Higher spin gauge field cubic interactions.

Transcript

1 Hgher spn gauge feld cubc nteractons. R. Manvelyan ( n collaboraton wth K. Mkrtchyan W. Rühl Plan:. Introducton and Motvaton. Selfnteracton. S - S -S Interacton 6. Concluson Phys. Lett. B 699 (0 87; Phys. Lett. B 696 (0 40; Nucl. Phys. B 844 (0 48; Nucl. Phys. B 86 (00 04 Tbls

2 References M. Ferz and W. Paul On relatvstc wave equatons for partcles of arbtrary spn n an electromagnetc feld Proc. Roy. Soc. Lond. A (99. C. Fronsdal Sngletons And Massless Integral Spn Felds On De Stter Space (Elementary Partcles In A Curved Space V Phys. Rev. D ( ; Massless Felds Wth Integer Spn Phys. Rev. D ( B. dewt and D.Z. Freedman Systematcs of hgher spn gauge felds Phys. Revew D ( A. K. H. Bengtsson I. Bengtsson and L. Brnk Nucl. Phys. B7 (98 ; Nucl. Phys. B7 (98 4 Phys. Rev. D ( F. A. Berends G. J. H. Burgers and H. van Dam Nucl. Phys. B 7 (986 49; Z. Phys. C 4 (984 47; Nucl. Phys. B 60 ( E. S. Fradkn and M. A. Vaslev On The Gravtatonal Interacton Of Massless Hgher Spn Felds Phys. Lett. B ( E. S. Fradkn and M. A. Vaslev Cubc Interacton In Extended Theores Of Massless Hgher Spn Felds Nucl. Phys. B (987 4 R. R. Metsaev Cubc nteracton vertces for massve and massless hgher spn felds Nucl. Phys. B ( [arxv:hep-th/054] Cubc nteracton vertces for fermonc and bosonc arbtrary spn felds arxv:07.56 [hep-th]. N. Boulanger S. Leclercq P. Sundell ``On The Unqueness of Mnmal Couplng n Hgher-Spn Gauge Theory'' JHEP 0808:056008; [arxv: [hep-th]]. R. Manvelyan K. Mkrtchyan and W. Rühl arxv: ;00.877; 00.58;N.P.B 86 (00-7. M. A. Vaslev Cubc Vertces for Symmetrc Hgher-Spn Gauge Felds n (AdS_d arxv: A. Sagnott and M. Taronna arxv:006.54v [hep-th] Strng Lessons for Hgher-Spn Interactons.

3 Introducton and Motvaton The constructon of nteractng hgher spn gauge feld theores (HSF has always been consdered an mportant task and was always n the center of attenton durng the last thrty years Partcular attenton caused the holographc dualty between the O(N sgma model n d= space and HSF gauge theory lvng n the AdS 4. Ths case of holography s sngled out by the exstence of two conformal ponts of the boundary theory and the possblty to descrbe them by the same HSF gauge theory wth the help of spontaneously breakng of hgher spn gauge symmetry and mass generaton by a correspondng Hggs mechansm. Does AdS/CFT works correctly on the level of loop dagrams n the general case and s t possble to use ths correspondence for real reconstructon of unknown local nteractng theores on the bulk from more or less well known conformal feld theores on the boundary sde? All these complcated physcal tasks necesstate quantum loop calculatons for HSF feld theory and therefore nformaton about manfest off-shell and Lagrangan formulaton of possble nteractons for HSF.

4 Introducton and Motvaton Important physcal tasks (AdS/CFT Quantum loop calculatons for HS Lagrangan formulaton of possble nteractons for HSF. Strng Theory ncludes a plethora of hgher-spn exctatons whose detaled behavor s largely unknown. There are consstent equatons for HSF n AdS (Vaslev Interactng Lagrangan for HS??? 4

5 Gauge nvarance Unque Cubc Interacton for arbtrary HS felds!!! 5

6 Power Expanson of Lagrangan and Gauge transformaton (0 ( δ δ δ = = ( + ( +... Gauge Symmetry δ = 0 Noether Equaton (0 ( ( ( ( δ + δ +...( = 0 6

7 Frst Nontrval Noether Equaton δ (0 ( ( ( + δ = 0 Is t possble so solve? In the case of general hgher spn cubc nteracton? In covarant way? The answer s YES!! 7

