YITP Workshop 2010 Six-point gluon scattering amplitudes from -symmetric integrable model Yasuyuki Hatsuda (YITP) Based on arXiv:1005.4487 [hep-th] in collaboration with K. Ito (TITECH), K. Sakai (Keio Univ.), and Y. Satoh (Univ. of Tsukuba)
AdS/CFT Correspondence Concrete example of gauge/string duality Maldacena ’97 N = 4 SU( N ) Type IIB Strings vs. AdS 5 × S 5 Super Yang-Mills on YM N = 4 π g s N = R 4 λ ≡ g 2 α 0 2 λ In the ’t Hooft limit: with held fixed, planar N → ∞ SYM/free string contribution is dominant The AdS/CFT correspondence is a strong/weak type duality Gauge theory side: weak coupling analysis String theory side: strong coupling analysis difficult to compare
Integrability in AdS/CFT AdS 5 × S 5 N = 4 Planar SYM and string theories have Minahan, Zarembo ’02 integrable structures Bena, Polchinski, Roiban ’03 Integrability is a power tool to analyze the spectrum of both theories Integrability also plays an important role in studying scattering amplitudes Thermodynamic Bethe ansatz (TBA) appears Alday, Gaiotto, Maldacena ‘09 Motivation: With the help of integrability, we would like to find a formulation that connects weak and strong coupling analyses
Alday-Maldacena Program How to compute gluon scattering amplitudes at strong coupling by using AdS/CFT Alday, Maldacena ‘07 There is a duality between gluon amplitudes and expectation values of null polygonal Wilson loops The expectation value of Wilson loop can be computated by the area of minimal surface of open string AdS bulk ‘T-dual’ AdS/CFT Boundary Wilson loop Gluons
Strategy Alday, Gaiotto, Maldacena ‘09 Start with classical strings in AdS Solve equation of motion with Solve a set of integral equations null polygonal boundary (TBA equations) Substitute a solution into action Compute free energy Area of minimal surface It is hard to construct solutions with polygonal boundaries Our goal is to know the area of minimal surface, not to construct solutions Alday, Gaiotto and Maldacena proposed a set of integral equations which determines the minimal area of the hexagonal Wilson loop in AdS(5)
String sigma-model in AdS(5) Pohlmeyer reduction EoMs + Virasoro constraints → Hitchin equations Stokes phenomenon for solutions of Hitchin equations Consider a solution in each Stokes sector Define new functions from such solutions These Y ’s satisfy some functional relations (Y-system) We can rewrite Y-system as a set of integral equations Such equations are of the form of Thermodynamic Bethe ansatz (TBA) equations We studied the TBA equations for six-point case in detail
Y-system and TBA equations ¡ ¢ f ± ≡ f θ ± π i 4 Y + 1 Y − Y-system: = 1 + Y 2 1 Y + 2 Y − = (1 + μ Y 1 )(1 + μ − 1 Y 1 ) Y 3 = Y 1 Y 2 Y 1 2 TBA equations: ² ( θ ) ≡ log Y 1 ( θ ) , ˜ ² ( θ ) ≡ log Y 2 ( θ ) Alday, Gaiotto, Maldacena ‘09 ² ) ² ( θ ) = 2 | Z | cosh θ + K 2 ∗ log(1 + e − ˜ + K 1 ∗ log(1 + μ e − ² )(1 + μ − 1 e − ² ) √ 2 | Z | cosh θ + 2 K 1 ∗ log(1 + e − ˜ ² ) ² ( θ ) = 2 ˜ + K 2 ∗ log(1 + μ e − ² )(1 + μ − 1 e − ² ) √ 1 2 cosh θ K 1 ( θ ) = 2 π cosh θ , K 2 ( θ ) = π cosh2 θ Z ∞ d θ 0 f ( θ − θ 0 ) g ( θ 0 ) f ∗ g = −∞
Minimal Area Although the area of the minimal surface is divergent, we can regularize it in a well understood way BDS conjecture Bern, Dixon, Smirnov ‘05 A = A div + A BDS − R remainder function R = R 1 − | Z | 2 − A free 3 X R 1 = − 1 Li 2 (1 − U k ) 4 k =1 x ij ≡ x i − x j cross ratios U 1 = x 2 14 x 2 , U 2 = x 2 25 x 2 , U 3 = x 2 36 x 2 36 14 25 x 2 13 x 2 x 2 24 x 2 x 2 35 x 2 46 15 26 µ Z ∞ A free = 1 2 | Z | cosh θ log(1 + μ e − ² ( θ ) )(1 + μ − 1 e − ² ( θ ) ) d θ 2 π −∞ ¶ √ 2 | Z | cosh θ log(1 + e − ˜ ² ( θ ) ) +2
Goal Our goal is to know the remainder function as a function of the cross ratios Three cross ratios are related to the Y-function µ (2 k − 1) π i ¶ U k = 1 + Y 2 − i ϕ ( k = 1 , 2 , 3) 4 Thus we can relate the cross ratios to three parameters in TBA systems in principle ( U 1 , U 2 , U 3 ) ↔ ( | Z | , ϕ , μ ) The TBA equations are easily solved numerically In some special cases, we can obtain analytical results
