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 KawaiLewellenTye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are biadjoint 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 0corrected biadjoint 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 biadjoint 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 YangMills theory [2, 3].
YangMills2
biadjoint scalar
we can write:
biadjoint scalar theory.
Closed string =
Open string2
0corrected biadjoint 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 eldtheory.
Review of the biadjoint scalar theory
to them as amplitudes of an
0corrected biadjoint 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)! YangMills 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 biadjoint amplitudes:
m(12354j12345) =
m(12354j12435) =
case [21].
studied in [32]. BerendsGiele 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 the leg e. In this way, we have introduced an
0se), where se = pe2=2 is
Let us now dissect each diagram in turn. Firstly,
m 0 (I6j126435) =
sin s12 sin s34 sin s345
m 0 (I6j126345) =
m 0 (I7j1276345) =
sin s12 sin s612
sin s12 sin s612
tan s34
tan s45
consideration is just a product of propagators given by sines. Secondly,
sin s12 sin s345
tan s34
tan s45
tan s67
tan s712
: (3.12)
sin s12 sin s345
= 2:
Here we have used (3.4) twice.
draw the permutation
on a circle, and then follow the points according to the other
w(I6j126345) =
the polygons form a loop.3 We have, for example:
= 0: (3.14)
with zero entries, which are easier to invert.
Diagonal amplitudes
Without loss of generality we can focus on the identity permutations, i.e.,
= ~ = In. So
= 1;
tan s12
tan s23
m 0 (I4jI4) =
m 0 (I3jI3) =
course, the m 0
( j ~) amplitudes could still be written in the expansion of sine propagators
each other.
0se). Hence,
to higher multiplicities.
m 0 (I5jI5) =
tan s12 tan s34
tan s23 tan s45
tan s45 tan s12
tan s51 tan s23
tan s34 tan s51
poles. There is also a contribution that stays
nite on all the factorization channels. It
m 0 (I6jI6) = BB
+ 13 other CC + BB
terms A
terms A
tan s12 tan s34 tan s56
tan s12
As we can see, the presence of the new
vevalent vertex introduced a hierarchy in the
0 ! 0 limit, as expected. There are no new vertices introduced at this stage.
contact terms appearing at higher multiplicities. We nd:
m 0 (I7jI7) = BB
terms A
+ 41 other CC + BB
terms A
m 0 (I8jI8) = BB
m 0 (I9jI9) = BB
+ 131 other CC + BB
+ 428 other CC + BB
terms A
3 = Ck 1;
terms depending on their leading 0 order. Finally:
the diagonal amplitudes m 0
( j ) can be calculated from a graph expansion, where each
conjecture that all the remaining vertices follow this pattern, i.e.,
where Ck is the kth Catalan number [34].
KLT relations
show how to construct KLT relations from the objects m 0
We have veri ed validity of this construction numerically for n
10 and it remains a
( j ~) introduced in this work.
conjecture for higher multiplicities.
relations according to
closed = A
Here, we treat m 0 ( j ~) as an (n
the permutations
3)! matrix with columns and rows labelled by
and ~. There is no restriction on the permutations we use for the open
an invertible matrix.
= ~ = (1234). We
can use the identity sin( z) =
= (z) (1
z) to rewrite the amplitude (3.16),
m 0 (I4jI4) =
0(s + t))
0s) sin(
0u) (1 + 0u)
closed = A
open(1234) m 01(1234j1234) Aopen(1234)
(1+ 0s+ 0t)
0u) (1+ 0u)
(1+ 0s+ 0t)
which is the VirasoroShapiro amplitude [36, 37].
We can also use any other basis for the KLT expansion, for example:
or alternatively,
closed = A
open(1234) BB
open(1324)
0t) Aopen(1234) Aopen(1324);
closed = A
open(1234) BB
open(1243)
0s) Aopen(1234) Aopen(1243):
C A
closed =
" open(12345)# 6
6
open(12435)
open(12435)
2 f(13254); (14253)g. Reinterpreted in our new language, the expression reads
open(14253)
open(14253)
sin s24 sin s35
= sin(
0s23) sin(
0s45) Aopen(12345) Aopen(13254) + (3 $ 4);
blocks, as in the example above.
2 f(13254); (14253)g and ~
2 f(12354); (12435)g. We then have:
closed =
open(13254)
open(14253)
open(12354)
open(12435)
open(14253)
0s23) sin(
sin( 0(s14 + s45))
0s45) open(13254) sin( 0s24) Aopen(12435)
A
sin s24 sin s35
open(12354)
open(12435)
sin( 0s14) Aopen(12354) ;
m 0 ( j ~) = 66
: (4.10)
columns are labelled by
and the rows by ~
2 f(123456); (124356); (132456); (134256); (142356); (143256)g
2 f(153462); (154362); (152463); (154263); (152364); (153264)g.
After a tedious but straightforward calculation we obtain:
to calculate the inverse of the rst block:
sin s12 sin s34 sin s345
sin s12 sin s345
tan s34
tan s45
sin s12 sin s345
tan s34
sin s12 sin s34 sin s345
sin s12 sin s35 sin s45
sin s12 sin s45 sin(s34 + s35)
sin s12 sin s35 sin(s34 + s45)
sin s12 sin s35 sin s45
closed =
0s12) sin(
0s45) Aopen(123456) sin(
0s35) Aopen(153462)
0(s34 + s35)) Aopen(154362) + P(2; 3; 4);
over permutations of f2; 3; 4g.
