STRING FIELD THEORY EFFECTIVE ACTION
FOR THE TACHYON AND GAUGE FIELDS
Marta Orselli
Based on:
• Phys. Lett. B543 (2002) 127, in collaboration with:
G. Grignani (Perugia University) , M. Laidlaw (UBC),
and G. W. Semenoff (UBC),
and
• hep-th/0311xxx, in collaboration with:
E. Coletti (MIT), V. Forini, G. Grignani (Perugia University)
and G. Nardelli (Trento University)
secondo incontro del P.R.I.N.
“TEORIA DEI CAMPI SUPERSTRINGHE E GRAVITA`”
Capri, October 2003
PLAN OF THE TALK
•
Motivations
•
Witten-Shatashvili String Field Theory (BIOSFiT)
•
RG
•
non-linear -function
•
=0
•
BIOSFiT
•
Tachyon and Abelian gauge fields
•
Conclusions
String dynamics
scattering amplitudes
Cubic SFT
MOTIVATIONS
Witten 1986
two formulations
CUBIC STRING FIELD THEORY
Abstract definition, complicated
star product. Can be quantized and
reproduce perturbative on-shell
amplitudes.
Witten-Shatashvili 1992
BOUNDARY STRING FIELD THEORY
or Background Independent open
String Field Theory. Directly tied
to world-sheet RG picture. Exact
results for tachyon condensation.
lead to
an effective action for the field representing the bosonic open string modes and
• provide a solution to the problem of what is the configuration space of string theory.
• provide a non-perturbative formulation of string theory.
motivations for our work
Establish a relationship between the effective actions of Cubic SFT and
Witten-Shatashvili SFT.
Correct the result found by Kutasov, Marino, Moore (hep-th/0009148)
linear function
wrong
integral
Study on the disk the relation between string dynamics and RG flow:
how the on-shell scattering amplitudes emerge from the fixed points of the theory.
Calculation beyond II order are very complicated. We arrive at the III order in
BIOSFiT.
Find a correct formulation for the effective action that could be extended to the
non-abelian case
should lead to derivative corrections to the BI action
WITTEN-SHATASHVILI STRING FIELD THEORY
An open bosonic string in 26 dim. contains a tachyon T, a massless gauge field A and
an infinite tower of massive fields.
tachyon
The theory is unstable
Sen’s conjectures on tachyon condensation (A. Sen 1999):
1:the form of the tachyon potential is:
U T   Mf T 
universal function
mass of the D-brane
2:there are soliton configurations of the tachyon field on unstable Dp-branes
– lower dim. branes - .
3:at the new vacuum there are no open string states; it describes the closed string
vacuum.
To demonstrate the validity of these conjectures one can use the Witten-Shatashvili
string field theory. In this theory, the configuration space of the open string field is
seen as the “space of all 2-dim. world-sheet field theories” on the disk. The worldsheet action and correlation functions are given by:
2
d
0 2 V  X 
general boundary perturbation
 S ws

....

dX
e
....
of ghost number 0
free action defining an open + closed
conformal background
V  b1O
Usually V is defined in terms of a ghost number 1 operator O
S ws  S0 
If V is constructed out of matter fields alone, then
 
O  cV
From the action
2
S ws  S0 
d
0 2 V  X 
The boundary term modifies the b.c. on X from the Neumann b.c. (follows from S0)

 r X    r 1  0 to “arbitrary” non-linear condition

 r X   
X

V X 

 
D
The space-time action S(O) is formally independent of the choice of a
particular open string background (Witten ’92) and it is defined trough its
derivative
K
dS O  
2
2
d
0 2
2
d 
0 2  dO Q, O  
Q is the BRST operator
Q
C D
J BRST
Since dO is an arbitrary operator, all solutions of the eq.n dS=0 correspond to
boundary deformations with {Q,  O}=0
2dim. theory is conformal (scale
invariant, =0)
valid string background.
V(X) can be expanded into “Taylor series” in the derivatives of X
V  X   T  X    A  X  X     B  X  X   X     C  X   2 X     ...
the action becomes the functional of the coefficients