Exact Result at Massless Limit TBA equations can be solved in the massless limit | Z | → 0 Alday, Gaiotto, Maldacena ‘09 In this limit, Y-functions are independent of θ Functional relations → algebraic equations Y 2 Y 2 2 = (1 + μ Y 1 )(1 + μ − 1 Y 1 ) 1 = 1 + Y 2 , µ φ ¶ µ 2 φ ¶ μ = e i φ Y 1 = 2 cos , Y 2 = 1 + 2 cos 3 3 The free energy is given by 6 − φ 2 A free = 1 π ( L μ ( Y 1 ) + L μ − 1 ( Y 1 ) + L 1 ( Y 2 )) = π 3 π µ µ ¶ µ ¶¶ L λ ( x ) ≡ 1 1 + λ − λ log x log − 2 Li 2 2 x x
In this limit, three cross ratios are all equal µ φ ¶ U 1 = U 2 = U 3 = 4 cos 2 3 We obtain the exact expression of the remainder function 6 + φ 2 R ( U, U, U ) = − π 3 π − 3 4 Li 2 (1 − U ) µ φ ¶ U = 4 cos 2 3
Analysis near Massless Limit We can also obtain analytical expression near | Z | ∼ 0 by using the CFT technique YH, Ito, Sakai, Satoh, arXiv:1005.4487 Recall that wide classes of 2d massive integrable models can be regarded as mass deformations of CFTs Zamolodchikov ‘87 Z ∆ ε = 1 d 2 x ε ( x ) ∆ ε = ¯ S = S CFT + λ 3 parafermion CFT in our case ( n = 6) The coupling constant is exactly related to the mass of TBA system Fateev ‘94 Γ ( x ) γ ( x ) ≡ Γ (1 − x ) λ = (0 . 44975388 . . . ) | Z | 4 / 3
Free Energy near CFT point Partition function ¿ iÀ Z h evaluate by CFT action d 2 x ε ( x ) Z = h 1 i = exp − λ 0 The free energy is perturbatively expanded as X ∞ ( − λ ) n (2 π ) − 4 3 n +2 − | Z | 2 + A free = A (CFT) free n ! n =1 Z n Y d 2 z j | z j | − 4 / 3 h V (0) ε (1) ε ( z 2 ) · · · ε ( z n ) V ( ∞ ) i 0 , connected × j =2 Due to -symmetry , the terms with odd n vanish ε → − ε The first non-trivial correction is n = 2 case
For n = 2, we can evaluate the correlation function exactly h V (0) ε (1) ε ( z 2 ) V ( ∞ ) i 0 , connected = | 1 − z 2 | − 4 2 φ 3 | z 2 | 3 π The first correction of the free energy: This result is in good agreement with the numerical result!
Remainder Function near CFT point To compute the remainder function, we need to know the behavior of R 1 3 X R 1 = − 1 Li 2 (1 − U k ) 4 k =1 Recall that µ (2 k − 1) π i ¶ U k = 1 + Y 2 − i ϕ ( k = 1 , 2 , 3) 4 We assume that the Y-function is expanded as X ∞ 4 Y ( n ) 3 n ˜ Y 2 ( θ ) = ( θ , φ ) | Z | 2 n =0
The first and second coefficients take the following forms µ 2 φ ¶ Y (0) ˜ ( θ , φ ) = 1 + 2 cos 2 3 µ 4 θ ¶ Y (1) ˜ ( θ , φ ) = y (1) ( φ )cosh 2 3 The perturbative expansion of R 1 ∞ X 4 R ( n ) 3 n ˜ R 1 = 1 ( ϕ , φ ) | Z | n =0 µ φ ¶ 1 ( ϕ , φ ) = − 3 R (0) ˜ 4 Li 2 (1 − 4 β 2 ) β = cos 3 R (1) ˜ 1 ( ϕ , φ ) = 0 1 ( ϕ , φ ) = 3(4 β 2 − 1 + log(4 β 2 )) R (2) ˜ y (1) ( φ ) 2 64 β 2 (4 β 2 − 1) 2
y (1) ( φ ) At present, we could not fix the function , but it can be evaluated numerically It should be fixed analytically In summary, the remainder function is expanded as R = R (0) + R (2) | Z | 8 3 + O ( | Z | 4 ) 6 + φ 2 R (0) = − π 3 π − 3 4 Li 2 (1 − 4 β 2 ) φ = 0 , ϕ = − π 48 µ 1 ¶ µ 1 ¶ 3 + φ 3 − φ R (2) = − C γ γ 3 π 3 π +3(4 β 2 − 1 + log(4 β 2 )) y (1) ( φ ) 2 64 β 2 (4 β 2 − 1) 2 µ φ ¶ β = cos 3
Comment on Large Mass Limit Large mass case: | Z | À 1 The TBA equations can be solved approximately ² ( θ ) = 2 | Z | cosh θ + (exponetial corrections) ² ( θ ) = 2 √ 2 | Z | cosh θ + (exponetial corrections) ˜ Free energy: modified Bessel fonction of the second kind h i √ √ A free ≈ 2 | Z | ( μ + μ − 1 ) K 1 (2 | Z | ) + 2 K 1 (2 2 | Z | ) π Similarly we can evaluate the remainder function
Summary Gluon scattering amplitude at strong coupling can be computed by the area of minimal surface with a null polygonal boundary The problem to determine the area of such minimal surface is mapped to a set of integral equations (TBA equations) We analyzed the TBA equations for six-point amplitudes in detail We obtained the analytical expression of the area up to an unknown function It is interesting to fix the analytic form of this unknown function Analysis of TBA equations for general n -point amplitudes Do TBA equations also appear if we consider -corrections?
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