= 4 Ten dimensional symmetry SYM correlators - Frank Coronado - ICTP - SAIFR

Ten dimensional symmetry
 of = 4 SYM correlators
 Frank Coronad
 McGill University

 [with Simon Caron-Huot]

 arxiv : 2106.03892

 Strings 2021 ICTP-SAIFR
 
 o

A triality in planar = 4 SYM
 Correlation function of protected (Square of) massless amplitude
 Dimension-2 operators

 2 p1
 tr(φ (x1))
 tr(φ 2(x2))

 p2
 [Eden, Korchemsky, Sokatchev; 2010]

 (xi − xi+1)2 → 0 pi ≡ xi − xi+1
 x1
 4D null limit (T-duality in AdS)
 x2 [Alday, Maldacena, 2007;
 [Alday, Eden, Korchemsky,
 Drummond, Henn, Korchemsky,
 Maldacena, Sokatchev; 2010]
 Sokatchev, 2008;
 Berkovits, Maldacena;
 Caron-Huot; Mason, Skinner]
 x3

 (Square of) null Wilson loop
 

Generalization for massive amplitude

 (Square of) massive amplitude

 (p1, m1)

 ?
 (p2, m2)

• Massive amplitude in the Coulomb branch (turned on VEVs for scalar elds).

• Amplitude (integrand) has a higher dimensional symmetry that acts on the vector
 (pi, mi) . [Alday, Henn, Plefka, Schuster; Caron-Huot,O’Connell; Bern, Carrasco, Dennen, Huang, Ita]

 fi

Generalization for massive amplitude
 (Square of) massive amplitude
 (p1, m1)

 ?
 • Need object with higher-dimensional structure
 • Candidates: BPS operators dual to KK modes (p2, m2)
 in AdS5 × S5
 Vector of six scalars • Massive amplitude in the Coulomb
 branch (turned on VEVs for scalar
 elds).
 • Amplitude (integrand) has a
 higher dimensional symmetry
 that acts on the vector (pi, mi) .
 4D position 6D null polarization vector
 y.y = 0
 # elds =
 Scaling dimension

 • In SUGRA a 10D symmetry emerges when [Caron-Huot, Trinh, 2018;
 summing all (four-point) correlators of Aprile, Drummond, Heslop, Paul]

 • This talk: similar 10D structure in a di erent
 coupling regime.
fi
 fi
 ff

Generalization of correlator/massive amplitude

 ∑
 The four-point function of the “master operator” (x, y) = k(x, y) has an
 • k
 emergent 10-dimensional structure that combines spacetime and R-charge
 distances:

 2 2 2 duality
 Xi,i+1 ≡ (xi − xi+1) + (yi − yi+1) = pi2 + mi2

 • The 10D null limit of the “master” correlator is equal to a massive amplitude in the
 Coulomb branch.
 Generating function of all (Square of) four-point
 four-point correlators massive amplitude

 (p4, m4) (p1, m1)
 (x1, y1)
 (x2, y2) 2
 (x4, y4) Xi,i+1 →0
 10D null limit ≡
 (x3, y3) massive on-shell condition
 (p3, m3) (p2, m2)

 • Checked at various loop orders.
 
 

Outline

• Ten dimensional structure of free correlators.
 Weak Coupling

• 10D symmetry of loop integrands.

• 10D null limit:
 massive amplitude = large R-charge correlator (octagon).

• Amplitude/octagon from integrability and massless limit.

Free correlators
 • Computed by Wick contractions: k
 k
 −yij2
 k ( xij )
 1 2 Tr(yi . Φ(xi))k Tr(yj . Φ(xj ))k
 ⟨ k(xi, yi) k(xj, yj)⟩ = 2
 + O(1/Nc) k k

 xij2 ≡ (xi − xj)2 , yij2 ≡ (yi − yj)2
 ∞
 The free four-point correlator of the “master operator” (x, y) =
 ∑ k(x, y)
 • k
 lij
 ∞ −yij2
 ( 2 )
 G free ≡ ⟨ (x1, y1) (x2, y2) (x3, y3) (x4, y4)⟩(0) = ∑ C{lij} ∏
 l 1≤i

Loop integrands
• Perturbative series in the 't Hooft coupling

• We can de ne an integrand by the Lagrangian insertion method:

• The ℓ-loop integrand is a (4 + ℓ)-point correlator evaluated at leading order:

 [Eden, Petkou, Schubert. Sokatchev]

• Advantage: integrand is a rational function with simple poles. It treats external and
 integration points almost in the same footing (e.g. has a full permutation
 symmetry). [Eden, Heslop, Korchemsky, Sokatchev, 2011]
 fi

• Decomposition in R-charge:

The number of inequivalent structures is nite and depends on the loop order.

• Saturation: thanks to planarity, a bridge becomes uncrossable when the number of
 propagators is larger than the loop order.

 l
 m

 2 l
 ⇠ (g )

 [Chicherin, Drummond, Heslop, Sokatchev, 2015]

• After saturation, we have an in nite tail forming a geometric series.
 fi
 fi

One-loop integrands
• At one loop, saturation implies that all R-charge structures are
 identical:

• The reduced integrands:

• Resumming the geometric series:

 • Higher-loop data shows similar pattern.

