Clustering in the N=4 SYM three point function D. Serban w/ Y. - - PowerPoint PPT Presentation
Clustering in the N=4 SYM three point function D. Serban w/ Y. Jiang, S. Komatsu, I. Kostov, arXiv:1604.03575 Overview Introduction: AdS/CFT integrability The three point function proposal by [Basso, Komatsu, Vieira, 15] Simple
w/ Y. Jiang, S. Komatsu, I. Kostov, arXiv:1604.03575
One loop dilatation operator = integrable spin chain
[Minahan, Zarembo, 02]
String sigma model classically integrable
[Bena, Polchinski, Roiban, 02] [Kazakov, Marshakov, Minahan, Zarembo, 04]
finite-gap solution of the classical sigma model solution in terms of Bethe Ansatz equations string solution
Z = Φ1 + iΦ2 X = Φ3 + iΦ4
Tr ZZZXXZZZXXXZXZZZZ . . .
Weak-strong coupling duality between N=4 SYM in 4d and string theory in AdS5 x S5 background [Maldacena; Witten; Gubser, Klebanov, Polyakov; 98] Solution for the spectrum in the planar limit at any value of λ through integrability
e.g. [Beisert, Staudacher, 03]
3 1 2 initial data: three states with definite conformal dimensions and psu(2,2|4) charges
| | | | Oα(x) , α = 1, 2, 3
and polarizations (or global rotations with respect to some reference BPS state, e.g Tr ZL)
α JA
basic building block; dual to three interacting strings
O X hO1(x1)O2(x2)O3(x3)i = C123() |x12|∆1+∆2−∆3|x13|∆1+∆3−∆2|x23|∆2+∆3−∆1 11
each characterised by a set of rapidities
At tree level the three point function can be computed using gaussian contraction
h ¯ Z(x)Z(y)i ⇠ 1 |x y|2 h ¯ X(x)X(y)i ⇠ 1 |x y|2
Spin chain language: the combinatorics can be expressed in terms of scalar products of states
use Algebraic Bethe Ansatz (ABA) to build and cut the chains into pieces “tailoring” of spin chains
In some special cases, at weak coupling, the correlators allow determinant representations [Foda,
11] whose semiclassical limit is rather straightforward [ESV, 11; Kostov,12; Kostov, Bettelheim, 14] log C123(g) ' I
C(12|3)
du 2π Li2
+ I
C(13|2)
du 2π Li2
1
2 3
X
a=1
Z
C(a)
dz 2π Li2
general structure in agreement with the strong coupling result [Kazama, Komatsu, (Nishimura),
13 & 16]
Generically, the three point function can be written as sum-over-partitions expressions coming from tailoring or from the recent proposal by [Basso, Komatsu, Vieira, 15] Resuming the sum over partitions for large numbers of magnons and taking the semiclassical limit is a major open problem for the three point function
[Basso, Komatsu, Vieira, 15]
the asymptotic part of the three point function can be written as a sum over partitions for the three groups of rapidities
s, u1 = α1 ∪ ¯ α1, u2 = α2 ∪ ¯ α2, u3 = α3 ∪ ¯ α3
1 1
2
2
3 3
, u2
, u3
[C•••
123]asympt =
X
↵i∪ ¯ ↵i=ui 3
Y
i=1
(−1)|↵1|+|↵2|+|↵3| w`31(α1, ¯ α1) w`12(α2, ¯ α2) w`23(α3, ¯ α3) × H(α1|α3|α2)H(¯ α2|¯ α3|¯ α1) .