8 Formalsm ( h za h za a a µ s ( ( µ µ s µµ µ s ; =. The most elegant and convenent way of handlng symmetrc tensors s by contractng t wth the s th tensoral power of a vector and star contracton a µ Tr : h ( z; a Trh ( z; a = a h ( z; a ss ( + Grad h z a Gradh z a a h z a : ( ; ( ; = ( ( ; a =. s µ a a µ! = Dv: h ( za ; Dvh ( za ; = ( a h ( za ;. s ( m ( m ( m ( mn ( a f ( za ; a g ( za ; = f ( za ; a m ( a g ( za ; a f ( a g ( a = f ( a b g ( a. ( m ( m m ( m ( m a a mm ( a Dualty relatons 8

9 Hgher spn gauge feld theores: Free feld Lagrangan and equaton of moton Free feld Lagragan of hgher spn s feld L0 ( h ( a = h ( a a ( а ( а а ( a + h a a 8 ss ( Invarant n respect to gauge transformaton a δ ε ( za ; = 0 (0 h ( za sa ( ε ( za ; = ; ( za ; = 0. a h ( za ; = h ( za ; sa ( D ( za ; 9

10 s ( ; = ( ; ( ( ; D za Dvh za a Trh za а D ( za ; = 0. Correct generalzaton of the Lorentz gauge condton n the case of s> could be only the so-called de Donder gauge condton D ( za ; = 0. In ths gauge free feld equaton of moton s very smple h ( za ; = 0 0

11 Equaton of moton a δl0 ( h ( a = ( ( a а ( a aδh ( a 4 Free feld equaton of moton Or equvalent a ( ( a = 0 4 a а ( a = 0

12 Hgher spn gauge feld selfnteracton: General settng of the problem Neother equaton for hgher spn symmetry δ Lh ( ( a δl( h ( a = ( 0 a δh a = δ h ( a For nonlnear Lagrangan L( h ( a = L ( h ( a + L ( h ( a + 0 And nonlnear gauge transformaton δh ( a = δ h ( a + δ h ( a +. Frst order equaton (0 ( δ L ( h ( a + δ L ( h ( a = 0 ( 0 (0

13 Noether equaton for Hgher spn selfnteracton δ a L ( h ( a = ( ( a ( a h ( a 4 (0 а aδ( 4 δ( h ( a = δ( h ( a + a δ( h ( a + ( a δh ( a + After δ a ( h ( a δ ( h ( a + а h( ( a ( D s δ + Noether equaton can be smplfed to δ L ( h ( a = ( a δh ( a. (0 a (

14 Selfnteracton for any spn Ascrbng the same dmensons to the free part of the Lagrangan that s quadratc n the felds and dervatves and to the nteracton cubc n the felds we arrve at the dea that: the number of dervatves n the nteracton should be s. Ths type of nteractng theores wll behave n the same way as gravty. The number of dervatves n the frst order varaton should be s-. For s= ths consderaton s of course n full agrement wth the lnearzed expanson of the Ensten-Hlbert acton. 4

15 Spn example lnearzed gravty Free feld Lagrangan for spn two gauge feld µν L0 = h ( hµν ( µ Dν + h( h ( D 4 where Dµ = ( h µ µ h δ h= h µ µ Whch s nvarant under gauge transformaton h ε (0 µν ( µ ν =. ( D =. µ Dµ 5

16 Selfnteracton for spn two gauge feld Trlnear selfnteracton Lagrangan ( αβ µν αµ βν L ( h = h h h + h h h µν µν ( D hµν h h µ hdν 4 Correspondng gauge transformaton α β µν α β µν δ h = ε h + γ h ρ ( ρ ( µν ρ µν ( µ ν ρ Geometrcal acton and transformaton we can get after feld redefnton h h + ( hh h g. µν µν 4 µν ( D µν γ = ε ( µν [ µ ν ] 6

17 Spn 4 case Spn four free gauge feld Lagrangan (4 L0 ( h = h αβγδ h αβ αβγδ + αβ Whch s nvarant n respect to gauge transformaton δ (0 h αβγδ 4 ( α ε βγδ = ε = 0. β αβ γ = h 4 D αβ = αβγ = h αβ ( D αβ αβγδ αβγδ ( α βγδ Dαβγ = ( h αβγ ( α h βγ h βγ h D β αβ = 0 = 0. h β β = α βγα 7

