Six-point gluon scattering amplitudes from -symmetric integrable - PowerPoint PPT Presentation

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? 