1

Massimo Taronna

Scuola Normale Superiore & INFN, Pisa

Based on:

Master Thesis(2009)[hep-th/1005.3061](M.T. Advisor: A.Sagnotti) hep-th/1006.5242: A.Sagnotti and M.T.

hep-th/1107.5843: M.T.

hep-th/11--.---: E. Joung and M. T.

Prague, 24/09/2011

Higher-Spin Interactions: Four-point functions and beyond

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It is a scheme based on the mechanical model of a vibrating relativistic string

Although very natural it raises several questions: Background (in)dependence(?) ....

Key ingredient for consistency: Infinite tower of HS excitations

The mechanical model hides the geometry The difficulty

Soft UV behavior, open-closed duality, planar duality, modular invariance, etc…

String Theory is a consistent Higher-Spin Theory!

What is String Theory?

Very general fact whenever an S-matrix theory is considered!

Global symmetries on the asymptotic (boundary) states are dual to bulk Geometry

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Plan String HS cubic couplings

Limiting couplings

Higher-Spin Noether Currents

Noether procedure A toy-model

S-matrix amplitudes or Lagrangians?

HS Four-point functions and corresponding couplings Colored spin-2 or gravity?

Lagrangian non-localities and non-local Geometry

(A)dS by radial reduction!

Outlook

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(Open) Bosonic-String S-matrix

Gauge fixed version of the Polyakov path integral

Vertex operators associated to asymptotic states via the state-operator isomorphism

Sopenj1¢¢¢jn =

Z

Rn¡3

dy4 ¢ ¢ ¢ dyn jy12y13y23j

£ h Vj1(y1)Vj2(y2)Vj3(y3) ¢ ¢ ¢ Vjn(yn) iTr(¤a1 ¢ ¢ ¢¤an) + (1 $ 2)

yij = yi ¡ yj

Chan-Paton factors

(L0 ¡ 1) jÁi = 0 L1 jÁi = 0 L2 jÁi = 0

Fierz-Pauli equations for massive fields!

Z »Z

dy4 : : : dyn exp

2

4¡1

2

nX

i 6=j

®0 p i ¢ p j ln jyij j ¡p

2a0»i ¢ p j

yij+

1

2

»i ¢ »jy2ij

3

5

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Generating Functions For external states of the first Regge trajectory of the open bosonic string

Unphysical dependence on the unintegrated yi ‘s

Convenient simplification: Commuting variables

Á i(p i; » i) =X

n

1

n!Ái ¹1¢¢¢¹n » ¹1

i : : : » ¹ni = Á i + » ¹

i Á i ¹ +1

2» ¹1

i » ¹2

i Á¹1¹2 + : : :

An = [Á 1(» 1) : : : Án(»n)] ? 1:::n Zn(» 0i)

Á(»1) ?1 Ã(» 01) = exp

µ@

@» 1¢ @

@» 01

¶Á(» 1)Ã(» 0

1)¯¯»=0

= Á à + Á¹ ù +1

2Á¹1¹2 ù1¹2 + : : :

Generating Function

One gets:

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Three-point Amplitudes

¡p 21 =

s1 ¡ 1

®0 ¡ p 22 =

s2 ¡ 1

®0 ¡ p 23 =

s3 ¡ 1

®0

Zphys » exp

(r®0

2

µ» 1 ¢ p 23

¿y23

y12y13

À+ » 2 ¢ p 31

¿y13

y12y23

À+ » 3 ¢ p 12

¿y12

y13y23

À¶+ (» 1 ¢ » 2 + » 1 ¢ » 3 + » 2 ¢ » 3)

)

Eliminate the unphysical dependence imposing the Virasoro constraints at the level of the generating function Z

On-shell Couplings: Star product with generating functions of fields

p i ¢ » i = 0 » i ¢ » i = 0

A§ = Á 1

Ãp 1;

@

@Ȥr

®0

2p 31

!Á 2

Ãp 2; » +

@

@Ȥr

®0

2p 23

!Á 3

Ãp 3; » §

r®0

2p 12

! ¯¯¯»= 0

Similar results for four-point amplitudes (also Superstring and Mixed-symmetry fields in progress)

As in any S-matrix theory one recover on-shell results (Geometry is hidden…)

Master Thesis(2009)[hep-th/1005.3061]

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Examples A0¡0¡s =

Ãr® 0

2

!s

Á1(p 1) Á2(p 2) Á3(p 3) ¢ ps12 J§(x; ») = ©

Ãx§ i

r®0

2»

!©

Ãx¨ i

r®0

2»

!Induced by a conserved current:

(Berends, Burgers and Van Dam, 1986; Bekaert, Joung, Mourad, 2009)

