Introduction AdS/CFT correspondence and Gluon amplitudes Numerial Solutions of minimal surface in AdS Conclusions and Outlook Discretized Minimal Surface and Gluon Scattering Amplitudes in N=4 SYM at Strong Coupling Katsushi Ito Tokyo Institute of Technology June 29– July 2, 2009@Integrability in Gauge and String Theory S. Dobashi, K.I. and K. Iwasaki, arXiv:0805.3594, JHEP 07 (2008)088 S. Dobashi and K.I., arXiv:0901.3046, Nucl. Phys. B819 (2009) 18 Katsushi Ito gluon amplitudes
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. . . . . .
IntroductionAdS/CFT correspondence and Gluon amplitudes
Numerial Solutions of minimal surface in AdSConclusions and Outlook
.
.
. ..
.
.
Discretized Minimal Surface and Gluon ScatteringAmplitudes in N=4 SYM at Strong Coupling
Katsushi Ito
Tokyo Institute of Technology
June 29– July 2, 2009@Integrability in Gauge and String TheoryS. Dobashi, K.I. and K. Iwasaki, arXiv:0805.3594, JHEP 07 (2008)088
S. Dobashi and K.I., arXiv:0901.3046, Nucl. Phys. B819 (2009) 18
Katsushi Ito gluon amplitudes
. . . . . .
IntroductionAdS/CFT correspondence and Gluon amplitudes
Numerial Solutions of minimal surface in AdSConclusions and Outlook
.
.Gluon Scattering Amplitudes in N = 4 SYM
Planar L-loop, n-point amplitude
A(L)n (k1, · · · , kn) = A(0)
n (k1, · · · , kn)M(L)n (ε)
the BDS conjecture Bern-Dixon-Smirnov, Anastasiou-Bern-Dixon-Kosower
lnMn(ε) =A2
ε2+
A1
ε
− 1
16f(λ)
nX
i=1
“
ln
„
µ2
−si,i+1
«
”2
− g(λ)
4
nX
i=1
ln
„
µ2
−si,i+1
«
+f(λ)
4F (BDS)
n (0) + C
For n = 4
F BDS4 =
1
2log2
“s
t
”
+2π2
3
For n ≥ 5, F BDSn (0) = 1
2
Pni=1 gn,i ( Mandelstam variables:
t[r]i ≡ (ki + ... + ki+r−1)
2)
gn,i = −[n/2]−1X
r=2
ln
−t[r]i
−t[r+1]i
!
ln
−t[r]i+1
−t[r+1]i
!
+ Dn,i + Ln,i +3
2ζ2
Katsushi Ito gluon amplitudes
. . . . . .
IntroductionAdS/CFT correspondence and Gluon amplitudes
Numerial Solutions of minimal surface in AdSConclusions and Outlook
.
.Test of the BDS conjecture (Weak Coupling)
explicit loop calculation
4-point up to 3-loops [BDS]5-point up to 2-loops [Cachazo et. al. , Bern et. al.]n(≥ 6)-point 1-loop [Bern et. al.]
Non-trivial dependence comes from the function of the cross-ratio
uij,kl =x2
ijx2kl
x2ikx2
jl
, (x2ij = t
[j−i]i )
For n = 6, u13,46, u24,15, u35,26 are independent cross ratios.
Katsushi Ito gluon amplitudes
. . . . . .
IntroductionAdS/CFT correspondence and Gluon amplitudes
Numerial Solutions of minimal surface in AdSConclusions and Outlook
Gluon amplitudes = Wilson loop with light-like boundaries
ends at r = R2
zIR→ 0 (zIR → ∞)
yµ: surrounded by the light-likesegments
Mn ∼ exp(−SNG)
k1
k2
r=0
y
SNG: the value of the Nambu-Goto action for the surface surrounded light-likesegments = Area of the minimal surfacestatic gauge: surface y0(y1, y2), r(y1, y2) parametrized by (y1, y2)
SNG =R2
2π
Z
dy1dy2
p
1 + (∂ir)2 − (∂iy0)2 − (∂1r∂2y0 − ∂2r∂1y0)2
r2
Euler-Lagrange equations
∂i
„
∂L
∂(∂iy0)
«
= 0, ∂i
„
∂L
∂(∂ir)
«
− ∂L
∂r= 0,
non-linear differential equations, difficult to solve
Test of the BDS conjecture at Strong Coupling
Katsushi Ito gluon amplitudes
. . . . . .
IntroductionAdS/CFT correspondence and Gluon amplitudes
Numerial Solutions of minimal surface in AdSConclusions and Outlook
.
.Alday-Maldacena’s Solution
4-point amplitude (s = t): s = −(k1 + k2)2, t = −(k1 + k4)