S  S T  X  , A  X  ,...
Goal: write S as an integral over the space-time (constant mode of X()) of some
local functional of T(X), A(X),…
X    X    
     0
with the condition
The action is a kind of field theory in space-time
More generally we can parametrize the space of boundary perturbations V by
couplings gi
i
V   g Vi
gi
i
The coefficients are couplings on the world-sheet theory
and are regarded as fields from the space-time point of view.
At the origin, gi=0, the theory is un-deformed
S ws  S0
and in linear approximation the deformation is given by the integral of Vi
S ws  S0  g i  Vi  S0  g i

V  g  g 0
i 
g
For arbitrary perturbation the theory is non-renormalizable, because the
Taylor expansion of V contains an infinite number of massive fields.
But for the case of the tachyon and gauge fields only, the theory is
renormalizable (perturbatively).
In this parametrization, the expression of the action is (Shatashvili ’93)
Witten-Shatashvili action

 
S g   1    i g  i  Z g 
g 

The derivative of the action with respect to the coupling has a zero exactly
where the theory is conformal
S g 
 Gij g 
i
g
j
g   0
this means

j
g   0
because the metric G has to be invertible and non-degenerate, otherwise we would
have an extra zero which cannot be interpreted as conformal field theory on the
world-sheet
at the fixed point
S g 
 0
 Z g 
This action seems to be only formally background independent. In the world-sheet
formalism background independence is manifest, it is lost once we compute the action
S perturbatively.
If the relation between the action S and the partition function Z is true to all
orders in coupling constant, then we recover the background independence.
It seems to depend on the choice of coordinates in the space of boundary interactions
(choice of contact terms).
If we ignore contact terms, then the -function is linear
Do not ignore contact terms (Shatashvili ’93)
Q depends on the couplings
The way to fix the structure of contact terms is that, since dS is a one-form,
whatever choice of contact terms we made in the computation, d of dS should be zero
K
d  dS  d
2
2
d
0 2
2
d 
0 2  dO Q, O  
 0
This leads to the formula with all non-linear terms for the -function
 i  1  i g i   ijk g j g k   ijkl g j g k g l  
where  is the anomalous dimension of the operator corresponding to the coupling gi,
 ijk is the contribution of the 3-point function and so on.
Only relevant coefficients in the formula for the -function are those which
satisfiy the “resonant condtion”
 j   k  i  1
It means that the -function cannot be reduced to the linear part of it by a field
redefinition and the non-linear terms cannot be removed.
It also means that in the expansion of S, coordinates should be chosen in such a
way that the corresponding metric G is invertible and non-degenerate.
WITTEN-SHATASHVILI ACTION
 
 

SS  11   TT  ZT
  A  Z T , A
TT 
A 

Z   [dX ] exp  S ws X 
Partition function
where the action is
2 d
1
S ws X    dd
 X  ,    X  ,   
T  X  
0
4
2
bulk action
  1
interactions
perturbatively super-renormalizable
The bulk excitations can be integrated out to get an effective non-local field
theory which lives on the boundary
X  X cl  X qu
field on the bulk
2
Z
0
  [dX j ] e
Z dir

d  1 j

j
 X i X T  X  J  X 
2  2

The absolute value of the
derivative operator is defined
by the Fourier transform
X cl  X bdry
X qu  0
  2 X cl  0
  [dX j ] e
i       
n
n
2

2
0
ein   
d  1 j

j
 X i X T  X ik  Xˆ 
2  2

2 d
Xˆ j  
X
0 2
zero mode
j
First order

2 d
Z 1 k 
1
   [dX ]  dk1 
T k1  e 0
0
Z dir
2
2
The functional integral over the
non-zero modes of X() gives
Green function
d  1 j

j
 X i X ik Xˆ ik1 X  1  
2  2

Z k 
   dXˆ j  dk1T k1  e
Z dir
1
   
G  1   2    log 4 sin 2  1 2 
 2 
When 1   2 the Green function is not defined
G
Introducing a cut-off , we set
0  2 log 
k12
 G 0  i  k1  k  Xˆ
2
ambiguity in
subtracting the
divergent terms
All the integrals are well defined even for 1   2 in the convergence
region, so we choose to regularize by analytic continuation.
The integrals over the zero modes give a D dim.  function and the result is
2
Z 1 k 
D
  2  T k  k 1
Z dir
From this expression we can identify the renormalized T in terms of the bare
coupling to the lowest order
2
TR k   T k  k
1 k 2
1
anomalous dimension of the tachyon
Second order
2 d d
Z 2  k 
ik1 X  1  ik2 X  2 
ikXˆ
1
2