= Á(x)¤Á(x) + i

r® 0

2» ¹hÁ¤(x)@¹Á(x) ¡ Á(x)@¹Á¤(x)

i+ : : :

The complex scalar HS currents

0-0-s:

A§s¡1¡1 =

çr

®0

2

!s¡2

s(s¡ 1) A1¹ A2 º Á¹º¢¢¢p s¡212

+

çr

®0

2

!s £A 1 ¢ A 2 Á ¢ p s

12 + sA 1 ¢ p 23 A2 º Á º¢¢¢p s¡112 + sA 2 ¢ p 31 A1 º Á º¢¢¢p s¡1

12

¤

+

çr

®0

2

!s+2

A 1 ¢ p 23 A 2 ¢ p 31 Á ¢ p s12

1-1-s:

The amplitudes can contain extra “stuff” that drops out in the massless limit where genuine Noether interactions ought to be recovered! Similar to a scaling limit

This coupling too is induced by a conserved current!

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HS cubic couplings A gauge invariant pattern is recovered!

Gauge invariant up to the linearized massless Eom’s. Off-shell completion uniquely fixed up to partial integrations and field redefinitions.

Non-abelian deformation of gauge symmetry!

The result is:

A = Á 1 (p 1; » 1) Á 2 (p 2; » 2) Á 3 (p 3; » 3)

? 123 exp

(r®0

2

h» 1 ¢ p 23 + » 2 ¢ p 31 + » 3 ¢ p 12 + » 1 ¢ » 2 » 3 ¢ p 12 + » 2 ¢ » 3 » 1 ¢ p 23 + » 3 ¢ » 1 » 2 ¢ p 31

i)

G=

Coupling function: completely arbitrary in Field Theory (?)

+

Lego bricks of any scattering

amplitude!

hep-th/1006.5242: A.Sagnotti and M.T.

Does string theory results from the spontaneous breaking of HS gauge symmetry?

D. Gross; E.S. Fradkin; …

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Cubic vertices V 3 » @ k 1 Á s 1 @ k 2 Á s 2

@ k 3 Á s 3

s1-s2-s3 couplings

V 3 » (G) 0 Á 1 Á 2 Á 3

Number of derivatives

S1+S2+S3

s 1 ¸ s 2 ¸ s 3

V 3 » (G) 1 Á 1 Á 2 Á 3 S1+S2+S3-2 A 1 ¢A 2 A 3 ¢ p 12 + cyclic

A 1 ¢ p 23 A 2 ¢ p31 A 3 ¢ p 12

p ij = p i ¡ p j

V 3 » (G) s 3 Á 1 Á 2 Á 3 S1+S2-S3

. . . . . .

Minimum number of derivatives

No minimal coupling of HS with gravity, but multipolar couplings are possible! Generalizes Weinberg-Witten Theorem…

G » p @ » 1 @ » 2 @ » 3

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Noether Procedure Perturbative approach to Geometry (non-linear gauge symmetry)

L = L (2) + L (3) + L (4) + : : :

±© = ± (0)© + ± (1)© + ± (2)© + : : :

Usual linearized gauge transformations: ± (0)© = @ ¤

Fully non-linear deformation that sums up to what we call

Geometry Free part:

L » ©¤©

Finding a solution order by order!

± (1) L (2) + ± (0) L (3) = 0

± (2) L (2) + ± (1) L (3) + ± (0) L (4) = 0

± (3) L (2) + ± (2) L (3) + ± (1) L (4) + ± (0) L (5) = 0

Increasingly complicated as soon as higher orders are considered…

…what is the logic behind it?

……

Non-homogeneous contributions!

±L (2) » ¤© ¼ 0

Free differential algebras over a nilpotent derivation Q capture exactly this structure (CSFT)

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A toy model: Scalar Yang-Mills We already know the answer! L =

1

4

¡F a¹º

¢2 ¡ 1

2(D¹©a)2 + : : :

Higher-order corrections

L (2) =1

2

¡Aa

¹ p 2 A a¹ ¡ ©a p 2 ©a¢

color ordering: manifest cyclicity!