18 Trlnear selfnteracton Lagrangan for spn 4 gauge feld We ntroduce some "coordnate system" for classfcaton of our nteracton nt (4 L = Lj ( h j= 0 + j where nt ( (4 4 ( ( (4 j ( (4 j j L h D h h. Interacton n de Donder gauge (4 nt (4 0 j j= 0 nt LdD ( h = L ( h. 8

19 9

20 ( = Γ 8 µ α ν γδλρ β µ ανλρ β γδ + hαβγδ h hµνλρ + ν hαβγδ h λ hµρ 4 nt (4 αβγδ µνλρ (4 µ α β γνλρ δ L00 h h h αβγδ µνλρ hαβγδ h hµνλρ L h h h h h nt (4 0 ( = α β γνλρ δ h δ α β ν λ h γρ αβγδ ν λρ αβγδ νλρ ν α µβγλ δ λ µαβγ ρ ν + h h hνλ h h µ αβγδ αβγ µ ν ( 4 h µαβγ h ν + h αβγ µ ν h λρ 0

21 nt (4 α β µ δ α µ γ L0 ( h = hαβγδ h h + hαβγδ h ν h γν βν δ µν µ βν δµ γδ h α h γ h h α β µ ν αβγδ αβγδ h ( h 4 4 α βγ δ µ hαβγδ ν h h µν nt (4 α µλ β ν νλ µ L0 ( h = µ ν hαβ h h λ µ h νh λ( h 4 4

22 nt (4 L0 ( h = D h αβγδ h νλρ + ρ D µ h αβγδ λ h D h h ( D h h α ν λρβ γ δ αβλ ρµγδ δ ν λ ραβγ αρ µ β γδ νλρ αβγδ αβγδ ρµ + ( D µ hαβγδ ν h ( D hαβγδ µ ν h 4 µν αβγδ µν αβγδ µν αβγδ µν αβγδ α ( D h µ hνβγδ + α ( D h β hµνγδ 4

23 nt (4 γδ γδ L ( h = h γ hµνλρ δ D + h γ δ hµνλρ D µ νλρ µ νλρ γδ γδ + µ h h ν D λρ h h µνλρ D 4 4 µνλρ γ δ µ ν λ γδρ 9 γδ γδ + h h D + h h D 4 γδ γδ + h ρhγµνλ δ D h ρ γ hδµνλ D γδ µ νλ µ γδ νλ ( D h h ( D h h µ ( D γδ h h νλ h γδ D µνρ ν γλ δµ + γ hδµνρ 4 h γδ D µνρ µ hγδνρ 8 λ µνρ µ ν λρ µ ρ γδνλ ρ γµνλ δ µ νλρ µ νλρ γ δµνλ λ γδµν

24 nt (4 µνρ δ 9 µνρ γδ L ( h = D ( h δ µ hνρ γ δ D h µ h 4 8 µνρ ν D h γ µ ν hδρ µ Dνργ h δ h γδ γδ µρ γδ µν γδ µν ( D γh δh ( D h γ δh 8 µ γδ γδ + ( D γh µν hδ ( D γ µ hδν h 9 γδ µ ν µ γδ ν ( D h h + ( D h h 4 µν µν ν µν γν δµ γδ µν γδ µ ν γδ + ( D h ( h + ( D h h 8 8 γδ γµ δν νρ ( D γδ h h µ γµ δ 4

25 nt (4 αβγ µνρ 9 αβγ µνρ L0 ( h = D D h D D h 4 µ ν ραβγ α µ βγνρ + D αβγ ρ D µν h ( D γδ D µνρ γ α βµνρ + γ hδµνρ L h h D D h D D ( h δ D µνρ D + h γδ D µνρ D 4 4 nt (4 γδ µνρ µνρ ( = γ δ µνρ + ( µνρ µ δνρ γ µ δνρ γδ µ νρ µ γδ νρ + h D D + h D D γ ν δµρ ν γ δµρ γδ µν γδ µ ν h ( D D + 6 h ( D D 4 + ( D ( D. h γδ ν γ δν µ νγδ γ δµν 5

26 Selfnteracton Lagrangan leadng term leadng s L( ( h ( z = ( δ z z δ( z z δ( z z s! z z z α β γ s αβγ + + = ( c( a( b( a b( b c( c a haz (; ( hbz ; ( hcz ; γ α β γ α β Correspondng frst order on feld gauge varaton leadng term leadng β s δ( h ( c; z = ( δ( z z δ( z z s!! z z α β γ s α βγ + + = ( c ( ( ( ( c ( c ( az ; ( hbz ;. γ α β γ α β a b a b b a ε where s s! = = α + β + γ. αβγ αβγ!!! 6