X
∪
Y H(α1|α3|α2)H(
explicit Ansatz for the hexagon amplitudes obeying form-factor-like axioms
[Basso, Komatsu, Vieira, 15]
the asymptotic contribution is “dressed” by the mirror (virtual) particles (one non-BPS state shown)
bottom mirror excitations physical excitations
[C•]bottom = Z 1
1
dw µ(w) eip(wγ)`B T(w) h6=(w, w) h(u, w3) ,
e.g. bottom channel: sum over the rapidities w and polarisations of the mirror particles emitted by one hexagon and absorbed by the other
generic su(2) three point function: six vertex partition function on a lattice with a conical defect no straightforward way to take the semiclassical limit methods based on Sklyanin’s Separation of Variables integral representation [Jiang, Komatsu, Kostov, DS, 15]
examples shown for su(2), one non-BPS case
[C•
123]asympt ⌘ A =
X
↵[¯ ↵=u
(1)|¯
↵| Y j2↵
eip(uj)`R Y
j2↵,k2¯ ↵
1 h(uk, uj)
hexagon 2-body amplitude: BES dressing phase
h(u, v)su(2) = u − v u − v + i✏ 1 s(u, v)(u, v)
tree level value symmetric part tree level: Izergin determinant
[Escobedo, Gromov, Sever, Vieira, 10-11] [Kostov, 12] [Kostov, Matsuo, 13] [Kostov, Bettelheim, 14]
Y Y Y Y Y Y Y _ Y _ Y _ Y _ Y _ Y _ Y _ Y
l B
3
L
1
L
2
L
2
L
1
L
3
L
+
R
l
2
L1 L
3
L
+
L
l
2
L1 L3 L
+
=
N
X
n=0
1 n! I
Cu n
Y
j=1
dzj 2⇡✏ F(zj)
n
Y
j<k
h(zj, zk) h(zk, zj)
F(z) = eip(z)` µ(z) h(z, u) , h(z, u) ⌘
N
Y
j=1
h(z, uj) µ(z) = (1 1/x+x)2 (1 1/(x+)2)(1 1/(x)2)
with:
A = X
↵[¯ ↵=u
(1)|↵| Y
j2↵
eip(uj))` Y
j2↵,k2¯ ↵
1 h(uj, uk),
sum over partitions coupled contour integrals
is ✏ ∼ 1/L1
integrals are coupled by the factors
∆all(zj, zk) ≡ h(zj, zk) h(zk, zj) =
tree-level contribution
∆all(u, v) = ∆(u, v)
(u − v)2 (u − v)2 + ✏2
∆(u, v) = ] =
→ x(u) = u + q u2 − (2g✏)2 2g✏
g = p λ 4π
is ✏ ∼ 1/L1 A =
N
X
n=0
1 n! I
Cu n
Y
j=1
dzj 2⇡✏ F(zj)
n
Y
j<k
∆(zj, zk),
manipulate the integrals by separating the contours
∆(u, v) = 1 + i✏/2 u − v − i✏ − i✏/2 u − v + i✏ .
catch the poles of this gives rise to clustered integrals, e.g.
I
Cu
I
Cu
dz1dz2 (2⇡✏)2 F(z1)F(z2)∆(z1, z2) = I
C1
I
C2
dz1dz2 (2⇡✏)2 F(z1)F(z2)∆(z1, z2) + I
C2
dz2 (2⇡✏)F(z2)F(z2 + i✏) (
[Moore, Nekrasov, Shatashvili, 98] [Borodin, Corwin, 11] [Bourgine, 14] [Menegelli, Yang, 14] [Basso, Sever, Vieira, 13], …
cluster of length 2
is ✏ ∼ 1/L1
In =
n
X
k=1
X
q1+···qk=n
Cq1,··· ,qk
n k
Y
j=1
I
Cj
dzj 2⇡✏ Fqj(zj) qj
n
Y
i<j
∆qi,qj(zi, zj).
result of clustering:
Fn(z) = F(z)F(z + i✏) · · · F(z + (n − 1)i✏)
∆mn(u, v) = 1 − mn m + n ✓ i✏ u − v + im✏ + i✏ u − v − in✏ ◆
Cq1,...,qk
n
= 1 d1!d2! · · · n! q1 · · · qk = n! Q
l ldldl! .
is the number of clusters of lengths
d q1 ≤ · · · ≤ qk (z ) is the wavefu
sfy q1+· · ·+qk = n.
the clusters lengths can be alternatively described by the number of clusters of each type
· · · { {q1, · · · , qk} = {1 · · · 1 | {z }
d1
, · · · , l, · · · , l | {z }
dl
, · · · } 7! ~ d = {d1, d2, · · · }.
is ✏ ∼ 1/L1
exact result:
Fn(z) = F(z)n + ✏n(n 1) 2 F(z)n−1@zF(z) + O(✏2) ( ∆mn(zi, zj) = 1 mn (zi zj)2✏2 + O(✏3) .
semiclassical expansion: the leading term exponentiates:
log A = I
Cu
dz 2⇡✏Li2 [F(z)] 1 2 I
C×2
u
dzdz0 (2⇡)2 log [1 F(z)] log [1 F(z0)] (z z0)2 + . . . .