Soft limits of closed strings
particles a; b 6= 1; n
2 f(1; !a; n
1; a; n)g
2 f(1; !~b; n
1; n; b)g;
where !a and !~b denote the permutations of the remaining n
4 labels. We can arrange
the matrix m 0
( j ~) so that a and b label its (n
4)! blocks. Let us consider the
soft limit of the particle n, pn =
! 0. We keep 0 nite.
particles in both orderings
. Therefore, there are no trivalent vertices involving
blocks go as O( 0) is the soft limit.
allowed to interact via higherorder vertices, but these give rise to
nite contributions.
Hence, the diagonal blocks behave as O(
1) in the soft limit.
More precisely, near
= 0 we have:
m 0 (1; !a; n
1; a; nj1; !~b; n
1; n; b) !
m 0 (1; !a; n
1; aj1; !~a; n
1; a) + : : : ;
4)! diagonal blocks on the right hand side becomes a small inverse KLT
matrix for n
1 particles.
as follows [9]:
open(1; !a; n
1; a; n) !
open(1; !a; n
1; a) + : : : ;
string amplitude with n
1 particles. Working to leading order and neglecting constant
factors, let us now collect all the terms together to obtain the result,
~ is the polarization vector of the massless state. In the last line we
amplitudes [39], coinciding with the pure gravity result.
Basis expansion
Open string partial amplitudes can be expanded in a basis of size (n
3)! [21, 45]. This can
way, utilizing the object m 0 ( j ~) introduced in this work.
Let us consider an (n
3)! vector A
3)! matrix m 0 ( j ~) with
open( ), and additionally (n
3)! + 1 matrix constructed from four blocks: an
and ~ ranging in a set forming a BCJ basis, an
3)! transposed vector m 0
= 0;
rst equality. Since the sets over which
determinant of m 0 ( j ~) is nonvanishing, so we conclude that:
and ~ range form a BCJ basis, the
which is a basis expansion for the open string amplitude A
It is a direct analogue of the expansion (2.8) for the
eldtheory amplitudes. Using the
open( ) in terms of a BCJ basis.
following, we illustrate the expansion (5.1) with a few examples.
open(1234)g.
This corresponds to taking
open(1243) in the oneelement basis
= f(1243)g,
= f(1234)g, and say
~ = f(1324)g. We obtain:
A slightly more involved example is:
open(1243) = BB
open(1324) = BB
open(1234):
open(1234):
open(1234)
C A
open(1234)
open(1234)
inverted only once.
to string theory.
Making the choice
f(13254); (14253)g we get:
tan s23 7 6 sin s23 sin s45
sin s24 sin s35
open(14253);
open(14253)
open(14253)
open(12354) = 66
= 6
tan s12
sin s12 sin s35
0(s12 + s23))
open(13254)
can be graphically summarized in a cartoon:
closed is understood as gluing of two open string partial
open( ~). The object stitching the two amplitudes together is
Here, the closed string amplitude Mn
amplitudes, A
the KLT kernel, m 01
gluing, with
and ~ each ranging over a set of (n
3)! permutations. Similar picture can
( j ~). The sum proceeds over all independent ways of performing the
be made to intuitively understand the change of basis relation (5.1).
argued that m 0
( j ~) can be understood as an interesting object on its own right. In fact,
present some supporting evidence for this conjecture below.
associated to Selberg integrals [49, 50].4
program of
light on the relations between the two.
4We thank Matilde Marcolli and Oliver Schlotterer for pointing out these references to us.
SUGRA = A
open( ) = Z
KLT
also equivalent to a change 0 !
0. In addition, it is known [8{11] that open string
a string theory KLT of open string amplitudes with a small \twist,"
Its disk integral representation, up to an overall factor, reads:
Z ( ) =
vol SL(2; R) z 1; 2 z 2; 3
Note that the two orderings play di erent roles. The permutation
gives a disk ordering
that is inherited by the open string. The ordering
enters the ParkeTaylor factor in
SUGRA = ASYM KLT
and (6.2) to obtain:
ASYM, it is a simple linear algebra exercise to combine it with (6.1)
m( j~) = Z( )
or equivalently
m 0 ( j ~) = Z
KLT
Due to the relation (6.4), the 0corrected biadjoint amplitudes m 0
disk orderings.
Furthermore, using (6.2), one can show that (5.1) implies,
which is distinct from the change of basis for the other permutation [9],
Z ( ) = m( j~) m 1(~j") Z ("):
from [62], one nds,
of (6.8).
P ( j ) =
in this note.
Acknowledgments
and Science.
Proof of the sign of m 0 ( j ~)
where w( j ~) is the relative winding number between permutations
and ~, see (3.13) for
su cient to show that (A.1) works for the eld theory biadjoint scalar.
Let us consider any Feynman diagram consistent with both orderings,
vertex carries a factor f abcf~a~~bc~. There are two options.
When the labels of the three
and ~. Each
minus sign.
We can now reshu e the permutation ~ into
by a series of ips. A single ip is a
which necessarily corresponds to
the overall sign of the initial con guration is ( 1)# ips.
For the diagonal con guration,
de nition (A.1) computes the correct sign.
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