V = 4 Scattering Amplitudes and the Deformed Graßmannian

N = 4 Scattering Amplitudes and the
Deformed Graßmannian
Matthias Staudacher
Institut für Mathematik und Institut für Physik
Humboldt-Universität zu Berlin & AEI Potsdam & CERN Geneva
Amplitudes 2014
12 June 2014
1
Based on
L. Ferro, T. L
! ukowski, C. Meneghelli, J. Plefka and M. Staudacher,
1212.0850 & 1308.3494
R. Frassek, N. Kanning, Y. Ko and M. Staudacher,
1312.1693
N. Kanning, T. L
! ukowski and M. Staudacher,
1403.3382
And to appear.
2
Related Work
D. Chicherin, S. Derkachov and R. Kirschner,
1306.0711 & 1309.5748
N. Beisert, J. Brödel and M. Rosso,
1401.7274
J. Brödel, M. de Leeuw and M. Rosso,
1403.3670
3
A Case for 3+1 Dimensions
Nature prefers Yang-Mills theory in exactly 1+3 dimensions:
Coordinates xµ, momenta pµ. So let us stay there!
Split index µ = 0, 1, 2, 3 into spinorial indices α = 1, 2 and α̇ = 1̇, 2̇ .
Interesting bijection R1,3 → H(2 × 2), pµ #→ pαα̇ = pµ (σ µ)αα̇ .
α̇α
Reverse map H(2 × 2) → R1,3, pµ #→ 12 Tr pαα̇ (σ̄ µ) .
Here σ µ = ( , #σ ) and σ̄ µ = ( , −#σ ) with Pauli matrices #σ . Explicitly:
!
"
p0 + p3 p1 − i p2
pαα̇ =
p1 + i p2 p0 − p3
Gluons are labeled by momenta pµ with p2 = pµpµ = det pαα̇ = 0 and
helicity ±1. Momentum factors: pαα̇ = λαλ̃α̇ , shorthand for = |λα%[λ̃α̇|.
4
Super-Spinor-Helicity and Amplitudes
There is a beautiful extension to maximally supersymmetric N = 4 theory:
One introduces for each leg j a Graßmann spinor ηjA where A = 1, 2, 3, 4.
!
!
A
With Pαα̇ = j λj,αλ̃j,α̇ and Qα = j λj,αηjA the (color stripped) tree
amplitudes for n particles are the known [ Drummond, Henn ‘08 ] distributions
n
1
···
2
··· ···
n 1
δ 4(Pαα̇)δ 8(QA
α)
=
Pn({λj , λ̃j , ηj }),
!12"!23" . . . !n − 1, n"!n1"
where !$m" = %αβ λ#,αλm,β and [$m] = %α̇β̇ λ̃#,α̇λ̃m,β̇ .
All external helicity configurations are generated by expansion in the ηjA.
Super-helicity k corresponds to the terms of order η 4k .
5
Graßmannian Integrals and Amplitudes, I
A Graßmannian space G(k, n) is the set of k-planes intersecting the origin
of an n-dimensional space. k = 1 is ordinary projective space.
“Homogeneous” coordinates are packaged into a k × n matrix C = (cai).
C and A · C with A ∈ GL(k) correspond to the same “point” in G(k, n).
α
α
α̇ A
Build super-twistors WjA = (µ̃α
j , λ̃j , ηj ) w. Fourier conjugates λj → µ̃j .
Graßmannian integral formulation of tree-level Nk−2MHVn amplitudes:
!
δ 4k|4k (C · W)
dk·nC
vol(GL(k)) (1 . . . k)(2 . . . k + 1) . . . (n . . . n + k − 1)
The (i i + 1...i + k − 1) are the n cyclic k × k minors.
Integration is along “suitable contours”.
[ Arkani-Hamed, Cachazo, Cheung, Kaplan ‘09 ]
6
Graßmannian Integrals and Amplitudes, II
For “most” points on G(k, n) we may use the GL(k) symmetry to write


C=
Ik×k

c1,k+1 c1,k+2 · · · c1,n
.. 