10D symmetry of loop integrands
• At each loop order, all (reduced) integrands form a geometric series that resums
 into a function which depends only on Xij2 ≡ xij2 + yij2 .

• This generating function can be uplifted from the known case by
 replacing all four-dimensional distances xij2 by ten-dimensional ones Xij2

 [Eden, Heslop,
 Korchemsky,
 Sokatchev; 2012]

• It inherits the full permutation symmetry of . .
 The dimension-2 operator and the chiral Lagrangian belong to the stress-tensor super-multiplet.

10D structure of four-point correlators
• We set the 6D null condition for the external points and turn o the R-charge of the internal points.

 and integrate:

• One and two-loop examples:

 x5
 x5 x6 βi ≥ 1

 • Similar checks up to 5 loops. [Chicherin, Georgoudis, Goncalves, Pereira, 2018] [Chicherin, Drummond, Heslop, Sokatchev, 2015]

 • Predictions at higher loops (up to ten loops from knowledge of the seed ) [Bourjaily, Heslop, Tran, 2016]

 • Higher loop integrals are hard to evaluate.

 • A tractable problem using integrability: correlators with large R-charge (octagons). [FC 2018]
 ff

10D null limit: octagon = amplitude
 K→∞
• The simplest correlators 1 2 O1 = Tr(X̄ 2K+a )
 factorized into squares a
 (octagons). O2 = Tr(X K Z̄ K Ȳ b ) + cyclic permutations

 O3 = Tr(Z 2K X a ) + cyclic permutations
 b
• Their integrands receive O4 = Tr(Z K X̄ K Y b ) + cyclic permutations
 contributions only from
 4 3

 1
 Only the terms of “G” with the four poles contribute to the simplest correlators. We
• X12 X23 X34 X41
 2 2 2 2

 can select them by taking the 10D null limit:
 2
 1 2
 a
 P
 m

 P •L4+` b
 m
 •L5 x5

 2
 lim
 Xi,i+1 !0
 G = = ..
 a,b,`
 ··

 ` x8
 .
 ·

 •L 8
 4 3
 •L7 •L6

 = O⇥O = M ⇥M

 • Octagon and amplitude are identical at the integrand and integrated level. They are IR nite.

 fi

Octagon from Integrability

Gluing two hexagons by summing over mirror particles: [Basso, Komatsu, Vieira 2013;
 Fleury, Komatsu; Eden, Sfondrini 2016;
 FC 2018]

 ψ : complete basis of states in the
 two-dimensional world sheet

Octagon is given by an in nite determinant [Kostov, Petkova, Serban;
 Belitsky, Korchemsky;
 2019-2021]
 fi

Steinmann condition for amplitudes:
 Double discontinuities Octagon has a vanishing double discontinuity
 vanish in overlapping discs disct = 0
 channels
 2 2
 with s = x13 , t = x24

 Weak coupling:
 Provides analytic results for
 = sum of det (unknown) conformal integrals
 such as shnets and
 deformations

 Strong coupling: = e −g Area [Alday, Maldacena, 2007;
 [Bargheer, FC, Vieira, 2019; Kruczenski]
 Belitsky, Korchemsky]

 2
 xi,i+1 →0

 4D null limit of
 octagon
 ?
 or massless limit of
 the amplitude

 Perimeter is null in AdS5 × S5 Four-cusped null Wilson loop
 
 fi

Massless limit
Coulomb-branch amplitudes with double logarithmic scaling:

 2
 mext
 i
 2
 = yi,i+1 = x2i,i+1

 2 int 1 (mint )2 (mint )2
 mint = yi2 6= 0 .. m !0 cusp log log
 2 x2 x2
 !0 !
 i
 ext . e 13 24
 m
 [Alday, Henn, Plefka, Schuster, 2007;
 Bern, Dixon, Smirnov, 2005]
 ..
 . m int
 !0
 ✓ ◆
 ext x2 2 2
 12 x23 x34 x41
 2
 .. m !0 1
 oct log2
 . ! e
 16
 (x213 x224 )2

 [FC, 2018]

Controlled by functions of the coupling satisfying a deformed BES equation:
 [Beisert, Eden, Staudacher, 2006]

 [Basso, Dixon, Papathanasious, 2020]

 ⟶ Γcusp
 2
 ⟶ Γoct = 2 log cosh(2πg) [Belitsky, Korchemsky, 2019]
 π

Correlator Amplitude
Summary
 2
 Xi,i+1 !0
 10D : G(X) [M (x, y)]2 = O2

 2 2
 yij !0 yij !0

 4D : G2222 (x) [M (x, 0)]2
 x2i,i+1 ! 0

 Outlook
 ?
• Can we relax the
 Half-BPS Coulomb-branch
 null condition? amplitudes
 correlators Octagons O
 yi2 =0 2
 Xi,i+1 =0
• Higher-points?

• Relation to 10D sym. in SUGRA?

• Integrability in the Coulomb branch?