Three-point correlators from string theory amplitudes

Transcription

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

2 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)

3 Introduction (cont) A key example: Konishi operator: O K = tr(φ I φ I ); Primary: [K µ, O K (0)] = 0; SO(6) singlet O K (x)o K (y) = Z x y 2 K (λ) λ = g 2 YMN

4 Introduction (cont) Besides the spectrum, to really solve the theory we need the three-point correlators Correlator for three local operators: O 1 (x 1 )O 2 (x 2 )O 3 (x 3 ) = C 123 x 12 2α3 x 23 2α1 x 31 2α2 α 1 = 1 2 ( ) α2 = 1 2 ( ) α3 = 1 ( ) 2 C 123 N 1 for N 1

5 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)

6 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 ( ), 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?

7 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: Gubser-Klebanov-Polyakov (1998) String size α = λ 1/4 1 Flat-space closed strings: m 2 = 4n/α = 4n λ 1/2 AdS/CFT dictionary: m 2 = 2 d 2 2 n λ 1/4 n = 1 for Konishi

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

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

10 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

11 3-point correlators Witten diagrams 3-point correlators in supergravity Witten (1998) Integral dominated by small region if i 1 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

12 3-point correlators Witten diagrams Integral dominated by small region if i 1 For Konishi operators treat as point-like outside the intersection region using AdS propagators Treat as strings in the intersection region

13 3-point correlators Witten diagrams Integral dominated by small region if i 1 For Konishi operators treat as point-like outside the intersection region using AdS propagators Treat as strings in the intersection region Small interaction region: use flat-space string vertex operators to find the couplings

14 3-point correlators Witten diagrams Integral dominated by small region if i 1 For Konishi operators treat as point-like outside the intersection region using AdS propagators Treat as strings in the intersection region Small interaction region: use flat-space string vertex operators to find the couplings Which vertex operators?

15 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 S cl (x µ, z) = 3 i=1 ( ) z ɛ i log z 2 + (x x i ) 2 Saddle point: Conservation of momentum: (See also Klose & McLoughlin (2011)) 3 Π µ,i = 0 Π z,i = 0 Π i Π i = 2 i Z 123 π 2 d 4 4 i=1 ( 1 2 3) d/4 α (α 1α 2α 3Σ d+1 ) 1/2 1 Σ= 1 ( ) 2 z,i α 1 1 αα 2 2 αα 3 3 ΣΣ x 12 2α 3 x23 2α 1 x31 2α 2 G123

16 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 C 123 = π 2 d 4 4 ( ) d/4 α (α 1 α 2 α 3 Σ d+1 ) 1/2 α 1 1 αα2 2 αα3 3 ΣΣ 1 G /2 Γ(α 1 )Γ(α 2 )Γ(α 3 )Γ(Σ d/2) π d/4 [Γ( 1 )Γ( 2 )Γ( 3 )Γ( d 2 )Γ( 2+ 2 d 2 )Γ( 3+ 2 d G 2 )]1/2 123 Freedman-Mathur-Matusis-Rastelli (1998) G 123 = V 123 ψ J1 ψ J2 ψ J3 C 123 = V 123 (AdS 5 S 5 Overlaps)

17 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 µi, Π zi ; J i ), M = 0 9 Flat-space factors of (2π) 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 = 4n/α = 4n λ J can be set to 0 for level 1, but not for level 0 The coupling factor uses the string result V 123 = 8π g 2 c α V (k 1)V (k 2 )V (k 3 ) Polchinski, String Theory, Vol 1, 2 g c = π 3/2 N 1 in AdS/CFT dictionary

18 Which vertex operators? N = 4 superconformal algebra in manifest SO(2, 4) form: M µν, M 1µ 1 2 (P µ K µ ) M 4µ 1 2 (P µ +K µ ) M 14 D, Q 1 ȧa (Q αa, S αa ), Q 2ȧa (ɛ αβ Sβ, a ɛ α β Qȧ ) α, α = 1, 2; ȧ = 1 4 β {Q 1 ȧ a, Q 2 ḃ b } = 1 2 δ a b M mn γ mn ȧḃ i 2 δ ȧ ḃ R IJ γ IJ ab, m, n = 1, 4