..
..
...

ck,k+1 ck,k+2 · · · ck,n
The Graßmannian integral simplifies to
'
(k
(n
n
k
+
)
,
*
dc
ai
a=1
i=k+1
A
4|4
A
caiWi
δ
Wa +
(1 . . . k)(2 . . . k + 1) . . . (n . . . n + k − 1) a=1
i=k+1
Fourier-transforming back to spinor-helicity
Nk−2MHVn amplitudes may be obtained.
7
space,
all
tree-level
Symmetries
The amplitudes enjoy N = 4 superconformal symmetry (A, B = 1 . . . 8):
J AB · An,k = 0 ,
with
J AB ∈ psu(2, 2|4)
However, there is also a “non-local” dual superconformal symmetry:
J˜AB · An,k = 0 ,
J˜AB ∈ psu(2, 2|4)dual
with
˜ one obtains Yangian symmetry.
Commuting J and J,
[ Drummond, Henn, Plefka ‘09 ]
∂
With twistor variables WjA and the “local” generators JjAB = WjA ∂W
B,
j
we can succinctly express it as
J
AB
=
n
!
JjAB ,
ˆAB
J
j=1
=
!
JiAC JjCB − (i ↔ j)
i<j
This is how integrability first appeared in the planar scattering problem.
8
Dual Graßmannian Integrals and Amplitudes
In the dual description one can employ 4|4 super momentum-twistors Zj .
With k̂ = k − 2, there is an equivalent “dual” description on G(k̂, n):
[ Mason, Skinner ‘09; Arkani-Hamed et.al. ‘09 ]
4
8
(QA
α)
δ (Pαα̇)δ
"12#"23# . . . "n1#
!
dk̂·nĈ
δ 4k̂|4k̂ (Ĉ · Z)
vol(GL(k̂)) (1 . . . k̂) . . . (n . . . k̂ − 1)
Note that the k = 2 MHV part factors out.
The fact that the two formulations are related by a simple change of
variables is due to dual conformal invariance, and thus Yangian invariance.
9
Deformed Symmetries
[ Ferro, L
! ukowski, Meneghelli, Plefka, MS ‘12 ]
Of particular interest is the central charge generator of gl(4|4):
C=
n
!
cj
with
cj =
λα
j
j=1
∂
A ∂
α̇ ∂
− λ̃j α̇ − ηj A + 2
α
∂λj
∂ηj
∂ λ̃j
For overall psu(2, 2|4) we have C = 0. So we can relax the “local”
condition cj = 0. This deforms the super helicities hj = 1 − 12 cj .
This yields something well-known: The Yangian in evaluation representation. Deforming the cj switches on the parameters vj . More below.
n
n
!
!
!
vj JjAB
JjAB , JˆAB =
JiAC JjCB − (i ↔ j) +
J AB =
j=1
i<j
10
j=1
Deformed Graßmannian Integrals
[ Ferro, L
! ukowski, MS, in preparation ]
One could then ask how the Graßmannian contour formulas are deformed.
c
The final answer is exceedingly simple, and very natural. With vj− = vj − 2j
!
dk·nC
δ 4k|4k (C · W)
−
vol(GL(k)) (1 ... k)1−v1− +vn− . . . (n ... k−1)1−vn− +vn−1
A derivation will be sketched below. With vj+ = vj +
!
cj
2
we can also write
dk·nC
δ 4k|4k (C · W)
+
+
1+vk−1
−vk+
vol(GL(k)) (1 ... k)1+vk+ −vk+1
. . . (n ... k−1)
Note that it is not really the Graßmannian space G(k, n) as such that is
deformed, but the integration measure on this space. GL(k) preserved!
11
Deformed Dual Graßmannian Integrals
[ Ferro, L
! ukowski, MS, in preparation ]
It is equally natural to ask how the dual Graßmannian integrals deform.
Using the parameters vj−, we found
δ 4(Pαα̇)δ 8(QA
α)
−
−
1−v1−k
+vn
!12"
×
!
