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JHEP05(2016)186
In the following in section II, we firs t review historically two independnt developments of
the string BCJ relations from field theory and from stri n g theory point of views.
2 Review of high energy string BCJ
The four point BCJ relations [
3] for Yang-Mills gluon color-stripped scattering amplit ude s
A were pointed out and calcul ate d in 2008 to be
tA(k
1
, k
4
, k
2
, k
3
) − sA(k
1
, k
3
, k
4
, k
2
) = 0, (2.1a)
sA(k
1
, k
2
, k
3
, k
4
) − uA(k
1
, k
4
, k
2
, k
3
) = 0, (2.1b)
uA(k
1
, k
3
, k
4
, k
2
) − tA(k
1
, k
2
, k
3
, k
4
) = 0, (2.1c)
which relates field theory scattering ampl i t ude s in t he s, t and u channels. In the rest of
this paper, we will discuss the relation for s and u channel amplitudes only. Other relations
can be si mi l ar l y addressed.
2.1 Hard string scatterings
For string theory, in constrast to the field theory BCJ relations, one has to de al with
scattering ampl i t ude s of infinite number of higher spin string states. The first “string BCJ
relation” discovered was the four point string BCJ relation in the HSS limit [
8] worked out
in 2006. For the tachyon state , one can express the open string s −t channel amplitude in
terms of the t − u channel amplitude [
8]
T
(4-tachyon)
open
(s, t) =
Γ
−
s
2
− 1
Γ
−
t
2
− 1
Γ
u
2
+ 2
=
sin (πu/2)
sin (πs/2)
Γ
−
t
2
− 1
Γ
−
u
2
− 1
Γ
s
2
+ 2
≡
sin (πu/2)
sin (πs/2)
T
(4-tachyon)
open
(t, u) (2.2)
where we have used the well known formula
Γ (x) =
π
sin (πx) Γ (1 − x)
. (2.3)
The string BCJ relation for tachyon derived above is valid for all energies.
For the N -point open st r i ng tachyon amplitudes B
N
of Koba-Nielson, some authors in
the early days of dual models, see for example [33] , discussed symmetry relations among B
N
functions with different c y c l i c order e x te r nal momenta by using monodr omy of countour
integration of the amplitudes. However, no discussion was addressed for string amplitudes
with infinite number of higher spin massive string states, on which we will discuss next.
Since there is no reliable saddle point to calculate s − t channel HSS amplitudes, for
all othe r higher spin string states at arbitrary mass levels, one first calculates the s − t
– 3 –