Off-shell construction of some trilinear higher spin gauge field interactions

K. Mkrtchyan

Plan:1. Introduction and Motivation2. Exercises on spin 0- and 1- field couplings with the

higher spin gauge fields3. Generalization to the 2-2-4 and 2-2-6 interactions4. 2S-S-S interaction Lagrangian5. Conclusions

R. Manvelyan, K.M.: arXiv:0903.0058 R. Manvelyan, K.M., W. Rühl: arXiv:0903.0243R. Manvelyan, K.M.: to appear in arXiv soon

‘Supersymmetries and Quantum Symmetries’, July-August 09, Dubna

1. Introduction and Motivation• The construction of interacting higher spin gauge field theories (HSF) has always been

considered an important task.

• Particular attention caused the holographic duality between the O(N) sigma model in d=3 space and HSF gauge theory living in the AdS4. This case of holography is singled out

by the existence of two conformal points of the boundary theory and the possibility to describe them by the same HSF gauge theory with the help of spontaneously breaking of higher spin gauge symmetry and mass generation by a corresponding Higgs mechanism.

• Does AdS/CFT work correctly on the level of loop diagrams in the general case and is it possible to use this correspondence for real reconstruction of unknown local interacting theories on the bulk from more or less well known conformal field theories on the boundary side?

All these complicated physical tasks necessitate quantum loop calculations for HSF field theory and therefore information about manifest, off-shell and Lagrangian formulation of possible interactions for HSF.

2

• F. A. Berends, G. J. H. Burgers and H. van Dam, “Explicit Construction Of Conserved Currents For Massless Fields Of Arbitrary Spin,” Nucl. Phys. B (1986) 429; F. A. Berends, G. J. H. Burgers and H. Van Dam, “On Spin Three Selfinteractions,” Z. Phys. C (1984) 247; F. A. Berends, G. J. H. Burgers and H. van Dam, “On The Theoretical Problems In Constructing Interactions Involving Higher Spin Massless Particles,” Nucl. Phys. B (1985) 295.

• E. S. Fradkin and M. A. Vasiliev, “On The Gravitational Interaction Of Massless Higher Spin Fields,” Phys. Lett. B (1987) 89. E. S. Fradkin and M. A. Vasiliev, “Cubic Interaction In Extended Theories Of Massless Higher Spin Fields,” Nucl. Phys. B (1987) 141.

• T. Damour and S. Deser, “Geometry of spin 3 gauge theories,” Annales Poincare Phys. Theor. , 277 (1987); T. Damour and S. Deser, “Higher derivative interactions of higher spin gauge fields,” Class. Quant. Grav. , L95 (1987).

• R. R. Metsaev, “Cubic interaction vertices for massive and massless higher spin fields,” Nucl. Phys. B (2006) 147 [arXiv:hep-th/0512342];R. R. Metsaev, “Cubic interaction vertices for fermionic and bosonic arbitrary spin fields,” arXiv:0712.3526 [hep-th].

• I. G. Koh, S. Ouvry, “Interacting gauge fields of any spin and symmetry,” Phys. Lett. B (1986) 115; Erratum-ibid. B (1987) 434

• O.A. Gelfond, E.D. Skvortsov and M.A. Vasiliev, “Higher Spin Conformal Currents in Minkowski Space”, arXiv:hep-th/0601106

• N. Boulanger, S. Leclercq, P. Sundell, “On The Uniqueness of Minimal Coupling in Higher-Spin Gauge Theory,” JHEP 0808:056,2008; [arXiv:0805.2764 [hep-th]].

• X. Bekaert, N. Boulanger, S. Cnockaert, S. Leclercq, “On killing tensors and cubic vertices in higher-spin gauge theories,” Fortsch. Phys. (2006) 282-290; [arXiv:hep-th/0602092].

• A. Fotopoulos, N. Irges, A. C. Petkou and M. Tsulaia, “Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex,” JHEP (2007) 021; [arXiv:0708.1399 [hep-th]].