−
−
1−vn−k
+vn−1
×
. . . !n1"
dk̂·nĈ
δ 4k̂|4k̂ (Ĉ · Z)
−
−
1−vn
+vn−1
vol(GL(k̂)) (1 ... k̂)
−
−
1−vn−1
+vn−2
. . . (n ... k̂ − 1)
There is a similar formula in terms of vj+.
Note that both the MHV-prefactor and the contour integral are deformed.
12
Why?
Why should we consider this deformation? Here are some of the reasons:
• It is fun!
• As we shall see, it is very natural from the point of view of integrability.
• In fact, constructing amplitudes by integrability (arguably) requires it.
• Amplitudes are related to the spectral problem, where it is indispensable.
• Most importantly: It promises to provide a natural infrared regulator!
The last point was our original motivation. Interestingly, we recently
learned that this deformation had been already studied as an infrared
regulator in twistor theory in the early seventies by Penrose and Hodges.
13
Meromorphicity Lost and Gained
Let us take another look at the deformed Graßmannian contour integral:
!
δ 4k|4k (C · W)
dk·nC
−
vol(GL(k)) (1 ... k)1−v1− +vn− . . . (n ... k−1)1−vn− +vn−1
Choosing the parameters vj− to be non-integer, we see that the poles in
the variables cai turn into branch points.
Important point: We can no longer use the BCFW recursion relations, as
they are based on the residue theorem, which does not apply anymore.
Sounds bad?
What we can hope to gain is complete meromorphicity in suitable combinations of the deformation parameters vj−. This should fix the contours.
14
A Toy Meromorphicity Experiment
Consider Euler’s first integral, the beta function B(v1, v2).
! 1
1
dc 1−v
c 1 (1 − c)1−v2
0
(v1 −1)!(v2 −1)!
(v1 +v2−1)! . The analytic continuation
Γ(v1 )Γ(v2 )
Γ(v1 +v2 ) . Meromorphic in both v1 and v2 .
For v1, v2 ∈ N Euler derived
arbitrary v1, v2 ∈ C is
This is not obvious from the integral. This problem was fixed by
!
1
1
dc 1−v
2πiv
2πiv
1
2
(1 − e
)(1 − e
) C c 1 (1 − c)1−v2
for
[ Pochhammer ‘90 ]
:
where the contour C goes at least two times through the cut:
[ Wikipedia, the free encyclopedia ]
15
fined
as
gian Invariants
as Spin
ChainasStates,
I
Yangian
Invariants
Spin Chain
States, I
Frassek, Kanning,
Kanning, Ko,
Ko, MS
MS ‘13;
‘13; Chicherin,
Chicherin,
Derkachov,
Kirschner
‘13 Chicherin,
[[ Frassek,
Derkachov,
‘13
]]
[ Frassek,
Kanning,Kirschner
Ko, MS ‘13;
Derkachov, Kirschner ‘13 ]
...
...
uct,How
generally
and systematically,
Yangian
invariants?
to
construct,
generally
and
systematically,
Yangian ⇤invariants?
,u
)proposed
. . . Ln (u,
v
n) =
to identify
them asto.special
spin-chain
|Ψ!.
.identify
.
.states
.
It was recently
proposed
them as .special
spin-chain states |Ψ!.
dromy eigenproblem s , v
1
1
sk , vk
s n , vn
sk+1 , vk+1
symmetry?
Yangian
appear
spin chains
with
How does
thefor
Yangian
appear
forgl(m|n)
spin chains
with gl(m|n) symmetry?
AB
!
AB
ocal”
generators
J
into
a
Lax
operator
L
(u,
v
):operator Lj (u, vj! ):
j
Package
the
“local”
generators
J
into
a
Lax
j
j
j
lternative way of defining Yangian invariance
for inhomogeneous spin chains
!
X
1
AB
AB AB
AB
M
(u)|
i
=
| i⇤, u
e
J
=
= (u v ) =+1 + u −
eAB
J
=
AB
j
i
!
vj
A,B
he monodromy matrix
is defined as
s, vi
AB
!
AB
!