  [dX ] 
dk
dk
T
k
T
k

e
e

e
1
2
1
2
0
Z dir
4 2 
The functional integral over X gives
Z
2 
k  
Z dir

2
0
where

d 1 d 2
D
2   1   2 








dk
dk
2


k

k

k
T
k
T
k
4
sin


1
2
1
2 R 1 R
2 

4 2
 2 

TR ki   T ki 
k1k 2
ki 2 1
Z 2  k  1
1  2k1k 2 
D
  dk1dk 2 2   k  k1  k 2 TR k1 TR k 2  2
Z dir
2
 1  k1k 2 
for
1  2k1k2  0
From this expression we can identify the renormalized T in terms of the bare
coupling to the second order in perturbation theory
TR k    k
2
1

1
D
1 2 k1k 2  1  2k1k 2  










T
k

dk
dk
2


k

k

k
T
k
T
k

1
2
1
2
1
2


2



2

1

k
k
1 2 

Third order
Z ( 3) ( k )
1
D
   dk1dk 2 dk3 2   (k  k1  k 2  k3 ) T (k1 )T (k2 )T (k3 ) I (k1 ,k2 ,k3 )
Z dir
3!
where I is the integral
 2   1   2 
2 2 k1k2  2 k2k3  2 k1k3
I k1 , k 2 , k3  
d

d

d


1
2
3 sin 
3

2
2 



k1k 2
 2   2   3 
sin  2 



k 2 k3
 2   1   3 
sin  2 



k1k3
The computation of I is highly non-trivial and the result is
I
(1  k1k 2  k 2 k3  k1k3 )(1  2k1k 2 )(1  2k 2 k3 )(1  2k1k3 )
(1  k1k 2 )(1  k 2 k3 )(1  k1k3 )(1  k1k 2  k 2 k3 )(1  k 2 k3  k1k3 )(1  k1k 2  k1k3 )
The convergence is for
1  k1k2  k2 k3  k1k3  0
1  2k1k3  0
all the expressions are understood to be completely symmetrized in the indices 1,2,3
The renormalized T in terms of the bare coupling to the third order is
TR k   

k 2 1

1
D
1 2 k1k 2  1  2k1k 2 
T k    dk1dk 2 2   k  k1  k 2 T k1 T k 2 
2
1  k1k2 
2


1

D
 2 1 k1k 2  k 2 k3  k1k3 


dk
dk
dk
2


(
k

k

k

k
)
T
(
k
)
T
(
k
)
T
(
k
)

I
(
k
,
k
,
k
)
1
2
3
1
2
3
1
2
3
1 2 3 
3! 

RG
STRING DYNAMICS
To compute the Witten-Shatashvili action we need the expression for the function of the tachyon field.
One of the most interesting topics of string theory is the relation between RG
and string dynamics.
The RG -function is defined as   
g i
i
 log 
A practical approach to off-shell string structure would be to obtain the
e.o.m. for the particle fields associated with the string modes and then to
reconstruct the corresponding action.
This action could be an appropriate tree-level action in a field theory
formulation of string theory. However, in general one has
S
 Gij 
i
g
j
where G is some metric.
It is very hard to construct the metric G
The Witten-Shatashvili action provides a prescription for the metric G in the
space of couplings. Then one needs the correct -function.
We managed to prove also a weaker form of the relationship between the RG and
string dynamics: the solutions of the RG fixed point eq.s can be used to generate
the open string scattering amplitudes.
The most general RG eq.s for a set of couplings is
dg i
 
 i g i   ijk g j g k   ijkl g j g k g l  
dt
t   log 
i
i  1  ki 2
The solution of this equations can be written as

g i t   eit g i 0  e
 j  k t
 eit
bare coupling
where

 ijk
g j 0g k 0  bijkl g j 0g k 0g l 0
 j  k  i
i
m
i
m
 j  k  i t
i t




2


2


e
e
jm kl
jm kl
b ijkl  
  ijkl 

  ijkl 
  j  k  l  i
  j  m  i
  j  k  l  i  k  l  m
2 ijm klm
   t

e j m
 j  m  i k  l  m 

 1  2k j k k  
 1 2 k j k k 
D

TR ki   
 ki  k j  k k   2
T ki    dk j dk k 2  T k j T k k 


 2 1  k j k k  
1

 2 1 k j k k  k k kl  k j kl 
D
  dk j dk k dkl 2   (ki  k j  k k  kl ) T (k j )T (k k )T (kl )
I ( k j ,k k , kl ) 
3!