• I. L. Buchbinder, A. Fotopoulos, A. C. Petkou and M. Tsulaia, “Constructing the cubic interaction vertex of higher spin gauge fields,” Phys. Rev. D (2006) 105018; [arXiv:hep-th/0609082]. 3

2. Exercises on spin 0- and 1- field couplings with the HS gauge fields

Starting action and variation for scalar field in

Notations

20 2

1 ( 2)( ) [ ]2 4

D D DS d z gL

1 2 1

1 2 1

1 ( ) ( ) ( )l

lz z z

1 1 2 1 1 2 1 2( ) ( ) ( )0 0(1)2l l l l lh l h

3 10

l

(1) (2)… … … … …

DAdS

22 1 2 1

1 2

( )1 (2 )0 ( 2 )

1

1( ) { [ ]

1

l

m m

m

D ml m

m

mS d z g

m

21 2 2

1 1 2 2

2( 2 1)2

2

1( 2)[ ] ( )}24

l

m

m m ml mm

mD DmL

1 2 1 1 2

(2 ) ( 1) {m m m m

m m

2 1 2 1 1 2 2( ) ( )2 m m m m m

m g g

2 1 2 1 1 2 22

( 2 2)( 2 4) }8 m m m m m

m D m D m gL

Field redefinition terms2

1 2 2

1 1 2 2

(2 )

2

1 ( )2( 2 1)

l

m

m m m

m

m

m hl m

Gauge Invariant Interaction of scalar and higher spin field

The gauge invariant action for a spin gauge field coupled to a scalar includes gauge invariant actions of a tower of all smaller even spin gauge fields coupled to the same scalar in an analogous way.

2 (2 )(2) (4) ( ) (2 )0 1

1

( ) ( ) ( )l

mGI m

m

S h h h S S h

( 2 )1 2

1 2

(2 )(2 ) (2 )1

1( )2

mm

m

mm D mS h d z ghm

(2) (4) ( )( )GIS h h h

l

Weyl invariant action for a higher spin field coupled to a scalar1 1 2 1( ) ( )0 ( 1)l l l lh l l g

1 2 1 2( )0 2( 2 4)l lh D l

1 2

1 2

1 l

l

( 2 )2

(2) (4) ( ) (2) ( ) (2 )1

1

( ) ( ) ( )m

lWI GI r m

m

S h h h S h h S h

1 2 1 2 1 2( ) ( ) ( )1 ( )2

l l lr TrF h h h

2 2 1 21 ( ) ) ( )(2

2 ( 1)( 3)2

l ll l D lh h

L

( ) (2) ( )1 ( )rS h h

2

1 2

2 1 2 1 1 2

1( )

0

12

l

l

m l m m m

m D

m

d z g r

1 2 1 2 2 2 1 2(2 ) ( )( 2 )2 (2 1) 1 2m m m mm ml mh m m C g m l

222 1

22

2 ( 1)( 1) ,2 ( 2) ( 1)

m Dm m

m Dm

mm m

222

2 122

2 1 ( 1)( 1)12 ( 2 )

m Dm m

m Dm

mC

m m

• We start this section constructing the well known interaction of the electromagnetic field in flat D -dimensional space-time with the linearized spin two field.

• Noether’s procedure

Spin 1- field coupling with the HS gauge fields

20

1 1 1 ( )4 2 2

L F F A A A

0 (2) ( )( ) 2 ( ) ( ) ( )h x x x x

1 0 (2)0 1( ) ( ) 0L A L A h

1A A C A

1C Noether’s procedureregulates the relation between gauge symmetries of different spin fields

8

1 ( )0

1( ) ,4

L A F F F F

1A F

1 1 1[ , ][ , ] ( )A A F x

(2) (2) (2)1

1( )2

L A h h

(2) 14

F F g F F

• Solution of Noether’s Equation

( ) ( )A A x A x

Reparametrization

Field dependent Gauge Variation

Energy-Momentum Tensor

- A L

9

• Now we turn to the first nontrivial case of the vector field interaction with a spin four gauge field

• Starting variation

Notations

1 ( )A x F

( )( ) (1) (0 1)

1 1 14 4

F FL F F F F

(1) (1)1 1( )4 2

F F F F F F ( )

