Three-point correlators from string theory amplitudes Joseph - PowerPoint PPT Presentation

Three-point correlators from string theory amplitudes Joseph Minahan Uppsala University arXiv:1206.3129 Till Bargheer, Raul Pereira, JAM: arXiv:1311.7461; Raul Pereira, JAM: arXiv:1407.xxxx Strings 2014 in Princeton; 27 June

Introduction Spectrum of local operators in N = 4 SYM effectively solved in the planar limit. Determined by: ◮ Integrability: Asymptotic Bethe ansatz Staudacher (2004), Beisert-Staudacher(2005), Beisert (2005), Janik (2006), Eden-Staudacher (2006), Beisert-Hernandez-Lopez (2006), Beisert-Eden-Staudacher (2006) . . . ◮ Finite size complications (winding effects). Handled by TBA, Y-system, Hirota, FiNLIE, Q-functions Ambjorn-Janik-Kristjansen (2005), Bajnok-Janik (2008), Gromov-Kazakov-Vieira (2009,2009), G-K-Kozac-V (2009), Arutyunov-Frolov (2008,2009), Bombardelli-Fioravanti-Tateo (2009), Frolov (2010), Gromov-Kazakov-Leurent-Volin (2011, 2013, 2014) . . .

Introduction (cont) A key example: Konishi operator: O K = tr ( φ I φ I ); Primary: [ K µ , O K (0)] = 0; SO (6) singlet Z λ = g 2 �O K ( x ) O K ( y ) � = YM N | x − y | 2∆ K ( λ )

Introduction (cont) Besides the spectrum, to really solve the theory we need the three-point correlators. Correlator for three local operators: C 123 �O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) � = | x 12 | 2 α 3 | x 23 | 2 α 1 | x 31 | 2 α 2 α 1 = 1 α 2 = 1 α 3 = 1 2 (∆ 2 + ∆ 3 − ∆ 1 ) 2 (∆ 3 + ∆ 1 − ∆ 2 ) 2 (∆ 1 + ∆ 2 − ∆ 3 ) C 123 ∼ N − 1 for N ≫ 1.

Introduction (cont) C 123 is protected for 3 chiral primaries. ◮ Chiral primary O C ( x ): [ Q , O C (0)] = 0 for half the Q ’s ◮ The gravity duals are K-K modes in the AdS 5 × S 5 type IIB supergravity ◮ Supergravity calculation shows that C 123 at large λ is the same as the zero-coupling result Lee, Minwalla, Rangamani and Seiberg (1998)

Introduction (cont) Nonchiral primaries are not dual to sugra states but to massive string states. ◮ “Heavy” operators: Dual to long classical strings that stretch across the AdS 5 × S 5 . ◮ Semiclassical string calculation for 3-point correlators . Janik-Surowka-Wereszczynski (2010) ◮ Two heavy, one light Zarembo (2010), Costa-Monteiro-Santos-Zoakos (2010), Roiban-Tseytlin (2010) . . . ◮ Three heavy Janik-Wereszczynski (2011), Buchbinder-Tseytlin (2011), Klose-McLoughlin (2011), Kazama-Komatsu (2011-13), . . . ◮ The Konishi operator is neither semi-classical nor light – blank it is dual to a short string state. ◮ Can one compute the 3-point correlators involving at least blankone Konishi operator for λ ≫ 1?

Introduction (cont) ◮ General idea: Since Konishi is short it doesn’t see the curvature of blank AdS 5 × S 5 ( R = 1) = ⇒ use the flat-space limit. ◮ Flat-space for the spectrum: √ α ′ = λ − 1 / 4 ≪ 1 Gubser-Klebanov-Polyakov (1998) String size ∼ m 2 = 4 n /α ′ = 4 n λ 1 / 2 Flat-space closed strings: m 2 = ∆ 2 − d ∆ ≈ ∆ 2 AdS/CFT dictionary: ∆ ≈ 2 √ n λ 1 / 4 n = 1 for Konishi

Introduction (cont) Back to the Gromov-Kazakov-Vieira plot: տ 2 λ 1 / 4 + 2 λ 1 / 4 + . . . . ↑ ↑ .. GKP 1-loop w-s .