Then
build
up
a
monodromy
matrix
M
(u,
{v
j }):
a monodromy matrix M (u, {vj }):
Lj (u, vj!i)
atrix around u ! 1 we find
...
!
!
M (u)M(u)
= L1=(u,
)
.
.
.
L
(u,
v
==
L1v(u,
v
)
.
.
.
L
(u,
n n nv)n )
1 1
...
⇤, u
...
...
1 AB
1 AB
(u) =
+ J
+ 2 Ĵ
+ ...
ation
is both
a tensor
product
and
a matrix
product.
u
u
Here
multiplication
is
both
a
tensor
product
and a matrix product.
ith the Lax operators
AB
s1 , v1
sk , vk
sk+1 , vk+1
! | i is
equivalent to demanding Yangian
invariance:
16X
s n , vn
1
2
(r)ÿ
(r)
(s) Mab
(r+s≠q) one
(q≠1)
(q≠1)
(r+s≠q)
yields the
[Mab generators
, Mcd ]min(r,s)
=ÿ of1 the Yangian,
Mcb where
Mad demands
≠ Mcb Mad2
.
(r)
(s)
(r+s≠q)
(q≠1)
(q≠1)
(r+s≠q)
(0)
q=1
[Mab , Mcd ] =
Mcb
M
Mad
.
Mab
= ”ab . ≠ Mcb
(3.27) (3.28)
ad
(3.28)
Yangian
Invariants
asof MSpin
Chain
The formulation of
the algebra in terms
satisfying
(3.28) isStates,
closely related II
to
q=1
(r) the commutation relations for these
Inserting the expansion (3.26) into (3.25), one obtains
ab
generators
(r)
The
formulation
the algebra
in terms
of M
(3.28)
closely reviews.
related toSetting
Drinfeld’s
first of
realization
of the
Yangian
[20].
See also [21]
forisfurther
ab satisfying
min(r,s)1(1)
2 MS
ÿ [20].
Drinfeld’s
realization
of(r)the(s)Yangian
See (q≠1)
[21]
for (r+s≠q)
further
reviews.
Setting
[also
Frassek,
Kanning,
Ko,
‘13;
Chicherin,
Derkachov, Kirschner ‘13 ]
(r+s≠q)
(q≠1)
r = s = 1first
in (3.28)
shows
generate
the
gl(n)
symmetry,
and
the
generators
[Mabthat
, Mcdthe
] = ≠M
M
M
≠
M
M
.
(3.28)
ab
cb
ad
cb
ad
(1)
r with
= s =r1 Ø
in (3.28)
shows that
the gl(n)
symmetry,
and
the the
generators
q=1 generate
ab
2 correspond
to the
its ≠M
Yangian
extension.
We
also note
that
monodromy
with
r Ø 2transform
correspond
Yangian
extension. (r)We this
also note symmetry,
that the monodromy
elements
in to
theofitsadjoint
representation
The formulation
the algebra
in terms of Mab of
satisfyinggl(n)
(3.28) is closely related to
elements transform
in the
adjoint representation
of See
thisalsogl(n)
symmetry,
Drinfeld’s
first realization
of the Yangian [20].
[21] for
further reviews. Setting
The Yangian generators, see above, appear by expanding at u = ∞:
(1)
(1)
r = s = 1 in (3.28)
shows
that the ≠Mab M
generate
[M
≠ Msymmetry,
, the generators
cb (u)”the
ad gl(n)
ad (u)”cband
(1) ab , Mcd (u)] =
AB
AB
AB
AB
= Mcb (u)”
with r Ø 2 [M
correspond
to(u)]
its Yangian
extension.
We
also note
ad ≠ M
ad (u)”
cb , that the monodromy (3.29)
ab , Mcd
elements transform in the adjoint representation of this gl(n) symmetry,
2
(3.29)
1
1 ˆ
M (u) = δ + J
+ ...