27 Cubc nteractons for arbtrary spns: Leadng terms (00 s I n δ δ δ n ( h ( a h ( b h ( c = C dz dz dz ( z z ( z z ( z z TQ ˆ( Q Q n h ( z ; ah ( z ; bh ( z ; c where TQ ˆ( Q Q n =( ( ( ( ( ( Q Q Q n n n a b b c c a a b c Number of dervatves n + n + n = n + Q + Q = s n + Q + Q = s Q = n ν Q = n ν n ν ν = / ( + s s s j k n + Q + Q = s Q = n ν Ijk are all dfferent 7

28 = max[ s + s s ] = s + s s mn j k max = s + s + s mn < < max ν = s s ν = s s ν =0 C s = n const n ν n ν ν n ν C C s = n s Q Q Q const = const s n s + s s + s / ( s Q Q Q ν =/ / j Q j = s ν 8

29 To derve the next term of nteracton contanng one dedonder expresson we turn to Lagrangan formulaton of the task and wll solve Noether's equaton ( 0 (0 δ + δ I = = ( h ( a ( h ( a h ( b h ( c = 0 where ( ( = ( * ( ( * ( 0 h a h a a a + a h a a a a 8 s ( s δ (0 s h a s a ε z a ( = ( ( ; = ( δ Shftng by trace term we obtan the followng functonal equaton: (0 δ I( h ( a h ( b h ( c = 0 + O( ( a 9

30 C Tˆ( Q n [( a h h h ( b h h h ( c ] { s } { n } j ε + ε + ε = 0 + O( ( a we see that all necessary nformaton we can fnd calculatng these commutators [ TQ ˆ ( n( a ] = QTQ ˆ ( Q Q n+ QTQ ˆ ( Q Q n+ j + ntq ˆ( Q Q n ( QTQ ˆ( Q Q n( b = ( h ( z; a= ( z; a + sd (0 ε ( z; a= δ D a a ( = sd + / ( a h ( z; a a = abc 0

31 Noether s Equaton leads to + s s s s s s n + + n + ( n v C ( n v C =0 + s s s s s s n + + n + ( n v C ( n v C =0 + s s s s s s n + + n + ( n v C ( n v C =0 Wth the followng soluton C s = n const n ν n ν ν ν

32 ( ( ( dzdzdzdzδ z z δ z z δ z z n (0 I = [ sn C TQ ˆ( n D h h s n j sn + C TQ ˆ( n h D h s n j sn s + C n TQ ˆ( j n h h D ] (0 I = snsn [ + C TQ ˆ( n D h D s n j snsn + C TQ ˆ( n D D h s n j snsn s + C n TQ ˆ( j n h D D ] snsnsn [ C TQ ( n D D D ] (0 ( I = s s s s ˆ + n j

33 (0 (0 I I = =0 In de Donder gauge Trace decouples!!! QQQ 8 (0 s I = Cn [ TQ ˆ( Q Q n h h h ] a b c I = Cn sq n [ + TQ ˆ( Q Q n D h h 4 b sq n + TQ ˆ( Q Q n h D h 4 c sq n + TQ ˆ( Q Q n a h h D ] 4

34 ( = s s s I Cn sq Q n [ + TQ ˆ( Q Q n D h h 8 b c sq Q n + TQ ˆ( Q Q n h D h 8 a c sq Qn + TQ ˆ( Q Q n a h b h D ] 8 I = Cn s sq nn [ + TQ ˆ( Q Q n h D D 4 a ssq nn + TQ ˆ( Q Q n D D h 4 c ssq nn + TQ ˆ( Q Q n D b h D ] 4 4

35 Trh D 0 0 hhh Dhh DDh DDD 000 D Trh h DD Trh 000 D Trh Trh Trh TrhTrh 5

36 5. Conclusons It was shown that there s a local hgher dervatve Lagrangan for HS nteractons n flat space-tme at least for frst nontrval order cubc couplngs. All possble cases of cubc nteractons between dfferent HS gauge felds ncludng selfnteracton of even spn felds are presented n one compact formula. These nteractons between HS gauge felds are unque therefore reproduce flat lmt of the Fradkn-Vaslev vertex for cubc couplngs of HS gauge felds to the lnearzed gravty and also nclude all lower spn cases whch are well known for many years. 6

37 Thank You for your attenton! 7