Introduction (cont) Back to the Gromov-Kazakov-Vieira plot: տ 2 λ 1 / 4 + 2 λ 1 / 4 + . . . . ↑ ↑ .. GKP 1-loop w-s . We would like a similar goal for 3-point correlators

3-point correlators – Witten diagrams 3-point correlators in supergravity Witten (1998) Freedman-Mathur-Matusis-Rastelli . (1998) : ◮ Boundary to bulk propagators meet at an intersection point. ◮ Integrate over the intersection point. ◮ Multiply by sugra coupling G 123

3-point correlators – Witten diagrams 3-point correlators in supergravity Witten (1998) Freedman-Mathur-Matusis-Rastelli . (1998) : ◮ Boundary to bulk propagators meet at an intersection point. ◮ Integrate over the տ intersection point. Integral dominated by ◮ Multiply by sugra small region if ∆ i ≫ 1 coupling G 123

3-point correlators – Witten diagrams ◮ For Konishi operators treat as point-like . outside the intersection region using AdS propagators. ◮ Treat as strings in the . intersection region. տ Integral dominated by small region if ∆ i ≫ 1

3-point correlators – Witten diagrams ◮ For Konishi operators treat as point-like . outside the intersection region using AdS propagators. ◮ Treat as strings in the . intersection region. տ ◮ Small interaction region: use flat-space string Integral dominated by vertex operators to find small region if ∆ i ≫ 1 the couplings.

3-point correlators – Witten diagrams ◮ For Konishi operators treat as point-like . outside the intersection region using AdS propagators. ◮ Treat as strings in the . intersection region. տ ◮ Small interaction region: use flat-space string Integral dominated by vertex operators to find small region if ∆ i ≫ 1 the couplings. ◮ Which vertex operators?

3-point correlators-particle path integrals in AdS Three incoming particles meet at a joining point: x µ ( s 0 )= x µ , z ( s 0 )= z . Z 123 ≡ . circular geodesics � � 3 � z ǫ S cl( x µ , z ) = − ∆ i log z 2 + ( x − x i ) 2 i =1 Saddle point: Conservation of momentum: (See also Klose & McLoughlin (2011)) 3 � � Π i · Π i = − ∆ 2 Π µ, i = 0 Π z , i = 0 i i =1 z , i 2 − d (∆ 1 ∆ 2 ∆ 3 ) d / 4 α α 1 1 α α 2 2 α α 3 3 Σ Σ Z 123 ≈ π 1 4 | x 12 | 2 α 3 | x 23 | 2 α 1 | x 31 | 2 α 2 G 123 ( α 1 α 2 α 3 Σ d +1 ) 1 / 2 ∆ ∆ 1 1 ∆ ∆ 2 2 ∆ ∆ 3 4 3 Σ= 1 . 2 (∆ 1 +∆ 2 +∆ 3 )

3-point correlators-particle path integrals in AdS Three incoming particles meet at a joining point: ( x µ ( s 0 )= x µ , z ( s 0 )= z . Z 123 ≡ . . . 2 − d (∆ 1 ∆ 2 ∆ 3 ) d / 4 α α 1 1 α α 2 2 α α 3 3 Σ Σ π 4 C 123 G 123 = 4 ( α 1 α 2 α 3 Σ d +1 ) 1 / 2 ∆ ∆ 1 1 ∆ ∆ 2 2 ∆ ∆ 3 3 2 3 / 2 Γ( α 1 )Γ( α 2 )Γ( α 3 )Γ(Σ − d / 2) ≈ 2 )] 1 / 2 G 123 π d / 4 [Γ(∆ 1 )Γ(∆ 2 )Γ(∆ 3 )Γ(∆ 1 + 2 2 )Γ(∆ 2 + 2 d 2 )Γ(∆ 3 + 2 d d − − − . Freedman-Mathur-Matusis-Rastelli (1998) G 123 = V 123 � ψ J 1 ψ J 2 ψ J 3 � C 123 = V 123 × ( AdS 5 × S 5 Overlaps)

String vertex operators: Strategy ◮ Let ∆ i ≫ 1. ◮ States are wave-packets with wavelength ∼ ∆ − 1 , spread ∼ ∆ − 1 / 2 ◮ ⇒ Treat as plane-waves in the intersection region Polchinski (1999) ◮ Momentum: k Mi = (Π µ i , Π zi ; � J i ), M = 0 . . . 9 ◮ Flat-space factors of (2 π ) 10 δ 10 ( k 1 + k 2 + k 3 ) replaced with AdS 5 × S 5 overlaps. ◮ Use level 1 (0) flat-space vertex operators for Konishi (chiral primaries). √ ◮ k 2 = − ∆ 2 + J 2 = − 4 n /α ′ = − 4 n λ ◮ � J can be set to 0 for level 1, but not for level 0. ◮ The coupling factor uses the string result 8 π V 123 = c α ′ � V ( k 1 ) V ( k 2 ) V ( k 3 ) � g 2 . Polchinski, String Theory, Vol 1, 2. g c = π 3 / 2 N − 1 in AdS/CFT dictionary .