19 Which vertex operators? N = 4 superconformal algebra in manifest SO(2, 4) form: M µν, M 1µ 1 2 (P µ K µ ) M 4µ 1 2 (P µ +K µ ) M 14 D, Q 1 ȧa (Q αa, S αa ), Q 2ȧa (ɛ αβ Sβ, a ɛ α β Qȧ ) α, α = 1, 2; ȧ = 1 4 β {Q 1 ȧ a, Q 2 ḃ b } = 1 2 δ a b M mn γ mn ȧḃ i 2 δ ȧ ḃ R IJ γ IJ ab, m, n = 1, 4 Define: Q L A Q 1 ȧa+γ 1 ḃȧ γ6 baq 2ḃb, Q R A i(q 1 ȧa γ 1 ḃȧ γ6 baq 2ḃb ), P m M 1,m, P J R J6 {Q L,R A, QL,R B } = 2ΓM ABP M + {QA, L QB R } = 0 10d N = 2 Super-Poincaré algebra A, B = 1 16, M = 0 9

20 Which vertex operators? N = 4 superconformal algebra in manifest SO(2, 4) form: M µν, M 1µ 1 2 (P µ K µ ) M 4µ 1 2 (P µ +K µ ) M 14 D, Q 1 ȧa (Q αa, S αa ), Q 2ȧa (ɛ αβ Sβ, a ɛ α β Qȧ ) α, α = 1, 2; ȧ = 1 4 β {Q 1 ȧ a, Q 2 ḃ b } = 1 2 δ a b M mn γ mn ȧḃ i 2 δ ȧ ḃ R IJ γ IJ ab, m, n = 1, 4 Define: Q L A Q 1 ȧa+γ 1 ḃȧ γ6 baq 2ḃb, Q R A i(q 1 ȧa γ 1 ḃȧ γ6 baq 2ḃb ), P m M 1,m, P J R J6 {Q L,R A, QL,R B } = 2ΓM ABP M + {QA, L QB R } = 0 10d N = 2 Super-Poincaré algebra A, B = 1 16, M = 0 9 Primary Operator: K µ O(0) = 0 S b αo(0) = S α b O(0) = 0 Flat-space: Q L = ±i Q R (sign depends on component)

21 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

22 String vertex operators Choosing components: Boundary: k = ( 0, i ; J) Qαa L = +i Qαa, R Q L α a = i Q Ṙ a α α, α are 4-d space-time spinors a are SO(6) spinor indices Bulk: k = ( k A ; J) Q L α a = +i QR α a, QL α a = i Q R α a α, α are spinors in 4-d space to k A a are SO(6) spinor indices

23 String vertex operators: Massless example First consider a twisted version: set Q L = i Q R for all spin comps Then untwist by rotating the righthand part of the state: T = exp(iπ(m M 2 3 ) R) Chiral primaries: Massless vertex ops Friedan-Martinec-Shenker (1985) k M = ( ; J) k 2 = 0 NS-NS : W1 = g c ε MN ψ M ψ N e φ φ e ik X, k M ε MN = 0 ( ) 1/2 R-R : W2 = g c t AB Θ A Θ B e 1 2 φ 1 2 φ e ik X, t /k = 0 α 2 Only solution to Q L = i Q R : ε T MN = η MN k M k N, t AB k k T = (C/k) AB Mix of dilaton and axion; descendant of the chiral primary (LMRS)

24 String vertex operators: Massless example Untwist: dilaton graviton, axion self-dual tensor Normalized vertex: W (k) = 1 4 (W 1(k) W 2 (k)) Amplitude: V 123 = 8π gc 2 α W (k α 1 α 2 α 3 Σ 5 1)W (k 2 )W (k 3 ) = 8πg c J1 2J2 2 J2 3 Using AdS/CFT dictionary and overlap integrals: C N (J αα1 1/2 1 1J 2 J 3 ) αα2 2 αα3 3 ΣΣ J J1 1 JJ2 2 JJ3 3 agrees with LMRS

