Exact Four-Point Functions: Genus Expansion, Matrix Model, and Strong Coupling T ILL B ARGHEER Leibniz Universität Hannover 1711.05326 , 1809.09145 : TB, J. Caetano, T. Fleury, S. Komatsu, P. Vieira 1904.00965 , 1909.04077 : TB, F. Coronado, P. Vieira + work in progress H IGH -E NERGY P HYSICS T HEORY S EMINAR N ORDITA , S TOCKHOLM D ECEMBER 2, 2019
General Idea In AdS 5 , string amplitudes can be cut into basic patches (rectangles, pentagons, or hexagons), which can be bootstrapped using integrability at any value of the ’t Hooft coupling. ◮ Amplitudes are given as infinite sums and integrals over intermediate states that glue together these integrable patches. ◮ This holds at the planar level as well as for non-planar processes suppressed by 1/ N c . ◮ State sums are especially efficient at weak coupling. Today: Focus on a regime where one can go to large orders in 1/ N c , or even re-sum the genus expansion, all the way from weak to strong coupling. Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 1 / 25
N = 4 SYM & The Planar Limit N = 4 super Yang–Mills: Gauge field A µ , scalars Φ I , fermions ψ α A . Gauge group: U ( N c ) / SU ( N c ) . Adjoint representation: All fields are N c × N c matrices. Double-line notation: Propagators: Vertices: � ∼ g 2 YM δ il δ kj = i l 1 Φ i I j Φ k Tr ( ΦΦΦΦ ) ∼ � g 2 J l j k YM • Diagrams consist of color index loops ≃ oriented disks ∼ δ ii = N c • Disks are glued along propagators → oriented compact surfaces Local operators: ◮ One fewer color loop → factor 1/ N c O i = Tr ( Φ . . . ) ∼ O i ◮ Surface: Hole ∼ boundary component Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 2 / 25
Planar Limit & Genus Expansion Every Feynman diagram is associated to an oriented compact surface. [ ’t Hooft 1974 ] Genus Expansion: Count powers of N c and g 2 YM for propagators ( ∼ g 2 YM ), vertices ( ∼ 1/ g 2 YM ), and faces ( ∼ N c ) Absorb one factor of N c in the ’t Hooft coupling λ = g 2 YM N c Use Euler formula V − E + F = 2 − 2 g ⇒ Correlators of single trace operators O i = Tr ( Φ 1 Φ 2 . . . ) : ∞ 1 1 ∑ �O 1 . . . O n � = G g ( λ ) N n − 2 N 2 g c g = 0 c 1 + 1 + 1 ∼ + . . . N 2 N 4 N 6 c c c Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 3 / 25
Spectrum: Planar Limit Goal: Correlation functions in N = 4 SYM Step 1: Planar spectrum of single-trace local operators Tr ( Φ . . . ) ◮ Spectrum of (anomalous) scaling dimensions ∆ ◮ Scale transformations represented by dilatation operator Γ ◮ Γ mixes single-trace (& multi-trace) operators ◮ Resolve mixing → Eigenstates & eigenvalues (dimensions) Planar limit: ◮ Multi-trace operators suppressed by 1/ N c ◮ Dilatation operator acts locally in color space (neighboring fields) Organize space of single-trace operators around protected states Tr Z L , Z = α I Φ I , α I α I = 0 (half-BPS, “vacuum”) . Other single-trace operators: Insert impurities { Φ I , ψ α A , D µ } into Tr Z L . Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 4 / 25
Planar Spectrum: Integrability Initial observation: One-loop dilatation operator for scalar single-trace [ Minahan Zarembo ] operators is integrable. Diagonalization by Bethe Ansatz. ◮ Impurities are magnons in color space, characterized by rapidity (momentum) u and su ( 2 | 2 ) 2 flavor index . su ( 2 | 2 ) 2 ⊂ psu ( 2, 2 | 4 ) preserves the vacuum Tr Z L � � ◮ Dynamics of magnons: integrability: → No particle production → Individual momenta preserved = → Factorized scattering ◮ Two-body ( → n -body) S-matrix completely fixed to all loops [ 2005 ][ 2006 ][ ] Beisert Janik Beisert,Hernandez Lopez 2006 ⇒ Asymptotic spectrum (for L → ∞ ) solved to all loops / exactly. Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 5 / 25
Finite-Size Effects Asymptotic spectrum solved by Bethe Ansatz. Resums ∞ Feynman diagrams that govern dynamics of ∞ strip: . . . . . . − → L → ∞ L Re-compactify: Finite-size effects. Leading effect: Momentum quantization constraint ≡ Bethe equations 1 = e ip j L ∏ S ( p k , p j ) j � = k Moreover: Wrapping interactions. ◮ No notion of locality for dilatation operator ◮ Previous techniques (Bethe ansatz) no longer apply Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 6 / 25
