4 232423 34 4 ) 4 4 4 2 3 4 d 2 2 34 4 3 4 4 2 3 4 )2 Lvi-Civit 2 symbol 4 3 Z 4ZZ p ) p 2 3 4 p Rcombintions Z 44 4consists AA AA 4xp g g A A) è Th ction of of d L(g, f,, D ghost-fr 2 3S S d x R L(g, f,, D ) 4 S d x g R L(g, f,, D A 2 4 A 3 2 3 4 S d4 x g 2 22R L(g, f,, Ddx ) d 2 3 products 4 th fiv indpndnt3wdg of nd 3 2 2 3 4 )
Mttr intrctions It hs bn shown tht drgt mssiv grvity rmins ghost-fr whn mttr filds coupl to n ffctiv virbin of th form dynmicl virbin q nondynmicl virbin or quivlntly to n ffctiv mtric 6 0 q v g 2 g 2 2 k, constnts è givs Lorntz brking mttr intrctions è would coupl to ll prticl sctors Cn mtch ths couplings to th SME to put bounds on th drgt Lorntz-brking mttr intrctions
V. Conclusions è Bckground filds in grvity brk diffs & LLI ithr spontnously or xplicitly è With xplicit brking, bckground is nondynmicl nd dos not stisfy Eulr-Lgrng qs è But with spontnous brking, th bckground is dynmicl nd stisfis Eulr-Lgrng qs Excittions occur with both typs of brking NG & Higgs-lik mods with spontnous brking xtr mtric/virbin mods with xplicit brking Stucklbrg introducs NG mods & diff invrinc Nothr idntitis provid consistncy conditions tht must b obyd whn th symmtry brking is xplicit
Mttr couplings in grvity thoris with bckground filds cn b nlyzd using th SME with ithr spontnous or xplicit spctim brking è With spontnous brking, SME cofficints occur vry nturlly s vv s of dynmicl tnsor filds nd inconsistncy issus r vdd è With xplicit brking, th SME cofficints cn b nondynmicl s long s th consistncy conditions hold (or Stucklbrg xcittions ct s NG filds) Furthr dvlopmnt of th grvity sctor of th SME will b prsntd by Q. Bily, M. Mws, J. Tsson, M. Sifrt, nd othrs (including xprimntl tlks).
Primry Rfrncs: V. A. Kostlcky nd S. Smul, Grvittionl Phnomnology in Highr-Dimnsionl Thoris nd Strings, Physicl Rviw D40 (988) 886. V. A. Kostlcky, Grvity, Lorntz Violtion, nd th Stndrd Modl, Physicl Rviw D69 (2004) 05009. RB nd V. A. Kostlcky, Spontnous Lorntz Violtion, Nmbu-Goldston Mods, nd Grvity, Physicl Rviw D7 (2005) 065008. RB, S.-H. Fung, nd V. A. Kostlcky, Spontnous Lorntz nd Diffomorphism Violtion, Mssiv Mods, nd Grvity, Physicl Rviw D77 (2008) 065020. RB, Explicit vrsus Spontnous Diffomorphism Brking in Grvity, Physicl Rviw D9 (205) 065034. RB, Spctim Symmtry Brking nd Einstin-Mxwll Thory, Physicl Rviw D92 (206) 08505. RB nd A. Shic, Nothr Idntitis in Grvity Thoris with Nondynmicl Bckgrounds nd Explicit Spctim Symmtry Brking, Physicl Rviw D94 (206) 04034.