+ J
as may be seen by expanding (3.25) in onlyuone of the twouspectral parameters. Condition
as may be seen by expanding (3.25) in only one of the two spectral parameters. Condition
(1)
[M ,factor
M (u)] =
(u)” monodromy
≠ M (u)” , (3.11) built(3.29)
(3.27) is satisfied up to a scalar
byM the
out of the Lax
cd
ad
ad
cb
ab
(3.27) is satisfied up to a scalar factor
by the cbmonodromy
(3.11)
built out of the Lax
AB
operators
(3.21).
We
use
a
normalization
of
the
Lax
operators
for which
(3.27)
as may
seenabynormalization
expanding (3.25) of
in only
of the
two spectral
Condition
operators (3.21).
Webeuse
theone
Lax
operators
forparameters.
which
(3.27)
holds. holds.
Finally,
come
to
objective
of monodromy
this
section,
the
theLax
main equation
(3.27)
isto
satisfied
upprimary
to a scalar
factor of
by this
the
(3.11)
built
out of
of
the
Finally,
wewe
come
thethe
primary
objective
section,
the
role
ofrole
the
main
equation
operators
(3.21).
We
a normalization
of the LaxLet
operators
for which
(3.27) in
holds.
(3.20)ofofsection
section
3.2
from
a use
Yangian
perspective.
us recall
(3.20)
the general
more general
(3.20)
3.2
from
a Yangian
perspective.
recall
in main
the equation
more
Finally,
we come
to the primary
objective ofLet
thisus
section,
the(3.20)
role of the
3
3
contextofof
current
section.
the
braLet
È–|,
theofset
ofineigenvalue
equations
context
thethe
current
section.
Omitting
the
bra È–|,
theus set
eigenvalue
equations
(3.20)
of section
3.2 from
aOmitting
Yangian
perspective.
recall
(3.20)
the more
general
Note that the deformation of the Jˆ
indeed appears naturally.
contextis
of the
current elegantly
section. Omitting theencoded
bra È–|, the set of as
eigenvalue equations
Yangian invariance
now
MabM
(u)|
Í = ”Íab=| ”Íab | Í
ab (u)|
Mab (u)| Í = ”ab | Í
3
(3.30)
(3.30) (3.30)
AB
AB
maybebe
graphically
represented
with
the
of(3.11)
(3.11)
may
graphically
with
the
help
of (3.11)
asas as
may berepresented
graphically
represented
with
thehelp
help
of
M
(u) · |Ψ" = δ
|Ψ"
or even M (u) · |Ψ" = |Ψ"
a
au u
⇤, u⇤,⇤,
a
| Í| Í
⇤, u a
a a
⇤,⇤,
u u
. . . 1.,.v.1
| Í
| Í
=
| Í | Í
=b =
...
bL , vLb
b
b
...
1 , we see that
L , vL
Expanding 1in, v
u≠1
(3.30) is equivalent to 1 , v1 , v
1 , v1
1 1
L , vL
(3.31)
.
...
1 , v1
b
.
. (3.31) (3.31)
L , vL
...
L , vL
L , vL
In usual spin chains we take the trace, and study Tr M
(u) · |Ψ" = t(u)|Ψ".
Expanding in
(r)
Mab | Í = 0
u≠1 ,
we see that (3.30) is equivalent to
≠1
(3.32)
Expanding in 3uIn [22], we
(3.30)
is equivalent
such see
a set that
of equations
is shown
to be satisfied byto
the physical vacuum state of integrable
two-dimensional quantum field theories. (r)
17
Mab | (r)
Í=0
(3.32)
Yangian
and Bethe
Ansatz, I
ngian Invariants
andInvariants
Bethe Ansatz,
I
[ Frassek, Kanning, Ko, MS ‘13 ]
[ Frassek, Kanning, Ko, MS ‘13 ]
Therefore,
of theansatz
algebraic
ansatz may be applied.
machinery
of the machinery
algebraic Bethe
mayBethe
be applied.
Already
in of
thegl(n)
simpler
case of
gl(n)
compact
much of the structure
simpler
case
compact
reps
much
of the reps
structure
of the
Arkani-Hamed,
Bourjaily,
Cachazo,
Goncharov, Postnikov,
Trnka ‘12 ] on-shell
diagramatics is found.
d, Bourjaily,
Cachazo,[ Goncharov,
Postnikov,
Trnka
‘12 ] on-shell
diagramatics
is found.