Mirror Theory [ Frolov ] Key to all-loop finite-size spectrum: Mirror map Arutyunov Double Wick rotation: ( σ , τ ) → ( i ˜ τ , i ˜ σ ) — exchanges space and time R L e ˜ e H L H R ˜ τ − → τ R σ ˜ L σ Magnon states: Energy and momentum interchange: ˜ E = ip , ˜ p = iE Finite size L becomes finite, periodic (discrete) time. Energy ∼ Partition function at finite temperature 1/ L , with R → ∞ . → Thermodynamic Bethe ansatz . [ Fioravanti,Tateo ][ Bombardelli ’09 Gromov,Kazakov Kozak,Vieira ’09 ][ Arutyunov Frolov ’09 ] Simplifications and refinements: ◮ Y-system (T-system, Q-system) [ Kozak,Vieira ’09 ][ Frolov ’09 ] Gromov,Kazakov Arutyunov ◮ Quantum Spectral Curve [ Leurent,Volin ’13 ][ Leurent,Volin ’14 ] Gromov,Kazakov Gromov,Kazakov ⇒ Scaling dimensions computable at finite coupling. Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 7 / 25
Three-Point Functions: Hexagons Differences: Topology: Pair of pants instead of cylinder Non-vanishing for three generic operators (two-point: diagonal) ⇒ Previous techniques not directly applicable Observation: O 1 The green parts are similar to two-point functions: Two segments of physical operators joined by parallel propagators (“bridges”, ℓ ij = ( L i + L j − L k ) /2). The red part is new: “Worldsheet splitting”, O 2 O 3 “three-point vertex” (open strings) Take this serious → cut worldsheet along “bridges”: [ Basso,Komatsu Vieira ’15 ] O 1 − → ⊗ O 2 O 3 Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 8 / 25
The Cut State Sum (Mirror Theory) On each bridge lives a mirror theory: Double-Wick (90 degree) rotation ( σ , τ ) → ( i ˜ τ , i ˜ σ ) R L e ˜ e H L H R ˜ τ − → τ R ˜ σ L σ In all computations, the volume R can be treated as infinite. ⇒ Mirror states are free multi-magnon Bethe states, characterized by rapidities u i , bound state indices a i , and flavor indices ( A i , ˙ A i ) . The mirror integration therefore expands to � ∞ ∞ ∞ m � u i = − ∞ d u i µ a i ( u i ) e − ˜ E ai ( u i ) ℓ b . ∑ ∏ a i = 1 ∑ ∑ d ψ b = M b m = 0 i = 1 A i , ˙ A i µ a i : measure factor, ˜ E = ip : mirror energy, ℓ b : length of bridge b (discrete “time”). Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 9 / 25
Hexagons & Gluing O 1 − → ⊗ O 2 O 3 ⊗ Glue hexagons along three mirror channels : [ Vieira ’15 ][ Komatsu,Vieira ’15 ] Basso,Komatsu Basso,Goncalves ◮ Sum over complete state basis (magnons) in the mirror theory ◮ Mirror magnons: Boltzmann weight exp ( − ˜ E ij ℓ ij ) , ˜ E ij = O ( g 2 ) → mirror excitations are strongly suppressed. Hexagonal worldsheet patches (form factors): ◮ Function of rapidities u and su ( 2 | 2 ) 2 labels ( A , ˙ A ) of all magnons. ◮ Conjectured exact expression, based on diagonal su ( 2 | 2 ) symmetry as well as form factor axioms. [ Basso,Komatsu Vieira ’15 ] Finite-coupling hexagon proposal: Supported by very non-trivial matches. Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 10 / 25
The Hexagon Form Factors Hexagon = Amplitude that measures the overlap between three mirror and three physical off-shell Bethe states. Worldsheet branching operator that [ Basso,Komatsu Vieira ’15 ] creates an excess angle of π . Explicitly: H ( χ A 1 χ ˙ A 1 χ A 2 χ ˙ A 2 . . . χ A n χ ˙ A n ) � � � χ A 1 χ A 2 . . . χ A n | S | χ ˙ A n . . . χ ˙ A 2 χ ˙ = ( − 1 ) F A 1 � ∏ h ij i < j ◮ χ A , χ ˙ A : Left/Right su ( 2 | 2 ) fundamental magnons ◮ F : Fermion number operator ◮ S : Beisert S-matrix x − i − x − x + j − 1/ x − x ± ( u ) = x ( u ± i u g = x + 1 1 2 ) , j i x ◮ h ij = , x − i − x + x + 2 − 1/ x + σ ij σ ij : BES dressing phase j 1 Example: ⊗ = Two magnons ( , ) S S Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 11 / 25
Planar Four-Point Functions: Hexagonalization Move on to planar four-point functions: One way to cut (now that three-point is understood): OPE cut Problem: Sum over physical states! ◮ No loop suppression, all states contrib. ◮ Double-trace operators. [ Komatsu ][ Sfondrini ] Instead: Cut along propagator bridges Fleury ’16 Eden ’16 − → Benefits: ◮ Mirror states highly suppressed in g . ◮ No double traces. Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 12 / 25
Hexagonalization: Formula H 4 H 3 1 2 H 2 ← → H 1 3 4 � � � = d ℓ c � O 1 O 2 O 3 ∏ c ∑ H 1 ( ψ 1 , ψ 2 , ψ 3 ) H 2 ( ψ 1 , ψ 2 , ψ 3 ) µ ( ψ c ) ψ c channels c ∈{ 1,2,3 } � � � = ∑ d ℓ c ∏ c ∑ � O 1 O 2 O 3 O 4 µ ( ψ c ) H 1 H 2 H 3 H 4 planar channels ψ c c ∈{ 1,...,6 } prop. graphs New Features: [ Fleury ’16 Komatsu ] ◮ Bridge lengths vary, may go to zero ⇒ Mirror corrections at one loop ◮ Mirror corrections may span several hexagons Till Bargheer — Nordita HEP-TH Seminar — December 2, 2019 13 / 25
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