25 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)) Q L = i Q R requires two types of NS-NS and one R-R V 1T (k) = g c ( 2 α ) σmn; MÑ ψm (z) X N ψ M( z) X Ñ e φ φe ik X, V 2T (k) = g c α MNL; MÑ L ψm (z)ψ N (z)ψ L (z) ψ M ( z)ψñ( z)ψ L ( z) e φ φ e ik X ( V 3T (k) = g 2 ) [ 1/2 c i X ( ) M α Θ ψm /k / ψ Θ ] e φ/2 α 16 C/k(ˆη MN 1 ˆΓ [ ( ) ] 9 M ˆΓN ) i X N Θ ψ N /k /ψθ e φ/2 e ikx α 16 where σ MN; MÑ = 1 (ˆη 2 M M ˆη NÑ + ˆη MÑ ˆη N M ) 1 ˆη 9 MN ˆη MÑ α MNL; MÑ L = 1 (ˆη 3! M M ˆη NÑ ˆη L L perms) ˆη MN η MN k Mk N k 2 ˆΓ M = Γ M /kk M /k 2

26 Results for three Konishi operators Untwist Normalized vertex: V (k) = 1 16 Compute V (k 1 )V (k 2 )V (k 3 ) ( ) V 1 (k) + V 2 (k) V 3 (k) ki = ( ; 0), = 2λ 1/4 2 + Various combinations: V 1(k 1)V 1(k 2)V 1(k 3), V 1(k 1)V 1(k 2)V 2(k 3), V 1(k 1)V 3(k 2)V 3(k 3), V 2(k 1)V 3(k 2)V 3(k 3), etc Nasty combinatorics: V 2 (k 1 )V 3 (k 2 )V 3 (k 3 ) is especially horrific

27 Results for three Konishi operators Untwist Normalized vertex: V (k) = 1 16 Compute V (k 1 )V (k 2 )V (k 3 ) ( ) V 1 (k) + V 2 (k) V 3 (k) ki = ( ; 0), = 2λ 1/4 2 + Various combinations: V 1(k 1)V 1(k 2)V 1(k 3), V 1(k 1)V 1(k 2)V 2(k 3), V 1(k 1)V 3(k 2)V 3(k 3), V 2(k 1)V 3(k 2)V 3(k 3), etc Nasty combinatorics: V 2 (k 1 )V 3 (k 2 )V 3 (k 3 ) is especially horrific But big simplification: V (k 1 )V (k 2 )V (k 3 ) = g 3 c C ( 3 N (4 35 π) 1/2 λ 1/4 4 Explicit λ dependence Suppression for large λ ) 3λ 1/

28 Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and J C π N 4 λ 2 J 2(1 J) (J 1 2 )J /2 1/2 (J )J+ /2+3/2 Extremal limit: Intersection point approaches the boundary Singular as J + /2

29 Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and J C π N 4 λ 2 J 2(1 J) (J 1 2 )J /2 1/2 (J )J+ /2+3/2 Extremal limit: Intersection point approaches the boundary Singular as J + /2 Analyze more closely: Use exact FMMR result ( 1 1)( 2 1)( 3 1) Γ(α 1 )Γ(α 2 )Γ(α 3 )Γ(Σ 2) C 123 = 2 5/2 π Γ( 1 )Γ( 2 )Γ( 3 ) 16 1 N λ3/8 as α 3 = 2J 0 2J G 123

30 Two chiral primaries and a Konishi Two chiral primaries with R-charge +J and J C π N 4 λ 2 J 2(1 J) (J 1 2 )J /2 1/2 (J )J+ /2+3/2 Extremal limit: Intersection point approaches the boundary Singular as J + /2 Analyze more closely: Use exact FMMR result ( 1 1)( 2 1)( 3 1) Γ(α 1 )Γ(α 2 )Γ(α 3 )Γ(Σ 2) C 123 = 2 5/2 π Γ( 1 )Γ( 2 )Γ( 3 ) 16 1 N λ3/8 as α 3 = 2J 0 2J No pole for 3 chiral primaries LMRS, D Hoker-FMMR (1999) Pole indicates mixing of O with double trace op O J J =: O J O J : G 123

31 Splitting at the crossover O K O J J Λ Splitting: δ = 16 N λ3/8 Interesting to compare with recent bootstrap results Beem-Rastelli-van Rees (to appear)

32 Splitting at the crossover O K O J J Λ Splitting: δ = 16 N M λ 3/8

33 Splitting at the crossover O K O J J Λ Splitting: δ = 16 N M λ 3/8 Interesting to compare with recent bootstrap results Beem-Rastelli-van Rees (to appear)

34 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 do 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 = 2 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 AdS 5 S 5