Wj in rep
the of
fundamental
rep of gl(n).
Letvariables”
us use “twistor
variables”
Wj in the
fundamental
gl(n).
istor
!
"
!
"
n = 2,
k = 1 two-site
The
simplest
n=
2, k = is1 the
two-site
invariant,
with C invariant,
= 1 c12with
, C = 1 c12 ,
s the
|Ψ2,1! "
#
1
dc12 n
1+s2 δ (W1 + c12 W2)
c12
1
2
A
1
2
B
2
C
our Here
is circular
around iszero,
and
sTwo-point
is ainvariant.
Dynkin
the contour
circular
zero,
and
s2label.
∈ Ndecomposition;
is a Dynkin
label.
2 ∈N
Figure 2:around
Yangian
A) Transposition
B) wilted on-shell
diagram; C) on-shell diagram with a perfect orientation.
4.2
n=3, k=1
For the three-particle invariant there are two non-trivial values of k. Let us first take the
permutation
✓
◆
18
1 2 3
=
= (13)(12) ,
(75)
for which k = 1. We specified here the decomposition of this permutation into transpos
The invariant (47) is given by
| i3,1 = B12 (y1 y2 )B13 (y2 y3 )|0i = B12 (s2 )B13 (s3 )|0i
Z
d↵1 d↵2 N |M
/
(W 1 + ↵1 W 2 + ↵2 W 3 )
1+s2 1+s3
↵1 ↵2
Z
dc12 dc13 N |M
/
(W 1 + c12 W 2 + c13 W 3 ) .
2 1+s3
c1+s
c
[ Frassek,
Kanning,
12
13
[ Frassek,
Kanning,
Ko, MS
‘13 ] Ko, MS ‘13 ]
Yangian
Invariants
andAnsatz,
Bethe
Ansatz,
II
Yangian
Invariants
Bethe
Ansatz,
II
Yangian
Invariants
andand
Bethe
II
[ Frassek, Kanning, Ko, MS ‘13 ]
This is exactly the deformed three-point MHV amplitude introduced in [7]. See Figur
the graphical representation of | i3,1 .
The
next
simplest
cases
are
theinvariants
three-site
invariants
with
simplest
cases
the
three-site
invariants
with
n3.=
3. n = 3.
tnext
simplest
cases
are are
the
three-site
with
n
=
The invariant is given by
1
| i = B (y
y )B (y
y )|0i = B ( s )B (s )|0i
!
"
Z
!
"
!
"
d↵ d↵
/,
(W + ↵ W )
(W + ↵ W )
k
=
1
one
gets,
with
C
=
For
1
c
c
k1=one
1 one
gets,gets,
withwith
C =C =1 c112 c12
, 12, 13 Z ↵ ↵
c13 c13
dc dc
(W + c2 W ) . 3
1
2 / 3
1 (W +2 c W 3)
c
c
#
#
A
B
C
This is again the deformed
three-point MHV amplitude
found in [7]. Together
with
dc12dcdc
1213dc13
n
n
is
the
building
block
for
all
deformations
of
on-shell
diagrams,
and
subsequently
all
tr
δ
(W
+
c
W
+
c
W
)
|Ψ! 3,1
δ
(W
+
c
W
+
c
W
)
|Ψ3,1
" ! " 1+s21+s
1
12
2
13
3
1
12
2
13
3
1+s
deformed
theYangian
graphicalinvariant.
representation
see Figure 4. decomposition; B)
2 3
3
Figure
3: amplitudes.
Three-point For
MHV
A) Transposition
c
c12 c12c1+s
on-shell diagram; C) on-shell diagram with a perfect orientation.
13 13
$ $
#% %
$
2
1
1
0
c
13
1 0 c4.313 n=3,
1 k=20, c13
er kforwhile
k
=
2
one
gets,
with
C
=
= 2 one
= with C
for gets,
k = 2with
one Cgets,
=1For theccase,of the three-particle invariant
, with k = 2 we have the permutation
0
0 1 c23 23
0 1 1 2 c23
✓
3
1
2◆
3
3
1 2 3
=
= (23)(12) .
