Three point correlators from

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 = 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: OK = tr(φI φI ); Primary: [K µ , OK (0)] = 0; SO(6) singlet OK (x)OK (y ) = Z |x − y |2∆K (λ) λ = gYM N Introduction (cont) Besides the spectrum, to really solve the theory we need the three-point correlators Correlator for three local operators: O1 (x1 )O2 (x2 )O3 (x3 ) = α1 = 12 (∆2 + ∆3 − ∆1 ) C123 ∼ N −1 for N |x12 α2 = 21 (∆3 + ∆1 − ∆2 ) C123 2α 2α 23 | |x31 | |2α3 |x α3 = 21 (∆1 + ∆2 − ∆3 ) Introduction (cont) C123 is protected for chiral primaries Chiral primary OC (x): [Q, OC (0)] = for half the Q’s The gravity duals are K-K modes in the AdS5 × S type IIB supergravity Supergravity calculation shows that C123 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 AdS5 × S 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 AdS5 × S (R = 1) =⇒ use the flat-space limit Flat-space for the spectrum: Gubser-Klebanov-Polyakov (1998) String size ∼ Flat-space closed strings: √ α = λ−1/4 m2 = 4n/α = 4n λ1/2 AdS/CFT dictionary: m2 = ∆2 − d∆ ≈ ∆2 √ ∆ ≈ n λ1/4 n = for Konishi Introduction (cont) Back to the Gromov-Kazakov-Vieira plot: + 2λ1/4 + λ1/4 ↑ ↑ GKP 1-loop w-s Introduction (cont) Back to the Gromov-Kazakov-Vieira plot: + 2λ1/4 + λ1/4 ↑ ↑ 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 G123 Which vertex operators? N = superconformal algebra in manifest SO(2, 4) form: Mµν , M−1µ ≡ √1 (Pµ −Kµ ) ˜˙ ), Qaa ˙ ≡ (Qαa , Sαa ˙ Q 2aa ≡( M4µ ≡ αβ Sβa , ˙ ˙ α˙ β˙ √1 (Pµ +Kµ ) ˜ a˙ ) α, α˙ = 1, 2; a˙ = Q β b ˙ M−14 ≡ −D , {Qa1˙ a , Q b b } = 21 δa b Mmn γ mn a˙ b − 2i δa˙ b RIJ γ IJ a , m, n = −1, Define: ˙ ˙ −1 −1 2bb 2bb , QAR ≡ i(Qaa ) , Pm ≡ M−1,m , PJ ≡ RJ6 QAL ≡ Qaa ˙ +γb˙ a˙ γba Q ˙ −γb˙ a˙ γba Q ⇒ {QAL,R , QBL,R } = 2ΓM AB PM + {QAL , QBR } = 10d N = Super-Poincar´e algebra Primary Operator: K µ O(0) = Flat-space: ⇒ A, B = 16, M = Sαb O(0) = Sα˙ b O(0) = Q L = ±i Q R (sign depends on component) String vertex operators ⇒ Mixing of NS-NS and R-R modes: Q L (|NS ⊗ |NS + |R ⊗ |R ) = |R ⊗ |NS + |NS ⊗ R Q R (|NS ⊗ |NS + |R ⊗ |R ) = |NS ⊗ |R + |R ⊗ NS Setting Q L = ±i Q R requires a mixture of both sets of fields String vertex operators Choosing components: a a L R Boundary: k = (0, i∆; J) ⇒ Qαa = +i Qαa , QαL˙ = −i QαR˙ α, α˙ are 4-d space-time spinors a are ⊥ SO(6) spinor indices a a Bulk: k = (kA ; J) ⇒ QαL a = +i QαR a , QαL˙ = −i QαR α , α˙ are spinors in 4-d space ⊥ to kA a are ⊥ SO(6) spinor indices String vertex operators: Massless example First consider a “twisted” version: set QL = i QR for all spin comps Then untwist by rotating the righthand part of the state: T = exp(iπ(M0 + M2 )R ) Chiral primaries:⇔ Massless vertex ops Friedan-Martinec-Shenker (1985) k M = (∆; J) ⇒ k2 = ˜ NS-NS : W1 = gc εMN ψ M ψ˜N e −φ−φ e ik·X , k M εMN = 1/2 1 ˜ B e − φ− φ˜ e ik·X , t /k = t AB ΘA Θ R-R : W2 = gc α2 Only solution to QL = i QR : εTMN = ηMN − kM k¯N k·k¯ , tTAB = (C/ k )AB Mix of dilaton and axion; descendant of the chiral primary (LMRS) String vertex operators: Massless example Untwist: dilaton → graviton, axion → self-dual tensor Normalized vertex: W (k) = − 14 (W1 (k) + √1 W2 (k)) 8π α1 α2 α3 Σ5 W (k )W (k )W (k ) = 8πg c gc2 α J12 J22 J32 Using AdS/CFT dictionary and overlap integrals: Amplitude: V123 = C123 ≈ αα1 αα2 αα3 ΣΣ (J1 J2 J3 )1/2 J12 J2 J3 N J1 J2 J3 agrees with LMRS String vertex ops: Level one (Untwisted) Relevant vert ops can be found in 1980’s literature (FMS (1985); Kostelecky-Lechtenfeld-Lerche-Samuel-Watamura (1987); Koh, Troost, van Proeyen (1987)) QL = i QR requires two types of NS-NS and one R-R α ˜ ˜ ˜ ¯ N e −φ−φ e ik·X , σMN;M˜ N˜ ψ M (z)∂X N ψ˜M (¯ z )∂X V1T (k) = gc V2T (k) = gc αMNL;M˜ N˜ L˜ ψ M (z)ψ N (z)ψ L (z) ψ M (¯ z )ψ N (¯ z )ψ L (¯ z ) e −φ−φ e ik·X V3T (k) = gc ˜ α 1/2 ¯ M Θ− ˜ i ∂X α 16 = αMNL;M˜ N˜ L˜ = α 16 (ˆ ηM M˜ ηˆN N˜ + ηˆM N˜ ηˆN M˜ ) − 91 (ˆ ηM M˜ ηˆN N˜ ηˆLL˜ − perms) 3! ηˆMN ≡ ηMN − ˜ ˜ ˜ ˜ e −φ/2 ψ˜M / k ψ/˜Θ ˆM Γ ˆN ) i ∂X N Θ− ×C / k (ˆ ηMN − 19 Γ where σMN;M˜ N˜ ˜ kM kN k2 ψN / k ψΘ / e −φ/2 e ikX ηˆMN ηˆM˜ N˜ ˆ M = ΓM − / Γ kk M /k Results for three Konishi operators Untwist ⇒ Normalized vertex: V (k) = 16 V1 (k) + V2 (k) + √1 V3 (k) Compute V (k1 )V (k2 )V (k3 ) ki = (∆; 0), ∆ = 2λ1/4 − + Various combinations: V1 (k1 )V1 (k2 )V1 (k3 ) , V1 (k1 )V1 (k2 )V2 (k3 ) , V1 (k1 )V3 (k2 )V3 (k3 ) , V2 (k1 )V3 (k2 )V3 (k3 ) , etc Nasty combinatorics: V2 (k1 )V3 (k2 )V3 (k3 ) is especially horrific Results for three Konishi operators Untwist ⇒ Normalized vertex: V (k) = 16 V1 (k) + V2 (k) + √1 V3 (k) Compute V (k1 )V (k2 )V (k3 ) ki = (∆; 0), ∆ = 2λ1/4 − + Various combinations: V1 (k1 )V1 (k2 )V1 (k3 ) , V1 (k1 )V1 (k2 )V2 (k3 ) , V1 (k1 )V3 (k2 )V3 (k3 ) , V2 (k1 )V3 (k2 )V3 (k3 ) , etc Nasty combinatorics: V2 (k1 )V3 (k2 )V3 (k3 ) is especially horrific 38 But big simplification: V (k1 )V (k2 )V (k3 ) = gc3 ⇒ C123 ≈ (4 · 35 π)1/2 λ1/4 N Explicit λ dependence Suppression for large λ 3λ1/4 Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and −J √ π √ 2−∆ J 2(1−J) (J − 12 ∆)J−∆/2−1/2 (J + 21 ∆)J+∆/2+3/2 C123 ≈ N λ Extremal limit: Intersection point approaches the boundary Singular as J →+ ∆/2 Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and −J √ π √ 2−∆ J 2(1−J) (J − 12 ∆)J−∆/2−1/2 (J + 21 ∆)J+∆/2+3/2 C123 ≈ N λ Extremal limit: Intersection