ki 2 1
1  2k j kk  0  1  ki  k j  kk  0  1  k j  1  kk  1  ki   j  k  i
2
2
2
2
2
2
We find
 
i
jk

i
jkl
2  2k j k k 
2 2 1  k j k k 
 ki  k j  k k 
 2  2k j kk  2k j kl  1  2kk kl 

1

 21  k j kk  k j kl  k k kl I 
 cycl  ki  k j  k k  kl 
2
2


3! 
  1  k j kk  k j k k   1  kk kl 

where
I
(1  k1k 2  k 2 k3  k1k3 )(1  2k1k 2 )(1  2k 2 k3 )(1  2k1k3 )
(1  k1k 2 )(1  k 2 k3 )(1  k1k3 )(1  k1k 2  k 2 k3 )(1  k 2 k3  k1k3 )(1  k1k 2  k1k3 )
NON-LINEAR -FUNCTION
T k   1  k 2 T k  

1
2  2k1k 2 
D








dk
dk
2


k

k

k
T
k
T
k
1
2
1
2
1
2
2
 2 1  k1k 2 
1
D


dk
dk
dk
2

 (k  k1  k 2  k3 ) T (k1 )T (k 2 )T (k3 )
1
2
3

3!

 2  2k j k k  2k j kl  1  2k k kl 


 cycl 
21  k j k k  k j kl  k k kl I   2
2


  1  k j k k  k j k k   1  k k kl 

FROM =0
SCATTERING AMPLITUDES
1  k T k   0
Lowest order equation
Next order
2
0
T k   T0 k   T1 k 
k1

k2

1
T1 k  
2 1 k 2


 dk1dk2 2 
D
The residue of the pole is the scattering
amplitude for 3 on-shell tachyons. In our notation is 1/2
T k   T0 k   T1 k   T2 k 
One more order
k
 
 k2
 k  k1  k 2 T0 k1 T0 k 2 
2
2 k 
  
 2 
T2 k  

1

 dk1dk2 dk3 2   (k  k1  k2  k3 )
D
3 k 2 1
T0 (k1 )T0 (k 2 )T0 (k3 )1  k1k 2  k 2 k3  k1k3 I (k1 ,k 2 ,k3 )
The residue of the pole is the
4 tachyon scattering amplitude
Using the on-shell condition we recover the
scattering amplitude for 4 on-shell tachyons
1
1  k1k 2  k 2 k3  k1k3 I (k1 ,k 2 ,k3 )   1 2 B1  2k1k 2 ,1  2k 2 k3   cycl.
3
2 
k2
k1
Veneziano Amplitude
k3
k2
k2
k3
k3
k1
k1

k


k



k

The 4 tachyon amplitude is the
sum of a contact graph and a
tachyon exchange graph
WITTEN-SHATASHVILI ACTION
 T k   1  k 2 T k  

 

S  1    T 
 Z T 

T


1
2  2k1k 2 
D








dk
dk
2


k

k

k
T
k
T
k
1
2
1
2
1
2
2
 2 1  k1k 2 
1
D


dk
dk
dk
2

 (k  k1  k 2  k3 ) T (k1 )T (k 2 )T (k3 )
1
2
3
3! 

 2  2k j k k  2k j kl  1  2k k kl 


 cycl 
21  k j k k  k j kl  k k kl I   2
2


  1  k j k k  k j k k   1  k k kl 


 ( 2  2k 2 ) 1
D
D


S  K 1   dk 2  T (k )T (k ) 2

dk
dk
dk
2

T (k1 )T (k2 )T (k3 ) (k1  k2  k3 ) 
1
2
3
2


(
1

k
)
3
!


 (1  2k2 k3 )(2  2k1k 2  2k1k3 )  (2  2k 2 k3 )(1  2k1k 2  2k1k3 )
 

 
4
(
1

k
k

k
k

k
k
)
I
(
k
,
k
,
k
)


cycl
.