3 1 2
C
# #
A
B
1323dc23
dc13dcdc
n δ n(W + c Wn) δ n (W + c W14 )
|Ψ
!
"
2c23
δ (W1 + 1c13W13
(W2 +MHV
W233invariant.
) 3 A) Transposition decomposition; B)
|Ψ3,2! 3,2
"
Figure
Yangian
3) δ4:3Three-point
1 21+s2
1+sc11+s
1+s
on-shell diagram; C) on-shell diagram with a perfect orientation.
c13 13c23 c23
3,2
12
1
2
23
1
3
1
2
N |M
1 s1 1+s3
1
2
13 23 N |M
1 s1 1 s2
13
23
1
1
12
1
13
2
3
1
23
N |M
2
N |M
2
3
2
23
3
3
3,2
4.4
n=4, k=2
All contours are closed and encircleThezero.
invariant with k = 2 is the first one which, interestingly, cannot be
ontours
closed
encircle
zero. solelyfour-point
ours
are are
closed
andand
encircle
zero.
by using representation labels. It corresponds to the deformation of the four-po
amplitude obtained in [8] and depends on a spectral parameter z. Let us show how it a
the context of this paper. The relevant permutation is
19
✓
◆
Bethe Ansatz, Permutations, and Yangian Invariants
Since we solve
M (u)on-shell
· |Ψ! = diagrams
|Ψ! and not Tr M (u) [Beisert,
· |Ψ!Broedel,
= Rosso]
t(u)|Ψ! the
an
invariance
of
dressed
angian invariance of dressed on-shell diagrams
[Beisert, Broedel, Rosso]
Bethe ansatz is more constraining. Apart from the Bethe roots, we find
angian
algebra
revisited
(generalization
of (4))
n
n
Yangian
algebra
revisited
(generalization
of (4))!
!
−
+
n
n
n
n
X
X
X
(u
−
v
)
(u
−
v
)
=
X
X
X
AB AB
AB AB
AB AB
AC AC
CB CB
j
j
AB (r)
J J= = Ji J,i ,
Ĵ Ĵ= =
Ji JJij Jj (i $(i j)$+j) + vi JiAB
v
J
(r)
i i
j=1
j=1
i=1
i=1 i=1
1i<n
1i<n
i=1
Thus,
Yangian
requires
the
existence
a spin
permutation
We map
the amplitude
toinvariance
antoinhomogeneous
spin
chain.
EachEach
siteofsite
the
chain
is is σ with
We
map
the amplitude
an inhomogeneous
spin
chain.
the
spin
chain
quipped
with with
its central
charge
si and
inhomogeneity
vi . Convenient
variables
equipped
its central
charge
si and
inhomogeneity
vi . Convenient
variables
+
−
vσ(j) s=
vsji
i
±
±
vi =
vi vi=±vi ±
2 2
Exactly
the
condition
of (r)
[ Beisert,
Rossodeformed
‘14 ] deformed
for deformed
on-shell
diagrams.
emanding
Yangian
invariance
(r)
for
three-point
amplitudes
leads
to relations
Demanding
Yangian
invariance
forBroedel,
three-point
amplitudes
leads
to
relations
Showed
to relations
diagramatics
ininto
[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka ‘12 ].
etween
vi and
relations
can be
a graphical
notation
between
vi srelation
and
si . These
canrecast
be recast
into
a graphical
notation
i . These
0
0
v+
1
v+
21
B
B #B
B #
@
@
v3
v13
1 1
v+
C3 C
C
# C
A# $
A
v12
v2
1
v2+
3
1
1
1
$
$
2
2
3
3
20
2
2
3
3
0
1
0
+
+
+
+
+
v1 v1v2 v2v3 v+
B B
C3
B
C
B
$
#
@ #@ # # # # A
v2 v2v3 v3v1 v1
1
C
C
A
Direct Construction of Yangian Invariants
ernative approach
[Chicherin, Derkac
[ Chicherin, Derkachov, Kirschner ‘13 ]
The Bethe ansatz is interesting, but constructing the states is hard.