point approaches the boundary Singular as J →+ ∆/2 Analyze more closely: Use exact FMMR result C123 = ≈ (∆1 − 1)(∆2 − 1)(∆3 − 1) Γ(α1 )Γ(α2 )Γ(α3 )Γ(Σ − 2) G123 Γ(∆1 )Γ(∆2 )Γ(∆3 ) 25/2 π 16 3/8 λ as α3 = 2J − ∆ → N 2J − ∆ Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and −J √ π √ 2−∆ J 2(1−J) (J − 12 ∆)J−∆/2−1/2 (J + 21 ∆)J+∆/2+3/2 C123 ≈ N λ Extremal limit: Intersection point approaches the boundary Singular as J →+ ∆/2 Analyze more closely: Use exact FMMR result C123 = ≈ (∆1 − 1)(∆2 − 1)(∆3 − 1) Γ(α1 )Γ(α2 )Γ(α3 )Γ(Σ − 2) G123 Γ(∆1 )Γ(∆2 )Γ(∆3 ) 25/2 π 16 3/8 λ as α3 = 2J − ∆ → N 2J − ∆ No pole for chiral primaries LMRS, D’Hoker-FMMR (1999) Pole indicates mixing of O∆ with double trace op OJ J¯ =: OJ OJ¯ : Splitting at the crossover OK OJ J¯ 11.0 10.5 10.0 9.5 9.0 500 Splitting: δ∆ = 600 700 800 900 1000 16 3/8 λ N Interesting to compare with recent bootstrap results Beem-Rastelli-van Rees (to appear) Λ Splitting at the crossover OK OJ J¯ 11.0 10.5 10.0 9.5 9.0 500 600 Splitting: δ∆ = 700 16 N √ 800 M λ3/8 900 1000 Λ Splitting at the crossover OK OJ J¯ 11.0 10.5 10.0 9.5 9.0 500 600 Splitting: δ∆ = 700 16 N √ 800 900 1000 Λ M λ3/8 Interesting to compare with recent bootstrap results Beem-Rastelli-van Rees (to appear) Discussion There has been much progress on 3-point correlators at low loop orders using integrability: Escobedo-Gromov-Sever-Vieira (2010,2011), Gromov-Sever-Vieira (2011), Georgiou (2011), Bissi-Harmaark-Orselli (2011), Gromov-Vieira (2011,2012), Kostov (2012, 2012), Serban (2012), Grignani-Zayakin (2012), Plefka-Wiegant (2012), Bissi-Grignani-Zayakin (2012), Foda-Jiang-Kostov-Serban (2013), Jiang-Kostov-Loebbert-Serban (2014), Caetano-Fleury (2014) We can also 3-point correlators containing an operator with nonzero spin Compares favorably with recent results using Mellin amplitudes on Regge trajectories Costa-Goncalves-Penedones (2012) Many possible generalizations: More massive operators at n = or higher Can study four-point correlators and duality of operator products The simplifications suggest an underlying symmetry playing an important role Perhaps these results can help lead us to the exact vertex operators in AdS5 × S ... + λ1/4 ↑ ↑ 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... propagators meet at an intersection point Integrate over the intersection point Multiply by sugra coupling G123 3 -point correlators – Witten diagrams 3 -point correlators in supergravity Witten... intersection point Integrate over the intersection point Integral dominated by small region if ∆i Multiply by sugra coupling G123 3 -point correlators – Witten diagrams For Konishi operators treat as point- like