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1
String Lessonsfor
Higher SpinsMassimo TaronnaScuola Normale Superiore & INFN, Pisa
Based on:Master Thesis(2009)[hep-th/1005.3061](M.T. advisor: Prof. A.
Sagnotti)&
hep-th/1006.5242: A.Sagnotti and M.T.
2
What is String Theory?
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
A possible way out
Probe String Theory within a regime where HS excitations are relevant
The difficulty
Soft UV behavior, open-closed duality, modular invariance, etc…
String Theory is a consistent Higher-Spin Theory!
3
Higher Spins and Field Theory
Starting Point Symmetry group of space-time
(Majorana, 1932; Dirac, 1936; Fierz-Pauli, 1939; Wigner, 1939; …)
Fundamental one-particle states are labeled by two quantum numbers
Mass:
m2 ¸ 0 Spin:
s = 0;12
; 1;32
; 2;52
; 3; : : :More labels
forD ¸ 5
There is at present no indication about the existence of a preferred subset of values!
The Challenge
Uncover the systematics of Higher Spins
4
Constructing a non-linear Gauge Theory
Postulate some non-abelian algebra and construct the corresponding invariant field theoryAnalyze vertices in a preexisting non linear theory
Classify all consistent deformations of the free theory (Nöther procedure)
Low spin High spin
YANG-MILLS
GENERAL RELATIVITY
SUGRA
VASILIEV’s SYSTEM
STRING (FIELD) THEORY
Bengtsson,; Ouvry and Stern (1986)
. . .M.T. (2009)
A. Sagnotti and M.T. (2010)
Berends, Burgers, van Dam(1985)
Barnich, Hanneaux (1993). . .
Bekaert (2008)Bekaert , Joung, Mourad
(2009)Manvelyan, Mktrchyan, Rühl
(2010)Fotopoulos, Tsulaia (2010)
5
Conjectures and Open problems
Does string theory result from the spontaneous breaking of HS gauge symmetry?
How unique is String Theory?
Using String Theory as a theoretical laboratory one can extract a lot of information about consistent
interactions of HS statesMassive scattering amplitudes and couplings do
not suffer from the usual problems encountered in the massless case
In the massless limit ordinary gauge invariance emerges!
D. Gross; E.S. Fradkin; …
6
Plan Open Bosonic String Tree-level S-matrix
Amplitudes String symbol calculus Generating function for 3-point correlation functions 3-point Amplitudes
String Higher-Spin Couplings Couplings 0-0-s Couplings 1-1-s General case Higher-Spin Conserved Currents
7
(Open) Bosonic-String S-matrix
The starting point is the gauge fixed version of the Polyakov path integral
Vertex operators associated to asymptotic states via the state-
operator isomorphism
Sopenj 1¢¢¢j n
=Z
Rn ¡ 3dy4 ¢¢¢dyn jy12y13y23j
£ hVj 1(y1)Vj 2(y2)Vj 3(y3) ¢¢¢Vj n (yn) iTr(¤a1 ¢¢¢¤an ) + (1$ 2)
yi j = yi ¡ yj
Chan-Paton factors
(L0 ¡ 1) jÁi = 0 L1 jÁi = 0 L2 jÁi = 0
8
Symbol CalculusIt is convenient to introduce the symbols of
the usual string oscillators
In terms of symbols the usual vertex operator correlation function that enters into the definition of the S-matrix
becomes
hÁ1(y1; p1) :: : Án(yn; pn) i = [Á1(»1) :: : Án(»n)] ? Z(»i )
Symbol of the vertex operator
? : Á(»1);Ã(»2) ! Á?Ã = expµ
@@»1
¢@
@»2
¶Á(»1)Ã(»2)
¯¯¯» i = 0
= Á(0) Ã (0) +Á(1) ¹ Ã (1)¹ +
12
Á(2) ¹ 1¹ 2 Ã ;(2)¹ 1¹ 2
+:::
Weyl-Wigner calculus!
Generating Function
Ái (pi ;»i ) =X
n
1n!