subleading terms easily computable:
Q A = X
k
1 k! X
q1,... ,qk k
Y
j=1
I
Cj
dzj 2⇡✏ Fqj(zj) q2
j k
Y
i<j
∆qi,qj(zi, zj)
A
A ' X
k
1 k!
k
Y
j=1
X
qj
I
Cj
dzj 2⇡✏ F(zj)qj q2
j
= exp I
Cu
dz 2⇡✏ X
q
F(z)q q2 = exp I
Cu
dz 2⇡✏Li2[F(z)]
A
mirror contribution given by similar integrals, but now all bound states of a constituents contribute mirror (virtual particle) dynamics:
x[a] = x(u + ia✏/2) x± = x(u ± i✏/2)
x[+a] ! 1/x[+a] x[−a] ! x[−a]
| |
2g'
x cut
[+a]
physical regime mirror regime
[-a]
x cut
u
[C•]bottom = X
~ n
B[~ n] Q
a na!
Q B[~ n] = (−1)n Z 1
1
Y
a na
Y
j=1
dza
j
2⇡✏ µ
a(za j ) g a(za j ) T a (za j ) ×
Y
a 1i<jna
H
aa(za i , za j )
Y
a<b 1ina 1jnb
H
ab(za i , zb j)
measure interaction with physical particles su(2|2) transfer matrix
Q B[~ n] = (−1)n Z 1
1
Y
a na
Y
j=1
dza
j
2⇡✏ µ
a(za j ) g a(za j ) T a (za j ) ×
Y
a 1i<jna
H
aa(za i , za j )
Y
a<b 1ina 1jnb
H
ab(za i , zb j)
H
ab(u, v) ≡ hab(u, v) hba(v, u) ,
Hγ
ab(u, v) ' ∆ab(u[−a], v[−b]) = ∆ab(u 1 2ia✏, v 1 2ib✏)
strong coupling limit:
✏ = 1 2g
the structure of the interaction between integrals is similar to weak coupling and gives rise to a similar clustering phenomenon, in which bound states cluster in larger bound states the new ingredient is the transfer matrix for the bound states a
Ta(u) ⌘ ga(u) Ta(u) = ˜ ga(u) ⇥ (a + 1) a f [a] a ¯ f [a] + (a 1)f [a] ¯ f [a]⇤ .
f(u) = eiG(x) , ¯ f(u) = eiG(1/x) ,
G(x) = 1 i X
j
ln x − x
j
x − x+
j
→ ✏ X
j
x0(uj) x − x(uj)
Sdet(1 z G)−1 = (1 zy1)(1 zy2) (1 zx1)(1 zx2) = X
a
za Ta Str(1 z G)−1 = z d dz log Sdet(1 z G)−1 = X
a
za ta.
The generating functional of transfer matrices at strong coupling:
G = e g diag(1, 1|f, ¯ f)
element of SU(2|2)
tn n = X
~ n : P
a na a=n
(1)k−1(k 1)! Y
a
Tna
a
na! , k
s ~ n = {n1, n2, · · · }
a set of bound states clusters into
{q1, · · · , qm} = {1 · · · 1 | {z }
d1
, · · · , l, · · · , l | {z }
dl
, · · · } 7! ~ d = {d1, d2, · · · }.