A more direct method uses an intertwiner, which in twistor variables reads
"u
!
∂
To prove (⇤) we use the fundamental relation
k
Bjk (u) = −W ·
∂W j
Li (u, vi )Lj (u, vj )Bij (vi vj ) = Bij (vi vj )Li (u, vj )Lj (u, vi )
Note u ∈ C. Representation
changing. Satisfies Yang-Baxter. Intertwines:
whichLcan
depicted
uj )L
)Bjk (uj − uk ) = Bjk (uj − uk )Lj (u, uk )Lk (u, uj )
j (u,be
k (u, ukas
vj
Graphical Depiction:
vi
u
=
u
vi
vj
Use to make a Bethe-like ansatz to construct the invariants |Ψ#.
Use intertwining
relation to
M (u)on
· |Ψ#
|Ψ#hand
iff for
“correct”
ūk .
Notice
that inhomogeneities
areshow
exchange
the =
right
side
of the equation
21
General Construction
[ Broedel, De Leeuw Rosso; Kanning, L
! ukowski, MS ‘13 ]
Every on-shell diagram corresponds to some permutation σ. [ Arkani-Hamed et.al. ‘12 ].
Resolve into “adjacent” transpositions: σ = τ1 . . . τP = (j1k1) . . . (jP kP )
Bethe-like ansatz
|Ψ! = Bj1k1 (ū1) . . . BjP kP (ūP )|0!
Bethe-like equations yield ūp = vτp(kp) −vτp(jp) with τp = (j1k1) . . . (jpkp).
22
Example
Let us
quickly look
at n = 4, k = 2:
General
construction
at work
[Kanning, TL, Staudacher]
Permutation:
!
σ =0
Permutation
4,2
Yangian invariant:
Yangian invariant
=@
1 2 3 4
3 4 1 2
1
2
3
4
3
4
1
2
"
= (12)(23)(12)(24)
1
A = (12)(23)(12)(24) .
|Ψ4,2! = B12(v1 − v2)B23(v1 − v3)B12(v2 − v3)B24(v2 − v4)|0!
v2 )B23 (v1
| i4,2 = B12 (v1
v3 )B12 (v2
v3 )B24 (v2
v4 )|0i
On-Shell
diagramatics:
Graphical representation
!
1
2
3
4
2
1
3
4
!
1
2
23
3
4
Contours
As pointed out by
[ Chicherin, Derkachov, Kirschner ‘13 ]
!
Bjk (u) = −W k ·
∂
∂W j
Bjk (u) acts like a BCFW shift:
"u
#
dα αW k ·∂ j
W
"
e
1+u
Cα
α
α
α̇ A
Recall super-twistors ZjA = (µ̃α
w.
Fourier
conjugates
λ
→
µ̃
,
λ̃
,
η
)
j
j
j.
j
j
This is however merely formal, unless the contour C is rigorously specified.
Note that
• a Hankel contour does not work, in general
• for u $= 0 BCFW recursion, based on residue theorem, no longer works
24
The Top-Cell
For the top-cell of the Graßmannian with general n, k the permutation
σ is just a cyclic shift by k. This allows to derive the general deformed
Graßmannian integral stated initially. [ Ferro, L! ukowski, MS, in preparation ]
Important: The top-cell is the deformed tree-level amplitude. BCFWdecomposition breaks down when deforming, as shown in [ Beisert, Broedel, Rosso ‘14 ].
But it is not needed!
5
4
6
3
1
5
2
4
5
(1)
4
6
3
3
1
6
5
2
4
1
2
(1)
(6)
(2)
6
3
(6)
(5)
(3)
(2)
(4)
5
1
4
6
2
3
1
5
4
6
2
3
1
5
(5)
4
(3)
6
3
1
2
(4)
2
25
[ Figure from arXiv: 1401.7274: Beisert, Broedel, Rosso ‘14 ]
Outlook
Establish that the deformed Graßmannian is useful for loop calculations.
26