(2) (3)14

F F F F

(1) (2)… … … … …

0 ( ) 0(1)4 2h h

Integrable Part

Field Redefinition

Variation Modification

(4) ((2) (4) ( 2) (2)4)1

12

( ) 14

hhL A h h

1 1( )2 8

A A h F h F

1 12

A F F

10

1 12

A F F

0 (4) ( ) 0(1)4 2h h

0 (2) ( ) 0 (2)(2) (3)2 2h h

4-1-1 Interaction

(4)( ) ( ) ( )

1 12 2

F F g F F g F F

(2) (4) (4) (2) )(4 2)1

(1 1( )4 2

L A h h h h

1 0 (2) (4)0 1( ) ( , ) 0L A L A h h

an additional spin two field coupling !!

(2) 14

F F g F F

11

Spin one gauge field couplings to the higher spin gauge fields

1 1

1 2 1

1 ,l

l lA F

11 2 ( )(2)1 00 1( ) ( , ,...., ) 0lL A L A h h

1 1 2 1 1 2 1 2( ) ( ) ( )0 0(1)2l l l l lh l h

Starting variation

2 1 2 1

1 2

21

01

( )( 2 )

1( ( ))

1m m

ml mm

mL A

m

1 2 2

2

1

1

1

2( 2 )

2

1

1 1 ( )1 m m m m

m

ml m F

m m Fm m

2 1 1

1

3 2

2 3( 2 )

2

11

1 1 ( )1 2 m m m m

ml

mm

m m Fm m

F

1

2 2 1 2 1

2 2( 2 1

2

1)

1 1 ( )1 2 1

m

m m m ml mm

mFm F

m l m

Field redefinition terms Variation ModificationIntegrable terms

13

1 2 1 1 1 2 1 2( ) ( 1) (

m m m m m m

mA F F

1 2 3 1 2 2 1 2

12 m m m m m

m g F F

1 2 3 1 2 24)

m m m

m g F F

1 2

1 2

2(2 )(2) (4) ( ) (2 )

11

1( ) ( )2

m

m

m m

m

L A h h h h Am

Final variation

1 1

1 2 1

1 l

l lA F

1 2 2

1 2 2 1 1

2

( 2 )1

1 1 ( )1

m

m m m ml mm

m m Fm m

Interaction

14

The gauge invariant action for the spin gauge field coupled to the spin 1 gauge field includes gauge invariant actions of tower of all smaller even spin gauge fields coupled to the same vector field in the same way.

1 2 3

2 3 1 2 1

2(2 )

1

1 1 ( )1 4

m

m m m m

m

m

m m h Fm m

1 2 2

2 2 1 2 1

2(2 )

1

1 1 ( )1 2( 2 1)

m

m m m m

m

m

m mA A h Fm l m

Field Redefinition

15

Generalization to the 2-2-4 and 2-2-6 interactions

(2) (2) (2) (2) (2) (2) (2) (2) (2)0

1 1( )2 2

L h h h h h h h h h

Starting point transformation and spin two Pauli-Fierz Lagrangian

1 (2)h

1 (2) 0 (2) (4)0 1( ) ( ) 0L h L h h

We solve the following Noether’s equation

1 ( )2R R ( ) 0

where is the spin two gauge invariant symmetrized linearized Riemann curvature

16

To integrate the Noether’s equation we submit to the following strategy:

1) First we perform a partial integration and use the Bianchi identity

to lift the variation to a curvature square term.

2) Then we make a partial integration again and rearrange indices using

to extract an integrable part.

3) Symmetrizing expressions in this way we classify terms as

• integrable

• integrable and subjected to field redefinition (proportional to the free field equation of motion)

• non integrable but reducible by deformation of the initial ansatz for the gauge transformation (again proportional to the free field equation of motion)

( )R R

( ) ( )

17

Solution for 4-2-2

(2) (4) (4) (4) (2)1 ( )

1( ) ( )4

L h h h h

(4) (2)( )

2( )3

h

After field redefinition

(2) (2) (4) (4) (4)( )

1 1 12 4 4

h h h h R h R

We arrive at the 4-2-2 gauge invariant interaction

with the following gauge transformations (2)