1

2Tr¡A¹ p 2 A¹ ¡ © p 2©

¢

A¹ = A¹ T a

p ¢ A = 0

± (1) L (2) + ± (0) L (3) = 0

p ¢ @A

p i ¢ @Ai L (3)(A1; A2; A3) ¼ 0The cubic coupling is the solution of simple linear homogeneous differential equations

Convenient simplification: the solution is a function of simple building blocks…

© » 1

G (0;1)123 = A 1 ¢ p 23 © 2 © 3 G (0;2)

123 = A 2 ¢ p 31© 3 © 1 G (0;1)123 = A 3 ¢ p 12© 1 © 2

G (1)123 = A 1 ¢A 2 A 3 ¢ p 12 + A 2 ¢ A 3 A 1 ¢ p 23 + A 3 ¢A 1 A 2 ¢ p 31

L (3) = a³G (0;1)

123 ; G (0;2)123 ; G (0;3)

123 ; G (1)123

´

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±(1)A =

³@A ± (0)L (3)

´¢ @A

p 2±

(1)© =

³@© ± (0)L (3)

´¢ @©

p 2

A toy model: Scalar Yang-Mills ± (1) L (2) + ± (0) L (3) = 0

± (2) L (2) + ± (1) L (3) + ± (0) L (4) = 0 p i ¢ @Ai L (4)(A 1; : : :) ¼ ¡ ± (1) L (3)

Starting from the quartic order the differential equation is non-homogeneous In general: not easy to find a solution…

…but there is a clear logic!

Split it into non-local contributions L (4) = L (4)part + L (4)

homo

p i ¢ @AiL (4)

part(A 1; : : :) ¼ ¡ ± (1) L (3)

p i ¢ @Ai L(4)homo(A 1; : : :) ¼ 0

The particular solution to the non-homogeneous equation is always

given by minus the current exchange contribution and is entirely specified

by the lower-point couplings!

The whole Noether procedure is reduced to linear homogeneous differential equations

hep-th/1107.5843: M.T.

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A toy model: Scalar Yang-Mills

L (4)part =

1

s¡¡

L (4)homo =

1

s++ +

We can characterize any contact Lagrangian quartic coupling as the counterterm compensating the violation of the linearized gauge invariance of

the current exchange part

Color-ordering contribution: only two channels contribute

hep-th/1107.5843: M.T.

1

u

1

u

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A toy model: Scalar Yang-Mills

1

s+

1

u±(0)4

= 2 A1 ¢A3 A2 ¢A4 ¡ A1 ¢ A4 A2 ¢ A3 ¡ A1 ¢ A2 A3 ¢A4

+ 2 (A1 ¢ A3 ©2 ©4 + A2 ¢ A4 ©1 ©3) ¡ A1 ¢ A2 ©3 ©4

¡ A2 ¢A3 ©1 ©4 ¡ A3 ¢A4 ©1 ©2 ¡ A4 ¢A1 ©2 ©3

¡ 2 A2 ¢p 4 + A1 ¢p 4 + A3 ¢p 4 ¡ 2 A1 ¢A3 A2 ¢p 4 + A1 ¢A2 A3 ¢p 4 + A2 ¢A3 A1 ¢p 4

Generating function: formal sum of all scattering

amplitudes with various external states

Two gauge bosons two scalars Four gauge bosons

Is the solution obtained at the cubic level

The procedure stops for YM: we recover the full 4-coupling in color-ordered form!

© » 1

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S-matrix amplitudes or Lagrangians? All the physical information encoded within the solution of homogeneous equations!

p i ¢ @Ai L (4)(A 1; : : :) ¼ 0

With the maximal transverse gauge-fixing coincides with the S-matrix amplitude

The Lagrangian coupling can be extracted subtracting the current exchange part…

…two faces of the same medal!

Gauge boson generating function:

G1234(p i; A i) = ¡ 1

sG12a ? a Ga34 ¡

1

uG41a ? a Ga23 + V (4)

1234

+ +Splits into 3 contributions

respectively linear, quadratic and quartic in Aµ

Generating functions are a convenient language to express these results

G123 = 1 + A 1¢p 23 + A 2¢p 31 + A 3¢p 12 + A 1¢A 2 A 3¢p 12 + A 2¢A 3 A 1¢p 23 + A 3¢A 1 A 2¢p 31

G1234(p i; A i) =

4Y

i=1

[A i(p i) ¢ » i + Á i(p i)] ?1234 G1234(p i; » i)

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Symbol Calculus Convenient simplification: A¹ ! »¹

Weyl-Wigner calculus!

An = [Á 1(» 1) : : : Án(»n)] ? 1:::n Zn(» 0i)

Generating Function

Á i(p i; » i) =X

n

1

n!Ái ¹1¢¢¢¹n » ¹1

i : : : » ¹ni = Á i + » ¹

i Á i ¹ +1

2» ¹1

i » ¹2

i Á¹1¹2 + : : :

Commuting auxiliary variables

YM: G123(p i; A i) =

3Y

i=1

[A i(p i) ¢ » i + Á i(p i)] ?1234 G123(p i; » i)

Straightforward to generalize to HS (most general QFT)

Á(»1) ?1 Ã(» 01) = exp

µ@

@» 1¢ @

@» 01

¶Á(» 1)Ã(» 0

1)¯¯»=0

= Á à + Á¹ ù +1

2Á¹1¹2 ù1¹2

+ : : :

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HS four-point functions ST suggests how to provide an answer to all orders:

G123= +

YM: Lego bricks of any scattering

amplitude!