1 2
2 3
1 3
1 2 3
2


(
1

k
k
)

(
1

k
k

k
k
)
2 3
1 2
1 3

 

K  Tp
normalization constant proportional to the tension of the Dp-brane
Exact tachyon potential
U (T )  e T (1  T )
Near the perturbative vacuum, T=0
For k=0
U (T )  1 
(Kutasov, Marino, Moore,hepth/0009148
Gerasimov, Shatashvili, hep1 3
2
th/0009103)
1
T  T  ...
2
3
2  The ratio of the cubic and quadratic term

U (T )  K 1  T 2  T 3  is precisely the one that comes from the
3 

expansion of the exact potential
BIOSFiT
CUBIC SFT
We can compare the Witten-Shatashvili action obtained up to the third order in the
tachyon field to the cubic string field theory action. We have found the off-shell
field redefinition which relates the two formulations. Here I only show how they are
related on-shell.
Near on-shell
S
K
1

D 2
D






dk
2

k

1
T
(
k
)
T
(

k
)

dk
dk
dk
2

T
(
k
)
T
(
k
)
T
(
k
)

(
k

k

k
)
1
2
3
1
2
3
1
2
3 
2  
3 

2
on-shell  constant
has a zero for k  1
This can be compared with the Cubic String Field Theory result. Near on-shell
1
1

D
D
S  2 2T25   dk 2  k 2  1(k )(k )   dk1dk 2 dk3 2  (k1 )(k 2 )(k3 ) (k1  k 2  k3 )
3
2

The required matching of the quadratic and cubic term implies
T
K  25
2
k  
1
 2
T k 
the field redefinition is
non-singular on-shell
This is in agreement with all the conjectures involving tachyon condensation
Provides a further verification of the validity of our expression for the
non-linear -function and the Witten-Shatashvili action.
Moreover this shows that, as expected, the Cubic String Field Theory provides an
effective action for the tachyon to which corresponds a non-linear -function
TACHYON AND GAUGE FIELDS
At the second order

1

2 k1k 2
2


k
k

1 2

2
1
2
D

  k  k  k 
TR k    k 1 T k    dk1dk 2 2 
1
2
2
 1  k1k 2 





k1k 2  k1 k 2
 1 2 k1k 2 
1 2 k1k 2  


 T k1 T k 2 
 A k1 A k 2 
2k1k 2  1

A

R
k   
k2
A

 


k    dk1dk2 2 D T k1 A k2  1 2 k k    k1k2  k1 k2 
1 2
 k  k1  k 2 
1

2 2 k1k 2   k1k 2  
2

 k1k 2 1  k1k 2 

T k   1  k 2 T k  
1
D


dk
dk
2

1
2
2
1

2 2 k1k 2   k1k 2 
2
  k  k  k 
1
2
 1  k1k 2 
1  2k k T k T k   A k A k 

1 2
1
2

1
2

k1k 2  k1 k 2

no T-A term
1

2 2 k1k 2   k1k 2 
D
2

2 




dk
dk
2


k

k

k
 A k   k A k    1 2
1
2
 k1k 2 1  k1k 2 
no T-T and A-A term
1  2k1k2 T k1 A k2   k1k2  k1 k2 
In the expressions for the -functions there are, as expected, only the terms
consistent with the twist symmetry.
At the third order we have been able to show that all the integrals (except for the
momentum dependence) can be expressed in terms of I
 2   1   2 
2 2 k1k2  2 k2k3  2 k1k3
I k1 , k 2 , k3  
d

d

d


1
2
3 sin 
3

2
2 




k1k 2
 2   2   3 
sin  2 



k 2 k3
 2   1   3 
sin  2 



k1k3
(1  k1k 2  k 2 k3  k1k3 )(1  2k1k 2 )(1  2k 2 k3 )(1  2k1k3 )
(1  k1k 2 )(1  k 2 k3 )(1  k1k3 )(1  k1k 2  k 2 k3 )(1  k 2 k3  k1k3 )(1  k1k 2  k1k3 )
CONCLUSIONS
 We obtained the non-linear expression for the -function
of the couplings.
 The string dymanics emerges from the -function fixed
points reproducing the open bosonic string scattering
amplitudes.
 We computed the Witten-Shatashvili action for the
tachyon and the Abelian gauge field up to the third order.
 The Witten-Shatashvili and the Cubic SFT formulations are
shown to be equivalent (at least up to the third order in the
tachyon and gauge fields) up to a field redefinition.
 Our result can be extended to the study of the non-Abelian
case.