Ái ¹ 1¢¢¢¹ n »¹ 1i : : : »¹ n
i = Á(0)i +»¹
i Á(1)i ¹ +»¹ 1
i »¹ 2i Á(2)
¹ 1¹ 2+:::
9
Z = i(2¼)d±(d)(J 0) Cexp
2
4¡12
nX
i6=j
®0pi ¢pj lnjyi j j ¡p
2a0 »i ¢pj
yi j+
12
»i ¢»j
y2i j
3
5
Generating Functions
For external states of the first Regge trajectory of the open bosonic string
Ái (pi ;»i ) =1X
n = 0
1n!
Ái ¹ 1¢¢¢¹ n »¹ 1i : : : »¹ n
i
Unphysical dependence on the yij
10
Three-point Amplitudes
¡ p21 =
s1 ¡ 1®0 ¡ p2
2 =s2 ¡ 1
®0 ¡ p23 =
s3 ¡ 1®0
Zphys = igo(2¼)d
®0 ±(d)(p1+p2+p3) exp
( r®0
2
µ»1 ¢p23
¿y23
y12y13
À+ »2 ¢p31
¿y13
y12y23
À+ »3 ¢p12
¿y12
y13y23
À¶+ (»1 ¢»2 + »1 ¢»3 + »2 ¢»3)
)
L0 constraint: mass parameterization for the
first Regge trajectory
The issue is now to impose the Virasoro
constraints at the level of the generating
function in order to select the physical information that it
containsL1 constraint: transversality
One thus obtains the physical generating function for three-point amplitudes
hxi = sign(x)
L2 constraint: tracelessness
[Á1(»1) Á2(»2) Á3(»3)] ? Zphys
11
Three-point AmplitudesThe amplitudes are given by the
* - product of the physical
generating function with the symbols of the vertex operators
Ái (pi ; »i ) =1X
n=0
1n!
Ái ¹ 1¢¢¢¹ n »¹ 1i : : : »¹ n
i The result is
A = igo
®0 (2¼)d±(d)(p1 + p2 + p3) fA +(p1;p2;p3)Tr [¤a1 ¤a2¤a3 ]+ A ¡ (p1;p2;p3)Tr [¤a2 ¤a1¤a3 ]g
A § = Á1
Ã
p1;@@»
§
r®0
2p31
!
Á2
Ã
p2; » +@@»
§
r®0
2p23
!
Á3
Ã
p3; » §
r®0
2p12
! ¯¯¯¯¯»= 0
Carrying out the * - product one
arrives at
Analogous simplifications for four-point amplitudes and string currents!
12
0-0-s CouplingsIn this case the current generating function is conserved and is given by
J = igo
®0 f J +(x;») Tr[ ¢¤a1 ¤a2 ] + J ¡ (x;») Tr[ ¢¤a2 ¤a1 ]g
J § (x;») = ©
Ã
x § i
r®0
2»
!
©
Ã
x ¨ i
r®0
2»
!coordinate space
(Berends, Burgers and Van Dam, 1986)
(Bekaert, Joung, Mourad, 2009)
= Á(x)¤Á(x) + i
r®0
2»¹
hÁ¤(x)@¹ Á(x) ¡ Á(x)@¹ Á¤(x)
i+ :: :
Example: the complex scalar
HS currents
13
1-1-s CouplingsThis coupling can be extracted from our general result and is given by
A §s¡ 1¡ 1 =
Ã
§
r®0
2
! s¡ 2
s(s ¡ 1) A1¹ A2º Á¹ º ¢¢¢ps¡ 212
+
Ã
§
r®0
2
! s£A 1 ¢A 2 Á¢ps
12 + sA 1 ¢p23 A2º Áº ¢¢¢ps¡ 112 + sA 2 ¢p31 A1º Áº ¢¢¢ps¡ 1
12
¤
+
Ã
§
r®0
2
! s+2
A 1 ¢p23 A 2 ¢p31 Á¢ps12
The on-shell gauge variation with respect to the spin-1 external legs vanishes identicallyOn the contrary the gauge variation with respect to the spin-s external leg gives
±H = p3 ¤
±A §s¡ 1¡ 1 =
Ã
§
r®0
2
! s¡ 2s(s ¡ 1)
2
£A 2 ¢p31 A1¹ ¤¹ ¢¢¢¢ps¡ 2
12 ¡ A 1 ¢p23 A2¹ ¤¹ ¢¢¢¢ps¡ 212
¤
» pi ¢pj ? To recover the massless limit one should take care of such hidden
subleading terms
Simplest non trivial example!