Noether procedure indeed is solved by any generating function satisfying:

Simpler since amplitude and couplings coincides at the

cubic level

p i ¢ @ »i L (3)(»1; »2; »3) ¼ 0 L (3) = a³G (0;1); G (0;2); G (0;3); G (1)

´

Something analog happens at the quartic order but not at the level of the couplings!

p i ¢ @»i A (4)(» 1; » 2; » 3; » 4) ¼ 0

u + + susA (4)(»i) =1

suexp

Gen. function modulo arbitrary functions of Mandelstam variables!

hep-th/1107.5843: M.T.

suG1234(p i; »i)

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Colored spin-2 or Gravity? Four spin-2 case: two different options!

At the fourth order two different color orderings are independent!

u + + sus

2

A1234=

Planar! (Open-string like):

A=

Non-planar (closed-string like):

+ cyclic

a(s; t; u)

su

a(s; t; u)

stuu + sus t + + sts+

Arbitrary function with no poles left free at this stage

Only pure YM contributes!

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Colored spin-2 or Gravity? Let us expand the result looking at the current exchange part!

A1234= ~a(t)

suu

2

+ s~a(t)

su

2

+ Local terms

A= ~a(t)

s

2

+ cyclic channels + Local terms

Can we read the exchanged particles? t®+ 3 t®+ 2

The second amplitude is the Gravitational one…

…but the first one is not!

Massless spin-2 not necessarily Gravity in HS theories (mixing?) But non-localities are needed on the Lagrangian side (see Vasiliev’s system…)

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Why non-localities? From the S-matrix perspective everything is standard…

…but if we extract the Lagrangian coupling explicit non-localities arise!

Crucial observation (as in the YM case):

V4 =

s1

s2

s4

s3

s1

s2

s4

s3

¡

The Noether procedure requires the full current exchange!

Non local 4-p coupling If the first term does not factorize on all available exchanges!

Only possible way out to bypass Weinberg argument!

…but this kind of structure forces infinitely many spins propagating!

NON-LOCAL Geometry!

smin =1

2(s1 + s2 + s3 + s4) ¡ 1 (2)

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Off-shell HS Couplings Strategy: Find counter terms proportional to traces and divergences ensuring gauge invariance up to full free Lagrangian Eom’s

A§ » exp

(r®0

2G123

)2

41 +® 0

2H12H13 +

Ãr® 0

2

!3

H21H32H13 + Cyclic

3

5

? 123 Á 1 (p 1; » 1) Á 2 (p 2; » 2) Á 3 (p 3; » 3)

Hij = (»i ¢ »j + 1) iDj ¡ p j ¢ »iAj

Generalized de Donder operator A-operator (~ trace, …)

The physical content is totally encoded into G!

± (0)(D ? Á) = p 2 ¤ ± (0)(A ? Á) = ¡ p ¢ @» ¤

The same procedure works also at higher orders! Nice free differential algebra behind

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(A)dS cubic vertices Remarkably from the flat space result one can extract all (A)dS vertices

Trick: ambient space formulation

General isomorphism between ambient space fields and (A)dS fields

(X ¢ @X ¡ ¢h) ©(X; ¥) = 0

X ¢ @¥ ©(X; ¥) = 0

One can write (A)dS Lagrangians using the measure

Zdd+1X ±

³pX 2 ¡ L

´

(A)dS gauge invariance becomes very simple ±©(X; ¥) = ¥ ¢ @X E(X; ¥)

Cubic (A)dS HS couplings coincides with the flat ones modulo a fixed boundary term!

G (A)dS123 = G °at

123 + @X ¢ (®1 ¥1 + ®2 ¥2 + ®3 ¥3 + : : :)

(to appear: E. Joung and M.T.)

This kind of approach can shed some new light on Vasiliev’ s system!

Similar logic to the massless limit of the ST vertex! The lower derivative tail disappears whenever Λ goes to zero while the G operator is disentangled!

Outlook All three-point couplings (abelian and non-abelian!)

Starting from ST all reference to the mechanical model completely eliminated

Four-point functions by Noether procedure

The maximally gauge fixed theory contain so far all the physical information

Behind ST there are key field theory properties!

Systematics of (A)dS cubic couplings

BEHIND THE CORNER:

Classifying the consistent coupling functions

Similar results on (A)dS by radial reduction

Loop computations can shed light on mass-generation!

Non-local HS Geometry! 23

Full systematics of HS theories (work in progress) For the near future:

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