( )h

0 (4) ( ) 0 (4)(1)4 2h h

Bell-Robinson current

18

Solution for 6-2-2

After field redefinition(2) (2) (6) (4)( )inth h R R h h

where

(6) (6) ( ) (6)16

1 1( )2 6intR R h h h R

(6) (6) (4) (6)1 1 1 53 12 4 6

h R h F h R h R

(6) ( ) (6) (6) (4) ( )5 1 1 16 12 12 4

h R h R h R h R

19

(4) (2)( )

2( )3

h

We arrive at the 6-2-2 gauge invariant interaction

with the following gauge transformations

(6)( )

(2) (4) (6) (4)(6) (4)1 ( )

1 1( )6 4

L h h h h h

(6)( ) ( ) ( )g

( )12g

1 (2) 4 13 3

h

0 (6) ( )6 ( )h x

20

1 2

( ) ( )

1

( ) ( ) ( )i

s

i

ss s

i

h z a a h z

1( 1 1) ( ) ( 1 1) ( )

1

( )( )( ) ( ) ( ) ( )

( )( )amm n m n m n m n

a b a bbn

af z a b g z a b f z a b g z a b

b

21

1( )

i

i

sa

a ais

� ( ) ( 1) ( )( ) ( ) ( ) ( )s s sGrad h z a Gradh z a a h z a

( ) ( 2) ( )1( ) ( ) ( )( 1)

s s saTr h z a Trh z a h z a

s s

( ) ( 1) ( )1( ) ( ) ( ) ( )s s saDiv h z a Divh z a h z a

s

4. 2S-S-S Interaction

1) Formalism

The most elegant and convenient way of handling symmetric tensors is by contracting it with the s’th tensorial power of a vector and star contraction

ia

12( 1)( 2 2) ( ) ( 2 2) ( )

12( 1)

( ) ( ) ( ) ( )m mm n m n m n m n m naa b a b

n n b

af a b g a b f a b g a b

b

1 1

1 1

( ) ( )( ) ( )( ) n s

n

s sn nz b a b b a a

De Witt-Freedman Christoffel Symbols- bitensors

Duality relations

21

( ) ( 1)( ) ( ) ( )s sh z a s a z a

( 1)( ) 0sa z a 2 ( )( ) 0,s

a h z a

Fronsdal fields , Equation and Lagrangian

( ) ( ) 2 ( )( ) 1( ) ( ) ( )( ) ( ) ( ) ( ) 02

s s sa a

s z a h z a a h z a a h z a F

Gauge field

Field Equation

( ) ( ) ( ) ( ) ( )0

1 1( ( )) ( ) ( ) ( ) ( )2 8 ( 1)

s s s s sa a a aL h a h a a h a a

s s

F F

2( ) ( ) ( ) ( )

0( ( )) ( ( ) ( )) ( )4

s s s sa a

aL h a a a h a F F

Lagrangian

22

( ) ( )

1

1( ) [( ) ( )( )] ( )s

s sa

n

z b a b a b h z an

( ) ( )

0

( 1)( ) ( ) ( ) ( ) ( )ks

s n k k k sa

k

z b a b a b h z ak

( ) ( )( ) ( ) ( ) ( ) 0s sb aa z a b b z a b

( ) ( )( ) ( )s sz a b z b a

HS Curvature

Primary Bianchi Identity

23

( )][ ( ) 0,s z a b

a b

( ) ( ) ( )( ) ( ) ( ) ( ) ( )s a s b ss z a b a z a b b z a b

( ) ( )1( 1)( ) ( ) [( ) ( )( )] ( )2

s sb a bs z a b b a b z a b

( ) ( ) ( )1 1( ) ( ) ( ) ( ) ( ) ( )2 2

s s sa b a b b az b a b z b a a z b a

( ) ( )1( ) ( ) ( ) ( )2

s sa az a a z a F F

Bianchi Identities (Secondary )