Which vertex operators? N = 4 superconformal algebra in manifest SO (2 , 4) form: 1 1 M µν , M − 1 µ ≡ 2 ( P µ − K µ ) M 4 µ ≡ 2 ( P µ + K µ ) M − 14 ≡ − D , √ √ aa ≡ ( ǫ αβ S a β ˜ α ˙ Q 1 aa ≡ ( Q α a , ˜ Q 2˙ β , ǫ ˙ Q a α a ) , β ) α, ˙ α = 1 , 2; ˙ a = 1 . . . 4 S ˙ ˙ ˙ b , b − i a a , Q 2 ˙ ˙ ˙ { Q 1 b b } = 1 b M mn γ mn b R IJ γ IJ 2 δ a 2 δ ˙ m , n = − 1 , . . . 4 a ˙ a ˙ a

Which vertex operators? N = 4 superconformal algebra in manifest SO (2 , 4) form: 1 1 M µν , M − 1 µ ≡ 2 ( P µ − K µ ) M 4 µ ≡ 2 ( P µ + K µ ) M − 14 ≡ − D , √ √ aa ≡ ( ǫ αβ S a β ˜ α ˙ Q 1 aa ≡ ( Q α a , ˜ Q 2˙ β , ǫ ˙ Q a α a ) , β ) α, ˙ α = 1 , 2; ˙ a = 1 . . . 4 S ˙ ˙ ˙ b , b − i a a , Q 2 ˙ ˙ ˙ { Q 1 b b } = 1 b M mn γ mn b R IJ γ IJ 2 δ a 2 δ ˙ m , n = − 1 , . . . 4 a ˙ a ˙ a Define: ba Q 2˙ bb , Q R ba Q 2˙ Q L A ≡ Q 1 γ − 1 a γ 6 A ≡ i ( Q 1 γ − 1 a γ 6 bb ) , P m ≡ M − 1 , m , P J ≡ R J 6 aa + aa − ˙ ˙ ˙ ˙ b ˙ b ˙ { Q L , R , Q L , R } = 2Γ M { Q L A , Q R ⇒ AB P M + . . . B } = 0 A B 10d N = 2 Super-Poincar´ e algebra A , B = 1 . . . 16 , M = 0 . . . 9

Which vertex operators? N = 4 superconformal algebra in manifest SO (2 , 4) form: 1 1 M µν , M − 1 µ ≡ 2 ( P µ − K µ ) M 4 µ ≡ 2 ( P µ + K µ ) M − 14 ≡ − D , √ √ aa ≡ ( ǫ αβ S a β ˜ α ˙ Q 1 aa ≡ ( Q α a , ˜ Q 2˙ β , ǫ ˙ Q a α a ) , β ) α, ˙ α = 1 , 2; ˙ a = 1 . . . 4 S ˙ ˙ ˙ b , b − i a a , Q 2 ˙ ˙ ˙ { Q 1 b b } = 1 b M mn γ mn b R IJ γ IJ 2 δ a 2 δ ˙ m , n = − 1 , . . . 4 a ˙ a ˙ a Define: ba Q 2˙ bb , Q R ba Q 2˙ Q L A ≡ Q 1 γ − 1 a γ 6 A ≡ i ( Q 1 γ − 1 a γ 6 bb ) , P m ≡ M − 1 , m , P J ≡ R J 6 aa + aa − ˙ ˙ ˙ ˙ b ˙ b ˙ { Q L , R , Q L , R } = 2Γ M { Q L A , Q R ⇒ AB P M + . . . B } = 0 A B 10d N = 2 Super-Poincar´ e algebra A , B = 1 . . . 16 , M = 0 . . . 9 α O (0) = � K µ O (0) = 0 S b Primary Operator: ⇒ S ˙ α b O (0) = 0 Q L = ± i Q R (sign depends on component) Flat-space:

String vertex operators ⇒ Mixing of NS-NS and R-R modes: Q L ( | NS � ⊗ | NS � + | R � ⊗ | R � ) = | R � ⊗ | NS � + | NS � ⊗ R � ց ւ . . Q R ( | NS � ⊗ | NS � + | R � ⊗ | R � ) = | NS � ⊗ | R � + | R � ⊗ NS � Setting Q L = ± i Q R requires a mixture of both sets of fields