14
Massless limit of the Theory
The procedure just outlined can be implemented for the general s1-s2-s3 coupling!
This amplitude is gauge invariant up to the linearized massless equations and determines uniquely its off-shell
completion, up to partial integrations and field redefinitions
m2 ! ¤
The result is:
A § = exp
( r®0
2
h(@»1 ¢@»2 )(@»3 ¢p12) + (@»2 ¢@»3 )(@»1 ¢p23) + (@»3 ¢@»1 )(@»2 ¢p31)
i)
£ Á1
Ã
p1; »1 +
r®0
2p23
!
Á2
Ã
p2; »2 +
r®0
2p31
!
Á3
Ã
p3; »3 +
r®0
2p12
! ¯¯¯¯¯» i = 0
G: non trivial gauge invariant operator
±i Ái (»i ) = pi ¢»i ¤ i (»i )
Coupling function: completely arbitrary in Field Theory (?)
15
Cubic verticesV3 » @k 1 Ás 1 @k 2 Ás 2 @k 3 Ás 3
s1-s2-s3 couplings
V3 » (G) 0 Á1 Á2 Á3
Number of derivativesS1+S2+S3
s1 ¸ s2 ¸ s3
V3 » (G) 1 Á1 Á2 Á3 S1+S2+S3-2 A 1 ¢A 2 A 3 ¢p12 + cyclic
A 1 ¢p23 A 2 ¢p31 A 3 ¢p12
pi j = pi ¡ pj
V3 » (G) s 3 Á1 Á2 Á3 S1+S2-S3
. . . . . .
Minimum number of derivatives
No minimal coupling of HS with gravity, but multipolar couplings
are possible!
G » p@»1 @»2 @»3
16
HS (bosonic) Conserved CurrentsThe couplings so far obtained are induced by
Noether currents
J § (x ; ») = exp
Ã
¨ i
r®0
2»®
£@³1 ¢@³2 @®
12 ¡ 2@®³1
@³2 ¢@1 + 2@®³2
@³1 ¢@2¤!
£ Á1
Ã
x ¨ i
r®0
2»; ³1 ¨ i
p2®0@2
!
Á2
Ã
x § i
r®0
2»; ³2 § i
p2®0@1
! ¯¯¯¯¯³ i = 0
Conserved up to massless Klein-Gordon equation, divergences and traces, but their completion turns out
to be completely fixed, up to redefinitions!
J (x ; ») = J (0)(x) + »¹ J (1)¹ (x) + :::
Spin one currents
HS currents
17
HS (fermionic) Conserved CurrentsAn educated guess is possible in the
fermionic case
J [0]§F F (x ; ») = exp
Ã
¨ i
r®0
2»®
£@³1 ¢@³2 @®
12 ¡ 2@®³1
@³2 ¢@1 + 2@®³2
@³1 ¢@2¤!
£ ¹ª 1
Ã
x ¨ i
r®0
2»; ³1 ¨ i
p2®0@2
!h1 + =»
iª 2
Ã
x § i
r®0
2»; ³2 § i
p2®0@1
! ¯¯¯¯¯³ i = 0
Conserved up to massless Dirac equation, divergences and (gamma-)traces, but their completion turns out to
be completely fixed, up to redefinitions!
J (x ; ») = J 0(x) + »¹ ( ¹Ã ° ¹ à + :::) + :: :
Spin one currents
HS currents
Analogous result for the currents with one fermion and one boson
Outlook All three-point couplings (both abelian and non-abelian!) Signals of non-abelian (unbroken) gauge symmetry at the
cubic order All four-point amplitudes (similar to the computation of
three-point amplitudes) All reference to the mechanical model completely
eliminated Educated guess for fermionic couplings and related
currentsOn-shell results contain (so far) all the physical information
For the first Regge trajectory of the open bosonic string:
BEHIND THE CORNER:
Consistent HS gauge theory
Non localities can play a crucial role
18V4 = V3
?¤
V3
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