From Primary Bianchi Identity

Similar to Ricci tensorIdentity 24

( ) ( ) 0s z b a

( ) ( )( ) ( 1) ( 3 ) ( )s sb z b a s s U a b s z a F

3

1( 3 ) [( ) ( )( )]s

ak

U a b s b a bk

Curvature and Fronsdal Equation

Trace

25

2( ) ( ) ( ) ( )

0( ( )) ( ( ) ( )) ( )4

s s s sa a

aL h a a a h a F F

( ) ( ) 2 ( 2) 2 2 ( 4)(1)( ) ( ) ( ) ( ) ( )s s s sh a h a a h a a h a …

( 2) ( )(1)

1( ) ( )2( 2 2)

s sah a h a

D s

( ) ( ) ( )(1) 0 (1)( ( )) ( ) ( )s s s

aL h a a h a F

We choose the second term of our field variation in a form that simplifies Lagrangian variation.

From double tracelessness of gauge field

26

To integrate the Noether’s equation we submit to the following strategy:

1) First we perform a partial integration and use the Bianchi identity

to lift the variation to a curvature square term.

2) Then we make a partial integration again and rearrange indices using

to extract an integrable part.

3) Symmetrizing expressions in this way we classify terms as

• integrable

• integrable and subjected to field redefinition (proportional to the free field equation of motion)

• non integrable but reducible by deformation of the initial ansatz for the gauge transformation (again proportional to the free field equation of motion)

( ) ( ) ( )( ) ( ) ( ) ( ) ( )s a s b ss z a b a z a b b z a b

( ) ( )( 1)( ) ( ) ( 1) ( 2 ) ( )s sbs z a b s s U a b s z a F

27

( ) 2 1 ( )(1) ( ) ( 2 )[ ( ) ( )]s s s

bh a U b a s z b z b a

2

( 1) 1( 2 ) ( ) ( )( )( 1)

s s

b b ak

U b a s as k

Is dual to

2

1 1[( ) ( )( )] ( 3 ) [( ) ( )( )]2

s

a ak

b a b U b a s b a bk

( ) 2 1 ( ) ( ) 2 1 ( ) ( )(1) 0

1( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 ( 1)(2 1)

s s s s s b s sb a b b a bL h a b b b a b a b b a b a

s s s

Using this technique we can guess the right variation

where

Then we bring variation of the Lagrangian to the following form

28

2(2 ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

2( 1)s s s b s s

a a bbz b b a b a b a b as

( ) (2 ) (2 ) (2 )1 ( )

1( ( ) ( )) ( ) ( )2

s s s sbL h a h b h z b z b

s

(2 ) (2 1)0 ( ) 2 ( ) ( )s sh z b s b z b (2 ) (2 1)

0 ( ) 4 ( ) ( )s sb bh z b s z b

( ) 2 1 ( )(1)

2

( 1) 1( ) ( ) ( )( ) [ ( ) ( )]( 1)

s ss s s

b b a bk

h a a z b z b as k

( ) ( ) (2 )(1) 0 0 1( ( )) ( ( ) ( )) 0s s sL h a L h a h b

Result for 2S-S-S Interaction

generalized Bell-Robinson current

29

30

1 2

1 2

(2 ) ...( ) (2 ) (2 )1 ( ) ...

1( )2

s

s

ss s sL h h hs

s-s-2s interaction result in the form that is convenient for comparison with already known cases

1 1

1 2 1 1 1 2 2 1 2 1 1 1 2 2

2... , ...(2 ) ( ) ( ) ( ) ( )

( ) ... ... , ... ... , ... , ... ...2( 1)s s

s s s s s s s s s s s

s s s s ss gs

Conclusions

We presented interaction Lagrangians for triplets of higher spin fields, a pair of which has equal spin S1 whereas the third has spin S2≥2S1. Besides the Lagrangians the next-to-leading order of the gauge transformations is given. The fields of smaller spins appear combined into currents of the Bell-Robinson form .

Remarkable is that for one such spin the interaction implies the existence of a whole ladder of interactions for smaller spins S2 ̶̶ 2n ≥2S1 .

Outlook• Constraction of Self Interactions using language of

deWit-Freedman curvature and Christoffel symbols• Several one loop calculations with these

interactions actual for AdS/CFT

31

Thank You For Your Attention

32