Inverse of the string theory KLT kernel
Received: April
Inverse of the string theory KLT kernel
Sebastian Mizera 0 1 2 3
0 Department of Physics & Astronomy, University of Waterloo
1 Waterloo , ON N2L 2Y5 , Canada
2 Perimeter Institute for Theoretical Physics
3 Waterloo , ON N2L 3G1 , Canada
The eld theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the 0-corrected bi-adjoint scalar amplitudes that are exact in 0. We nd compact expressions in terms of graphs, where the standard Feynman propagators 1=p2 are replaced by either 1= sin(
kernel; Scattering Amplitudes; Bosonic Strings; Superstrings and Heterotic Strings
-
0p2=2) or 1= tan(
0p2=2), as determined by a
recuropen string partial amplitudes in a BCJ basis.
Contents
1 Introduction 2 3 4
Review of the bi-adjoint scalar theory
Diagrammatic rules for m 0 ( j ~)
O -diagonal amplitudes
Diagonal amplitudes
KLT relations
Soft limits of closed strings
Basis expansion
Future directions
A Proof of the sign of m 0 ( j ~)
Introduction
\square" of the Yang-Mills theory [2, 3].
Yang-Mills2
bi-adjoint scalar
we can write:
bi-adjoint scalar theory.
Closed string =
Open string2
0-corrected bi-adjoint scalar
an interesting structure. We will explore it in this work.
0s34) tan(
0s51) tan(
0s12) sin(
0s345) tan(
0s12) tan(
0s23) tan(
0s45) tan(
the amplitudes1 we will study,
m 0 (1234j1234) =
m 0 (123456j126345) =
m 0 (12345j12345) =
theory in the in nite tension limit, 0 ! 0.
of closed string amplitudes from the open string ones.
theory and eld-theory.
Review of the bi-adjoint scalar theory
to them as amplitudes of an
0-corrected bi-adjoint scalar theory, since they compute physical quantities at
aa~ b~b cc~, where f abc and f~a~~bc~ are the structure constants
T a n ) Tr(T~a~ ~1 T~a~ ~2
where the sum goes over all the permutations
and ~ modulo cyclicity. The partial
appropriate propagator,
m( j ) =
owing through the edge e. In particular,
overall sign can be determined using the rules described in [6].
We will introduce an
Examples of the amplitudes are as follows:
interaction of the form f abcf~a~~bc~
of the two avour groups.
decomposition:
mfull =
; ~ 2 Sn=Zn
m(123j123) = 1;
m(1234j1234) =
m(12345j13254) =
m(12345j14253) = 0;
m(1234j1243) =
m(12435j14253) =
m(12435j13254) = 0:
corresponding amplitudes vanish.
derivation [6] leads to the eld theory KLT relation,
GR = A
which we take as a de nition of the operation
AfYuMll = P
2Sn=Zn Tr(T a 1 T a 2
YM( ), analogous to (2.1). By m 1
by permutations
larly. The sum over repeated indices
and ~ respectively. The vectors A
and ~ is implied. Here
YM( ~) are de ned
simiand ~ range over sets of
3)! permutations forming a BCJ basis, in which case the (n
3)! matrix
KLT
GR is a pure gravity
amorderings to be
2 f(12345); (12435)g and ~
2 f(13254); (14253)g. We can then use the
GR =
" YM(12345)#| "1=s23s45
YM(12435)
1=s24s35
YM(14253)
= s23s45 A
YM(13254) + s24s35 A
YM(14253):
as long as the matrix is invertible.
relations are linking di erent theories, the kernel stays the same.
YM( ) = m( j ~) m 1( ~j ) A
3)! matrix and
Here, A
YM( ) on the left hand side is a vector of size p, m( j ~) is an p
3)! matrix,
YM( ) is a vector of size (n
combination of (n
3)! Yang-Mills partial amplitudes. The latter forms a basis.
For instance, let us expand A
YM(12354) in the basis fA
YM(13254); A
YM(14253)g. We
can reuse the matrix from (2.7) to write:
YM(12354) =
1=s12s45
1=s23s45
1=s12s35
s12 + s23
YM(13254)
#| "1=s23s45
1=s24s35
YM(14253):
Here, we have also used two extra bi-adjoint amplitudes:
m(12354j12345) =
m(12354j12435) =
case [21].
studied in [32]. Berends-Giele recursion relations were given in [33].
YM(14253)
closed = A
In our normalization,
m 0 (123j123) = 1;
m 0 (1234j1234) =
m 0 (12345j13254) =
m 0 (12345j14253) = 0;
tan( 0s12)
tan( 0s23)
sin( 0s23) sin( 0s45)
m 0 (1234j1243) =
m 0 (12435j14253) =
sin( 0s24) sin( 0s35)
m 0 (12435j13254) = 0:
m 0 ( j ~) =
= ~
1 i<j m pai paj . It is straightforward to
invariants are always multiplied by the factor
0, from now on we will set
0 = 1 for the
n labels. Then, some more interesting examples become:
m 0 (I6j126435) =
m 0 (I6j126345) =
m 0 (I7j1276345) =
sin s12 sin s34 sin s345
sin s12 sin s345
tan s34
sin s12 sin s345
tan s34
tan s45
tan s45
tan s67
tan s712
propagators. This motivates a separate discussion of the two cases.
work, using the diagrammatic rules described below.
O -diagonal amplitudes
We can compute the o -diagonal terms, i.e., 6
m 0 (I6j126345) and m 0 (I7j1276345), we have respectively:
Here, the permutations
and ~ are drawn with black and red lines respectively. The
us all the diagrams that are planar with respect to both
and ~ [6].
the norm of the momentum
owing through th (...truncated)