e B~
d =
Y
l
1 dl! ⇥
m
Y
j=1
Z dzj 2⇡✏ tqj(zj) q2
j
[C•]bottom = X
~ d
B~
d = exp
Z dz 2⇡✏ X
n
tn(z) n2
[Beisert, 06] [Kazakov, Sorin, Zabrodin, 07]
log[C•]bottom = Z 1
1
dz 2⇡✏ ⇣ Li2 h ei(ˆ
p(2)(1/x)+ˆ p(3)(1/x)ˆ p(1)(1/x))i
− Li2 h ei(˜
p(2)(1/x)+˜ p(3)(1/x)˜ p(1)(1/x))i⌘
− Z 1
1
dz 2⇡✏ ⇣ Li2 h ei(ˆ
p(2)(x)+ˆ p(3)(x)ˆ p(1)(x))i
− Li2 h ei(˜
p(2)(x)+˜ p(3)(x)˜ p(1)(x))i⌘
,
˜ p(1)(x) = 1
2∆ p(x) − Gu(x),
˜ p(k)(x) = 1
2Lk p(x),
k = 2, 3 , ˆ p(1)(x) = 1
2∆ p(x) ,
ˆ p(k)(x) = 1
2Lk p(x),
k = 2, 3 ,
sphere part AdS part
[Basso, Komatsu, Vieira, 15]
beginning of the expansion obtained in
= I
U
dz 2⇡✏ ⇣ Li2 h ei(ˆ
p(2)(x)+ˆ p(3)(x)ˆ p(1)(x))i
− Li2 h ei(˜
p(2)(x)+˜ p(3)(x)˜ p(1)(x))i⌘
U: unit circle in the Zhukovsky plane x(z)
[C•]bottom = exp Z 1
1
dz 2⇡✏ X
n
e gn 1 + 1 f n ¯ f n n2
[Kazama, Komatsu, Nishimura, 16]
matches part of the result of
su(2) and sl(2) sectors
[Kazama, Komatsu, 13; Kazama, Komatsu, Nishimura, 16]
agreement with strong coupling computations from string sigma model
log[C•••
123]asympt su(2)
= −1 ✏ I
Cu1∪u2
du 2⇡ Li2 h ei˜
p(1)
L +i˜
p(2)
L i˜
p(3)
R
i − 1 ✏ I
Cu3
du 2⇡ Li2 h ei˜
p(3)
R +i˜
p(2)
L i˜
p(1)
L
i , (2.7) log[C•••
123]asympt sl(2)
= 1 ✏ I
Cu1∪u2
du 2⇡ Li2 h eiˆ
p(1)
L +iˆ
p(2)
L iˆ
p(3)
R
i + 1 ✏ I
Cu3
du 2⇡ Li2 h eiˆ
p(3)
R +iˆ
p(2)
L iˆ
p(1)
L
i . (2.8)
log[C•
123]bottom su(2)
= 1 ✏ I
U
du 2⇡ ⇣ Li2 h ei(ˆ
p(2)+ˆ p(3)ˆ p(1))i
− Li2 h ei(˜
p(2)+˜ p(3)˜ p(1)(x))i⌘
, log[C•
123]bottom sl(2)
= −1 ✏ I
U
du 2⇡ ⇣ Li2 h ei(ˆ
p(2)+ˆ p(3)ˆ p(1))i
− Li2 h ei(˜
p(2)+˜ p(3)˜ p(1)(x))i⌘
,
AdS part S part
su(2)L su(2)R ' so(4) 2 so(6)
I II
the combinatorics of clusters can be compactly packed as (generalised) Fredholm determinants
1 ✏n
n
Y
j<k
(zj − zk)2 (zj − zk)2 + ✏2 = det
j,k
✓ i zj − zk + i✏ ◆
measure of integration as a Cauchy determinant:
In = (−1)n n! Z
C n
Y
j=1
dzj 2⇡i det ✓ F(zj) zj − zk + i✏ ◆
so that
[Bettelheim, Kostov, 14]
A = Det(1 − K)
K (u) = I
C
dz K(u, z) (z), K(u, v) = 1 2⇡i F(u) u − v + i✏
with K the integral kernel:
Tr log(1 K) =
∞
X
n=1
Z dz Kn(z, z) = 1 ✏ I
C
dz 2⇡ X
n≥1
Fn(u) n2 ,
leading approximation:
Kn(z1, z2) = 1 2⇡i Fn(z1) z1 z2 + in✏,
| | log[C•
123]bottom su(2)
→
1
X
k=0
1 k! X
|I|=k
Z
U−
Y
{i,a}2I
dza
j
2⇡i det
k⇥k
T[a]
a
(za
j )
za
i − zb j − ia✏.
a similar procedure can be applied for the contribution of the mirror particles:
) = Det I +
1
X
a=1
Ka !
Ka(z1, z2) = 1 2⇡i T[a]
a
(v) u − v − ia✏
with
1
X
a=0
T[a]
a
D2a =
1 − gf 1 D22 1 − gf 1 D2 ¯ f
with D the shift operator
[C•
123]bottom su(2)
= Det ⇣ sdet
, G = diag(X, X | Y, ¯ Y ).
functional determinant matrix determinant
! e ⇥
We have computed, in some simple cases:
To be done:
channels?)
Similarities with amplitudes at strong coupling
[Basso, Sever, Vieira, 13] [Fioravanti et al. 15; Belitsky, 15]